[{"id": 1, "type": "game", "source": "achi", "section": "overview", "text": "Achi\nA Ghanaian three-in-a-row game, a close relative of Three Men's Morris.\nSolution status: Weakly solved (small enough for exhaustive analysis). Game-theoretic value: Draw **[verify]**. Players: 2. Type: Partisan positional / sliding game."}, {"id": 2, "type": "game", "source": "achi", "section": "Description", "text": "Achi — Description\n\nA traditional game of Ghana played on a 3×3 grid of points with both diagonals\nmarked (giving the centre point a high connectivity). Each player has **four**\npieces. Players place all their pieces alternately, then enter a movement phase,\nsliding a piece along a marked line to an adjacent empty point; three in a row\nalong a marked line wins."}, {"id": 3, "type": "game", "source": "achi", "section": "Solution status", "text": "Achi — Solution status\n\nAchi is **small enough to be solved by exhaustive search** — its state space is\nonly a few thousand positions, comparable to [Three Men's Morris](three-mens-morris.md)\nand [Nine Holes](nine-holes.md). It is generally reported, like its close\nrelatives, to be a **draw** with perfect play, the first player's central\nadvantage notwithstanding.\n\n> **[verify]** — This archive has not located a single canonical primary\n> source giving Achi's exact game-theoretic value; the \"draw\" claim is by\n> analogy with the closely related morris games and from general references.\n> A direct exhaustive-search citation should be added."}, {"id": 4, "type": "game", "source": "achi", "section": "Consensus on optimal play", "text": "Achi — Consensus on optimal play\n\n- **Take the centre on move 1** — the centre point connects all four lines (horizontal, vertical, two diagonals), giving more winning threats than any corner or edge point.\n- **Fill corners before edges** — corners connect 3 lines each; edges connect only 2, making corners more strategically valuable in the placement phase.\n- **Deny your opponent two-in-a-row** — because the board is tiny (4 pieces each, 9 points), a single unchallenged two-in-a-row often converts directly to a win.\n- **In the movement phase, use the centre as a pivot** — the centre connects to all other points; controlling it in the sliding phase gives mobility advantage.\n- **Mirror or block immediately** — with optimal play by both sides the game is a draw; any passive move that allows an unblocked two-in-a-row is fatal."}, {"id": 5, "type": "game", "source": "achi", "section": "Engines & current best play", "text": "Achi — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The state space is small enough that a complete minimax solve is trivial to implement; no dedicated competitive engine has been published."}, {"id": 6, "type": "game", "source": "achi", "section": "Complexity", "text": "Achi — Complexity\n\nA few thousand positions."}, {"id": 7, "type": "game", "source": "achi", "section": "References", "text": "Achi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Achi_(game)) ([archive](http://web.archive.org/web/20251222103232/https://en.wikipedia.org/wiki/Achi_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 8, "type": "game", "source": "achi", "section": "See also", "text": "Achi — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Nine Holes](nine-holes.md) · [Tic-tac-toe](tic-tac-toe.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 9, "type": "game", "source": "amazons", "section": "overview", "text": "Amazons\nA territorial game of queens that shoot arrows; a favourite CGT research\nSolution status: Unsolved (standard 10×10); small boards analysed. Game-theoretic value: Unknown for the standard board. Players: 2. Type: Partisan combinatorial game."}, {"id": 10, "type": "game", "source": "amazons", "section": "Description", "text": "Amazons — Description\n\nPlayed on a 10×10 board, each side with four \"amazons\" that move like chess\nqueens. After moving, an amazon **shoots an arrow** (also moving queen-like)\nthat permanently blocks a square. Blocked squares and pieces obstruct movement.\nA player who cannot move loses — so the game is about walling off territory."}, {"id": 11, "type": "game", "source": "amazons", "section": "Solution status", "text": "Amazons — Solution status\n\nAmazons is **unsolved** on its standard 10×10 board. It is, however, heavily\nstudied in combinatorial game theory because the endgame *fragments into\nindependent regions*, each with an exact CGT value, making it an ideal showcase\nfor [hot-game](../lexicon/README.md#temperature--hot-game) and thermography\ntechniques. Endgames and small boards have been analysed exhaustively, and the\ngeneralised (n×n) game is known to be PSPACE-complete\n([Hearn & Demaine, 2009](../references.md#hearn-demaine2009)). But the value of\nthe standard initial position is not known. Complexity estimates place Amazons\nbetween Othello and chess/Go."}, {"id": 12, "type": "game", "source": "amazons", "section": "Consensus on optimal play", "text": "Amazons — Consensus on optimal play\n\n- **Control territory early, not pieces** — the object is to leave your opponent without moves, so expanding your reachable squares matters more than capturing or threatening amazons directly.\n- **Shoot arrows that restrict the opponent** — the arrow after each move is as important as the move itself; a well-placed arrow that limits an enemy amazon's future options is often stronger than a distant territorial gain.\n- **Keep your amazons mobile** — amazons trapped behind their own arrows become worthless; avoid self-blocking by thinking two moves ahead about where you will shoot next.\n- **Fragment the board in your favour** — when the board breaks into independent regions, each region has a CGT value; aim to create more and larger regions on your side than on your opponent's.\n- **In the endgame, count liberties** — once regions are isolated, the player whose region contains more \"moves remaining\" (mobility surplus) wins; thermographic analysis from CGT guides exact endgame play.\n- **Opening: anchor amazons near the corners** — moving toward the corners early gives your amazons protected territory to develop from without being cut off."}, {"id": 13, "type": "game", "source": "amazons", "section": "Engines & current best play", "text": "Amazons — Engines & current best play\n\n- **Strongest known program(s):** Amazons Engine / various research bots (e.g., Galactic, 8Q3, NAgents) — minimax/alpha-beta with CGT-based endgame solvers\n- **Strength:** Super-human; top engines consistently outperform top human players.\n- **Where the proof / tablebase lives (if solved):** Not applicable — standard 10×10 board is unsolved; CGT endgame analysis covers many late-game positions.\n- **Notes:** Amazons is a leading benchmark for combining heuristic search (opening/midgame) with exact CGT calculation (endgame); competition results appear in the Computer Olympiad proceedings."}, {"id": 14, "type": "game", "source": "amazons", "section": "Complexity", "text": "Amazons — Complexity\n\nState-space ~10^40, game-tree ~10^212 (figures of the order cited in\n[van den Herik et al., 2002](../references.md#vandenherik2002) and subsequent\nliterature)."}, {"id": 15, "type": "game", "source": "amazons", "section": "References", "text": "Amazons — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Game_of_the_Amazons) ([archive](http://web.archive.org/web/20260511121116/https://en.wikipedia.org/wiki/Game_of_the_Amazons))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [Hearn, R. A. & Demaine, E. D. (2009). *Games, Puzzles, and Computation*.](../references.md#hearn-demaine2009)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 16, "type": "game", "source": "amazons", "section": "See also", "text": "Amazons — See also\n\n- [Lines of Action](lines-of-action.md) · [Clobber](clobber.md) · [Go](go.md)\n- Lexicon: [temperature / hot game](../lexicon/README.md#temperature--hot-game) · [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 17, "type": "game", "source": "anti-reversi", "section": "overview", "text": "Anti-Reversi\nReversi played to lose — fewer discs wins. Largely unsolved despite the\nSolution status: Unsolved (6×6 solved **[verify]**; 8×8 open). Game-theoretic value: Unknown. Players: 2. Type: Partisan placement game (misère)."}, {"id": 18, "type": "game", "source": "anti-reversi", "section": "Description", "text": "Anti-Reversi — Description\n\nThe misère cousin of [Othello](othello.md): the rules are identical, but **the\nplayer with the fewest discs at the end wins**. As with most misère games,\nstrategy diverges sharply from the normal-play version."}, {"id": 19, "type": "game", "source": "anti-reversi", "section": "Rules", "text": "Anti-Reversi — Rules\n\n1. Same board, setup, and movement rules as [Othello](othello.md):\n   - 8×8 board with the standard four-disc cross opening.\n   - On a turn a player places a disc that **must** bracket at least one\n     opposing run of discs, flipping all bracketed discs.\n   - If no legal placement exists, the player passes; if neither player can\n     move, the game ends.\n2. At the end, the player with the **fewer** discs on the board wins."}, {"id": 20, "type": "game", "source": "anti-reversi", "section": "Solution status", "text": "Anti-Reversi — Solution status\n\nAnti-Reversi is **not formally solved** on the standard 8×8 board. There are\nstrong programs and some claim of a 6×6 solution, but the headline result for\n[Othello](othello.md) — Takizawa's 2023 weak solution showing a draw — has no\nanalogous published proof for the anti-variant. The misère nature of the game\ninverts the usual evaluations; opening theory is markedly different."}, {"id": 21, "type": "game", "source": "anti-reversi", "section": "Consensus on optimal play", "text": "Anti-Reversi — Consensus on optimal play\n\n- **Avoid big flip chains** — in normal Othello you want to flip many discs at once; in Anti-Reversi, large flips move discs to your colour, which is bad. Prefer moves that flip as few discs as possible.\n- **Surrender edge and corner squares** — in normal Othello corners are gold; here, landing on a corner anchors your disc permanently (it cannot be flipped back), which is a liability. Avoid corners unless forced.\n- **Aim for fewer discs throughout, not just at the end** — disc count shifts dramatically in late-game mass flips; trailing in disc count mid-game is usually good.\n- **Force the opponent to flip your discs** — set up positions where the opponent's only legal moves are ones that convert your discs to theirs.\n- **Parity still matters** — like normal Othello, the final sequence of forced moves is often decisive; maintaining move-parity in the last region can determine who makes the last large flip."}, {"id": 22, "type": "game", "source": "anti-reversi", "section": "Engines & current best play", "text": "Anti-Reversi — Engines & current best play\n\n- **Strongest known program(s):** No widely-distributed dedicated Anti-Reversi engine known to the cataloguer; general Othello/Reversi engines can be adapted with inverted evaluation.\n- **Strength:** Not benchmarked publicly.\n- **Where the proof / tablebase lives (if solved):** — (8×8 unsolved; 6×6 claimed but no canonical public source)\n- **Notes:** Misère Othello strategy is essentially the inverse of normal Othello; engine-derived opening books exist in competitive Anti-Othello circles but are not publicly documented in the literature."}, {"id": 23, "type": "game", "source": "anti-reversi", "section": "Complexity", "text": "Anti-Reversi — Complexity\n\nSame as Othello."}, {"id": 24, "type": "game", "source": "anti-reversi", "section": "References", "text": "Anti-Reversi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Reversed_Othello)\n- [Takizawa (2023). *Othello is Solved*.](../references.md#takizawa2023) (related)\n- [Feinstein (1993). *Amenor Wins World 6×6 Championships*.](../references.md#feinstein-othello6x61993)"}, {"id": 25, "type": "game", "source": "anti-reversi", "section": "See also", "text": "Anti-Reversi — See also\n\n- [Othello](othello.md) · [Quixo](quixo.md) · [Losing chess](losing-chess.md)\n- Lexicon: [misère play](../lexicon/README.md#misere-play)"}, {"id": 26, "type": "game", "source": "arimaa", "section": "overview", "text": "Arimaa\nA game deliberately designed to be hard for computers — playable with a chess\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 27, "type": "game", "source": "arimaa", "section": "Description", "text": "Arimaa — Description\n\nInvented by Omar Syed in 2002, playable on a chessboard with chess pieces.\nPlayers first **set up** their pieces freely on their two home ranks. Each turn\nconsists of **up to four steps**, which may move several pieces; stronger pieces\ncan **push and pull** weaker enemy pieces. Pieces are lost by being forced onto\ntrap squares. A player wins by getting a rabbit (pawn) to the far rank."}, {"id": 28, "type": "game", "source": "arimaa", "section": "Solution status", "text": "Arimaa — Solution status\n\nArimaa is **unsolved**, and was *engineered* to resist computer solving and\nstrong computer play. The free setup phase and the up-to-four-steps move\nstructure give an enormous branching factor — on the order of tens of thousands\nof moves per turn — which crippled the brute-force search that dominates\nchess engines. A standing \"Arimaa Challenge\" prize ran from 2004; bots finally\ndefeated top human defenders in **2015**, ending the challenge. But strong (and\nnow winning) computer play is, as always,\n[not a solution](../lexicon/README.md#solving-vs-strong-play): the game-theoretic\nvalue of Arimaa is unknown and the game is nowhere near solved."}, {"id": 29, "type": "game", "source": "arimaa", "section": "Consensus on optimal play", "text": "Arimaa — Consensus on optimal play\n\n- **Control the traps** — the four trap squares are the primary way pieces are eliminated; placing friendly pieces adjacent to your own traps (as \"hosts\") protects them, while attacking the defenders of enemy traps is the main tactical theme.\n- **Setup determines the game** — the free placement phase is crucial; standard setups place the stronger pieces (elephant, camel) behind rabbits near the traps they will anchor, and cats/dogs as trap defenders.\n- **The elephant dominates — push yours forward** — the elephant cannot be pushed or pulled; advancing it aggressively ties down the opponent's elephant in defense and creates a positional wedge.\n- **Rabbit advancement is the winning condition** — every strategic decision filters through \"can I advance a rabbit to the 8th rank?\"; keeping rabbit lanes open while blocking opponent rabbit paths is the strategic core.\n- **Immobilisation wins without captures** — a player with no legal moves loses; surrounding the opponent's pieces (especially their camel or elephant) to create a goal-through-immobilisation threat is a major tactical weapon.\n- **Tempo matters enormously** — with up to four steps per turn, wasting steps on non-threatening moves hands the initiative to your opponent; experienced players maximise \"goal threats\" per step."}, {"id": 30, "type": "game", "source": "arimaa", "section": "Engines & current best play", "text": "Arimaa — Engines & current best play\n\n- **Strongest known program(s):** bot_sharp (later variants) and Ziltoid — MCTS-based with neural networks; defeated top humans in 2015.\n- **Strength:** Super-human since 2015.\n- **Where the proof / tablebase lives (if solved):** — (unsolved; no tablebase)\n- **Notes:** The Arimaa Challenge (2004–2015) specifically tested whether bots could beat top human players; after that barrier was crossed, neural-network-based engines rapidly improved further."}, {"id": 31, "type": "game", "source": "arimaa", "section": "Complexity", "text": "Arimaa — Complexity\n\nState-space ~10^43; the per-turn branching factor (tens of thousands) makes the\neffective game-tree complexity enormous."}, {"id": 32, "type": "game", "source": "arimaa", "section": "References", "text": "Arimaa — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Arimaa) ([archive](http://web.archive.org/web/20260505162323/https://en.wikipedia.org/wiki/Arimaa))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 33, "type": "game", "source": "arimaa", "section": "See also", "text": "Arimaa — See also\n\n- [Chess](chess.md) · [Go](go.md) · [Lines of Action](lines-of-action.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 34, "type": "game", "source": "atoll", "section": "overview", "text": "Atoll\nAn island-connection game by Mark Steere — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 35, "type": "game", "source": "atoll", "section": "Description", "text": "Atoll — Description\n\nAtoll (Mark Steere) is a hexagonal-board connection game with **island\nendpoints**: instead of two opposite *edges* of the board, each player has\ntwo **specific cells** (the \"islands\") that they must connect."}, {"id": 36, "type": "game", "source": "atoll", "section": "Rules", "text": "Atoll — Rules\n\n1. Hexagonal grid board with four marked cells (two per player), located near\n   the four corners or on opposing sides.\n2. Players alternate placing one stone of their colour on any empty cell. (One\n   stone may not be played adjacent to a same-colour stone if doing so would\n   violate the no-clumping rule — variants differ; **[verify]** the canonical\n   version.)\n3. The first player to form a connected chain of their colour linking **both**\n   of their islands wins.\n4. Draws are not possible."}, {"id": 37, "type": "game", "source": "atoll", "section": "Solution status", "text": "Atoll — Solution status\n\nAtoll is **unsolved**. The two-target connection objective is closer to\n[Bridg-it](bridg-it.md) and [TwixT](twixt.md) than to Hex, and no formal\nsolution has been published."}, {"id": 38, "type": "game", "source": "atoll", "section": "Consensus on optimal play", "text": "Atoll — Consensus on optimal play\n\n- **Build toward both islands simultaneously** — connecting your two islands requires a spanning path; advancing a chain that serves neither island wastes tempo and leaves you vulnerable to being cut.\n- **The virtual connection principle applies** — as in Hex, two groups that share two disjoint paths to each other are \"virtually connected\" and cannot both be cut; recognise these structures to play confidently without fully bridging gaps yet.\n- **Cutting between opponent's islands is the primary attack** — find the narrowest crossing between the opponent's two islands and contest it; a stone planted in that corridor forces the opponent to detour.\n- **Centralise early** — cells near the centre of the board lie on more potential paths between any pair of islands; central stones are harder to render irrelevant than peripheral ones.\n- **The strategy-stealing argument applies** — an extra stone is never a liability in a connection game, so first player has at least a draw theoretically; this implies the pie rule swap is appropriate for fair play."}, {"id": 39, "type": "game", "source": "atoll", "section": "Engines & current best play", "text": "Atoll — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Atoll has a modest online player community but no dedicated published engine; general connection-game heuristics from Hex programs offer the strongest available guidance."}, {"id": 40, "type": "game", "source": "atoll", "section": "Complexity", "text": "Atoll — Complexity\n\nComparable to Hex on similar board sizes."}, {"id": 41, "type": "game", "source": "atoll", "section": "References", "text": "Atoll — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Atoll_(game))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework)"}, {"id": 42, "type": "game", "source": "atoll", "section": "See also", "text": "Atoll — See also\n\n- [Hex](hex.md) · [TwixT](twixt.md) · [Bridg-it](bridg-it.md) · [Havannah](havannah.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 43, "type": "game", "source": "atomic-chess", "section": "overview", "text": "Atomic chess\nChess variant where captures cause an explosion — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 44, "type": "game", "source": "atomic-chess", "section": "Description", "text": "Atomic chess — Description\n\nAtomic chess uses the standard chessboard and pieces, but every capture\ntriggers an **explosion** that removes both the capturing piece and all\nnon-pawn pieces on the eight adjacent squares. The king cannot capture\n(suicide), and detonating the enemy king wins."}, {"id": 45, "type": "game", "source": "atomic-chess", "section": "Rules", "text": "Atomic chess — Rules\n\n1. Same setup and base movement as orthodox chess.\n2. Whenever a piece captures, the capturing piece is also removed; all pieces\n   on the 8 squares around the capture square are removed **except pawns**.\n3. The king may not capture (it would self-destruct).\n4. You may **explode** the opposing king by capturing any piece adjacent to it;\n   doing so wins the game immediately.\n5. The kings may legally stand on adjacent squares — they cannot capture each\n   other.\n6. Stalemate, threefold and 50-move rules carry over from chess."}, {"id": 46, "type": "game", "source": "atomic-chess", "section": "Solution status", "text": "Atomic chess — Solution status\n\nAtomic chess is **not solved**. Quick decisive games are possible because\nexplosions reshape the position drastically; opening tactics are sharp but no\nformal solving result exists."}, {"id": 47, "type": "game", "source": "atomic-chess", "section": "Consensus on optimal play", "text": "Atomic chess — Consensus on optimal play\n\nPractical wisdom from strong online play:\n\n- **King safety dominates** — because any capture next to the king explodes it, the king is far more exposed than in chess. Castling is often *avoided*; many strong games keep the king on its starting square (or move it to f1/f8) to keep adjacent squares free.\n- **Pawn shields are deadly to you, not the opponent** — a pawn directly in front of your king means the opponent can blow you up with any capture on that square.\n- **Avoid early queen exchanges**; queens cannot capture without exploding themselves and are often used as long-range detonators.\n- **Known losing first moves for White** include 1.Nf3 in some lines (engine analysis on Lichess); 1.e4 and 1.d4 are the standard practical choices.\n\nNo proven game-theoretic value."}, {"id": 48, "type": "game", "source": "atomic-chess", "section": "Engines & current best play", "text": "Atomic chess — Engines & current best play\n\n- **Strongest known programs:** [Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish)) (Stockfish fork supporting atomic and most chess variants, open source); used by [Lichess](https://lichess.org/variant/atomic) ([archive](http://web.archive.org/web/20260313075304/https://lichess.org/variant/atomic)) for its server analysis.\n- **Strength:** Super-human at fast time controls; opening theory is shallow compared to chess but deeply explored by engines.\n- **Notes:** No tablebases exist for atomic-specific endgames — explosion semantics break standard chess endgame reductions."}, {"id": 49, "type": "game", "source": "atomic-chess", "section": "Complexity", "text": "Atomic chess — Complexity\n\nSimilar to chess."}, {"id": 50, "type": "game", "source": "atomic-chess", "section": "References", "text": "Atomic chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Atomic_chess) ([archive](http://web.archive.org/web/20260407014026/https://en.wikipedia.org/wiki/Atomic_chess))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 51, "type": "game", "source": "atomic-chess", "section": "See also", "text": "Atomic chess — See also\n\n- [Chess](chess.md) · [Crazyhouse](crazyhouse.md) · [Three-check chess](three-check-chess.md) · [Losing chess](losing-chess.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 52, "type": "game", "source": "atropos", "section": "overview", "text": "Atropos\nA Sperner-lemma-flavoured colouring game — its decision problem is\nSolution status: Strongly solved as a theory (PSPACE-complete). Game-theoretic value: Position-dependent. Players: 2. Type: Partisan combinatorial game (graph colouring)."}, {"id": 53, "type": "game", "source": "atropos", "section": "Description", "text": "Atropos — Description\n\nAtropos is a board game inspired by **Sperner's lemma**. The board is a\ntriangulated triangle whose corner vertices are pre-coloured 1, 2, 3 (one each).\nPlayers alternately colour internal vertices subject to constraints, and the\nlosing player is the one forced to complete a tricoloured (1-2-3) triangle —\nSperner's lemma guarantees such a triangle eventually exists."}, {"id": 54, "type": "game", "source": "atropos", "section": "Rules", "text": "Atropos — Rules\n\n1. A triangular grid (a triangulation of a large triangle) is given, with each\n   corner labelled 1, 2, 3 and edge vertices restricted to the two\n   adjacent-corner colours.\n2. Players alternate colouring an internal vertex with one of {1, 2, 3}; the\n   first move colours any vertex adjacent to the last move. (Atropos enforces\n   a chain rule: each move must be next to the previous move.)\n3. A player who is **forced to create a tricoloured triangle** loses."}, {"id": 55, "type": "game", "source": "atropos", "section": "Solution status", "text": "Atropos — Solution status\n\nSolved as a theory: Burke & Teng (2008) **[verify]** proved Atropos is\n**PSPACE-complete**, by reduction from a Boolean-formula game. Small instances\nare solvable by direct backward induction, but no efficient algorithm exists\nin general."}, {"id": 56, "type": "game", "source": "atropos", "section": "Consensus on optimal play", "text": "Atropos — Consensus on optimal play\n\n- **Avoid completing the third colour in a near-Sperner triangle** — whenever a triangle already shows two of the three colours on its vertices, colouring the third vertex with the missing colour hands your opponent the losing condition; scan for such triangles before every move.\n- **Force your opponent into constrained positions** — because each move must be adjacent to the previous one (the chain rule), steer the chain toward dense regions where your opponent will have fewer safe colour choices.\n- **Colour ambiguously where possible** — choosing a colour that does not immediately threaten any near-complete Sperner triangle maximises your future options and minimises risk.\n- **Control the last few uncoloured vertices** — the endgame typically funnels down to a small cluster; the player who can force their opponent to colour the final triangle-completing vertex wins; work backward from likely endgame configurations.\n- **PSPACE-hardness means no simple heuristic suffices on large boards** — on small (≤ size-3 or size-4) triangulations, exhaustive backward induction is feasible and should be used; for larger boards there is no known efficient strategy."}, {"id": 57, "type": "game", "source": "atropos", "section": "Engines & current best play", "text": "Atropos — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Atropos is primarily a complexity-theory research object; PSPACE-completeness (Burke & Teng, 2008) means computing optimal play on large boards is intractable in general."}, {"id": 58, "type": "game", "source": "atropos", "section": "Complexity", "text": "Atropos — Complexity\n\nPSPACE-complete in the size of the triangulation."}, {"id": 59, "type": "game", "source": "atropos", "section": "References", "text": "Atropos — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Atropos_(game))\n- [Hearn & Demaine (2009). *Games, Puzzles, and Computation*.](../references.md#hearn-demaine2009)\n- [Schaefer (1978). *On the complexity of some two-person perfect-information games*.](../references.md#schaefer1978)"}, {"id": 60, "type": "game", "source": "atropos", "section": "See also", "text": "Atropos — See also\n\n- [Sim](sim.md) · [Generalized Geography](geography.md)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete)"}, {"id": 61, "type": "game", "source": "awari", "section": "overview", "text": "Awari (Oware)\nThe classic two-row mancala — strongly solved in 2002, every one of its\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan sowing (mancala) game."}, {"id": 62, "type": "game", "source": "awari", "section": "Description", "text": "Awari (Oware) — Description\n\nPlayed on two rows of six pits, with 48 seeds (4 per pit at the start). On a\nturn a player picks up all seeds from one of their pits and **sows** them one\nper pit, counter-clockwise. If the last seed lands in an enemy pit bringing it\nto 2 or 3 seeds, those are captured (and possibly preceding pits too). The\nplayer capturing more than 24 seeds wins."}, {"id": 63, "type": "game", "source": "awari", "section": "Solution status", "text": "Awari (Oware) — Solution status\n\nAwari is **strongly solved**. [Romein & Bal (2003)](../references.md#romein-bal2003)\ncomputed the game-theoretic value of **every one of its ~889 billion positions**\nby massively parallel [retrograde analysis](../lexicon/README.md#retrograde-analysis)\non a cluster — building, in effect, a complete tablebase for the whole game. The\nresult: **with perfect play Awari is a draw** (each side captures exactly 24\nseeds).\n\nBecause the entire state space is stored, optimal play is available from *any*\nposition, not just the opening — the [strong-solution](../lexicon/README.md#strongly-solved)\nstandard. Awari was, at the time, one of the largest games to be solved this\ncompletely."}, {"id": 64, "type": "game", "source": "awari", "section": "Consensus on optimal play", "text": "Awari (Oware) — Consensus on optimal play\n\n- **Both sides capturing exactly 24 seeds is the perfect-play outcome** — every deviation from the drawn line eventually hands the opponent a material edge; the value is 24-24 with best play.\n- **Deny grand slams** — a move that would leave the opponent with no seeds on their side is illegal if the opponent has no seeds; under legal-play rules, plan ahead to avoid giving your opponent no valid pits to sow from, which forfeits your capture rights.\n- **Count seeds before sowing** — the exact landing pit of the last seed determines captures; precise arithmetic about pit counts (especially pits holding 12+ seeds that wrap the whole board) separates strong from weak play.\n- **Capture chains compound** — a sow can trigger a cascade of captures in preceding pits if each of those pits also holds exactly 2 or 3 seeds after the final seed lands; spotting multi-pit capture chains is a core tactical skill.\n- **Preserve your own pit count** — keeping seeds spread across your pits maintains future flexibility; having one or two fat pits is predictable and allows your opponent to count your landing squares accurately."}, {"id": 65, "type": "game", "source": "awari", "section": "Engines & current best play", "text": "Awari (Oware) — Engines & current best play\n\n- **Strongest known program(s):** Romein & Bal's 2002 retrograde tablebase — complete lookup for all positions.\n- **Strength:** Perfect (optimal play from every position is computable in constant time by tablebase lookup).\n- **Where the proof / tablebase lives (if solved):** Romein & Bal (2003) — see [../references.md#romein-bal2003](../references.md#romein-bal2003); the full ~889 billion-position database was held at Vrije Universiteit Amsterdam.\n- **Notes:** At time of publication this was one of the largest completely solved games; the tablebase is too large for casual download but the result (draw) is definitively established."}, {"id": 66, "type": "game", "source": "awari", "section": "Complexity", "text": "Awari (Oware) — Complexity\n\n~8.9 × 10^11 positions — all enumerated in the 2002 solution."}, {"id": 67, "type": "game", "source": "awari", "section": "References", "text": "Awari (Oware) — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Oware) ([archive](http://web.archive.org/web/20260503155212/https://en.wikipedia.org/wiki/Oware))\n- [Romein, J. W. & Bal, H. E. (2003). *Solving Awari with Parallel Retrograde Analysis*.](../references.md#romein-bal2003)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 68, "type": "game", "source": "awari", "section": "See also", "text": "Awari (Oware) — See also\n\n- [Kalah](kalah.md) · [Bao](bao.md) · [Nine Men's Morris](nine-mens-morris.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 69, "type": "game", "source": "backgammon", "section": "overview", "text": "Backgammon\nThe ancient race game — superhuman computer players exist and equity is known\nSolution status: Unsolved (superhuman play exists; equity known approximately). Game-theoretic value: Unknown exactly — first player has a small known equity edge. Players: 2. Type: Stochastic race game."}, {"id": 70, "type": "game", "source": "backgammon", "section": "Description", "text": "Backgammon — Description\n\nTwo players race 15 checkers each around a 24-point board in opposite\ndirections, moving according to the roll of two dice; a lone checker (\"blot\") can\nbe hit and sent back. The doubling cube adds a wagering dimension. Backgammon has\n**no hidden information** but is **stochastic** — every move depends on a dice\nroll — which places it outside the perfect-information solving framework that\ncovers most of this archive."}, {"id": 71, "type": "game", "source": "backgammon", "section": "Solution status", "text": "Backgammon — Solution status\n\nBackgammon is **not solved** and, realistically, cannot be solved in the strong\nsense: the dice give each ply ~21 distinct roll outcomes, each with ~20 legal\nplays, so the game tree is astronomically branchy and there is no single\n\"game-theoretic value,\" only an **expected equity**. What exists instead is\n**superhuman approximate play**: starting with [TD-Gammon](../references.md#tesauro1995)\n(Tesauro, 1992–95), which learned near-expert evaluation from self-play,\nneural-network engines (GNU Backgammon, XG, etc.) now exceed the best humans and\nagree closely on equities. Endgame (\"bearoff\") databases *are* solved exactly by\nretrograde analysis. The opening-position equity is known to be a small edge for\nthe player on roll — but this is a precise statistical estimate, not a proof."}, {"id": 72, "type": "game", "source": "backgammon", "section": "Consensus on optimal play", "text": "Backgammon — Consensus on optimal play\n\n- **Prime your opponent** — building a consecutive wall of 6 points (a \"prime\") that an opponent's checker cannot pass is the single most powerful strategic structure; a checker trapped behind a 6-prime is out of the game until the prime breaks.\n- **Hit loose blots early, especially on your home board** — sending an opponent checker to the bar when you have a strong home board forces them to re-enter from scratch; timing hits with a strong board maximises this penalty.\n- **Double aggressively, take marginal cubes** — the doubling cube swings equity dramatically; engines show that players double too late and drop too readily; the correct take/drop threshold is around 25% winning chances (accounting for gammon chances).\n- **Race equity: use pip count** — in pure racing positions (no contact), the player ahead in raw pip count has winning equity proportional to the lead; top players count pips mentally to calibrate cube decisions.\n- **Anchor on opponent's high points to survive backgame** — when behind in a race, establishing an \"anchor\" (your own point deep in the opponent's home board) gives re-entry from hits, delays bearoff, and threatens counterplay.\n- **Bearoff accuracy is exact** — retrograde databases give perfect play for all bearoff positions; memorise the key bearoff equities (e.g., single vs. two-checker bearoffs) to avoid errors in the final race."}, {"id": 73, "type": "game", "source": "backgammon", "section": "Engines & current best play", "text": "Backgammon — Engines & current best play\n\n- **Strongest known program(s):** eXtreme Gammon (XG) and GNU Backgammon ([https://www.gnu.org/software/gnubg/](https://www.gnu.org/software/gnubg/) ([archive](http://web.archive.org/web/20260512225409/http://www.gnu.org/software/gnubg/))) — neural-network evaluation with rollout verification; TD-Gammon (Tesauro, 1992–95) was the original breakthrough.\n- **Strength:** Super-human; top engines exceed world-champion humans.\n- **Where the proof / tablebase lives (if solved):** Bearoff databases (exact, up to ~15 checkers per side) are widely distributed; full game is unsolved.\n- **Notes:** TD-Gammon's self-play learning (reinforcement learning from temporal differences) was a landmark in game AI; modern engines use similar but deeper neural networks with vastly more rollout data."}, {"id": 74, "type": "game", "source": "backgammon", "section": "Complexity", "text": "Backgammon — Complexity\n\nState-space on the order of 10^20 positions; the effective game-tree complexity\nis enormous because of the dice branching, which is the fundamental obstacle to a\nformal solution."}, {"id": 75, "type": "game", "source": "backgammon", "section": "References", "text": "Backgammon — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Backgammon) ([archive](http://web.archive.org/web/20260513154445/https://en.wikipedia.org/wiki/Backgammon))\n- [Tesauro (1995). *Temporal Difference Learning and TD-Gammon*.](../references.md#tesauro1995)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 76, "type": "game", "source": "backgammon", "section": "See also", "text": "Backgammon — See also\n\n- [EinStein würfelt nicht!](einstein-wurfelt-nicht.md) · [Yahtzee](yahtzee.md)\n- Lexicon: [chance element](../lexicon/README.md#chance-element) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 77, "type": "game", "source": "bao-la-kujifunza", "section": "overview", "text": "Bao la Kujifunza\nBeginner's variant of Bao — a teaching mancala, unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan mancala."}, {"id": 78, "type": "game", "source": "bao-la-kujifunza", "section": "Description", "text": "Bao la Kujifunza — Description\n\nBao la Kujifunza (\"Bao for learning\") is the simplified variant of\n[Bao](bao.md) used to teach the full game. It uses the same 4×8 board but\nomits the complex \"namua\" reserve and the special \"nyumba\" home-square rules,\nleaving a recognisably simpler relay-sow capture system."}, {"id": 79, "type": "game", "source": "bao-la-kujifunza", "section": "Rules", "text": "Bao la Kujifunza — Rules\n\n1. Board: 4×8, 32 pits. Each side controls the two rows closest to them.\n2. All 64 seeds start distributed evenly (commonly 2 seeds per pit); there is\n   no off-board reserve.\n3. On a turn the player picks up all seeds from one of their pits and sows\n   counterclockwise (within their two rows).\n4. If the last seed lands in a non-empty pit, the player picks up its contents\n   and continues sowing (**relay sowing**); if it lands in an empty pit, the\n   turn ends.\n5. **Capture**: when the relay ends in a front-row pit and the opposing\n   front-row pit (across the board) is non-empty, the player captures those\n   seeds and sows them into their own back row starting from a designated\n   end.\n6. A player who cannot move loses."}, {"id": 80, "type": "game", "source": "bao-la-kujifunza", "section": "Solution status", "text": "Bao la Kujifunza — Solution status\n\nBao la Kujifunza is **not solved**. It is smaller than the full Bao game but\nno game-theoretic value has been computed."}, {"id": 81, "type": "game", "source": "bao-la-kujifunza", "section": "Consensus on optimal play", "text": "Bao la Kujifunza — Consensus on optimal play\n\n- **Initiate relay chains from your front row** — relay sowing only captures if it terminates in a front-row pit opposite a non-empty enemy pit; seeds in the back row set up future front-row attacks but do not threaten immediately.\n- **Target loaded front-row pits** — look for opponent front-row pits with many seeds; terminating your relay there strips those seeds and sows them beneficially into your own back row.\n- **Keep relay chains going** — a relay that continues through multiple pits in a single turn is much more powerful than a simple drop; pick up pits that land in non-empty squares to maximise the cascading effect.\n- **Leave your opponent with thin or empty pits** — a player with no legal move loses; exhausting the opponent's front row denies them capture threats and moves them toward immobility.\n- **Preserve seeds in your back row** — back-row seeds are harder for your opponent to capture; building a seed reserve in the back row gives you future relay material even when the front row is depleted."}, {"id": 82, "type": "game", "source": "bao-la-kujifunza", "section": "Engines & current best play", "text": "Bao la Kujifunza — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Bao la Kujifunza is primarily a pedagogical stepping stone to full Bao; no dedicated computational analysis or competitive engine has been published."}, {"id": 83, "type": "game", "source": "bao-la-kujifunza", "section": "Complexity", "text": "Bao la Kujifunza — Complexity\n\nSmaller than full Bao but still substantial."}, {"id": 84, "type": "game", "source": "bao-la-kujifunza", "section": "References", "text": "Bao la Kujifunza — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Bao_(game)) ([archive](http://web.archive.org/web/20260210162711/https://en.wikipedia.org/wiki/Bao_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 85, "type": "game", "source": "bao-la-kujifunza", "section": "See also", "text": "Bao la Kujifunza — See also\n\n- [Bao](bao.md) · [Awari](awari.md) · [Kalah](kalah.md) · [Toguz Kumalak](toguz-kumalak.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 86, "type": "game", "source": "bao", "section": "overview", "text": "Bao\nThe deepest mancala — a four-row East African game of legendary complexity,\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan sowing (mancala) game."}, {"id": 87, "type": "game", "source": "bao", "section": "Description", "text": "Bao — Description\n\nPlayed on **four** rows of eight pits (two rows per player), with an additional\nprivate store pit (*nyumba*, \"the house\"). Bao has an elaborate rule set: a\ndistinct opening \"namua\" phase in which reserve seeds are entered one per turn,\nmulti-lap sowing, captures that feed seeds into the player's own rows, and\ndirection choices. It is widely regarded as the most strategically deep of all\nmancala games."}, {"id": 88, "type": "game", "source": "bao", "section": "Solution status", "text": "Bao — Solution status\n\nBao is **unsolved**. Its four-row board, multi-lap sowing (a single move can\ncascade around the board many times), and the two-phase structure with reserve\nseeds give it a state space and branching factor far beyond the two-row\nmancalas. Where [Awari](awari.md) was strongly solved by enumerating ~10^12\npositions, Bao's complexity has kept it out of reach; it has a competitive\nhuman community and some playing programs, but no game-theoretic value has been\nestablished and even strong computer play is comparatively undeveloped."}, {"id": 89, "type": "game", "source": "bao", "section": "Consensus on optimal play", "text": "Bao — Consensus on optimal play\n\n- **Protect the nyumba throughout the namua phase** — the home pit (nyumba) accumulates seeds during opening; a well-defended nyumba gives a powerful late-game resource while an exposed one is a vulnerability to opponent captures.\n- **Control the front-row inner pits** — the two inner pits of each player's front row are the capture trigger points; keeping seeds in these pits maximises capture opportunity while denying the opponent access reduces their attacking options.\n- **Initiate multi-lap chains that end on opponent-facing pits** — the power of Bao lies in relay sowing; a move that cascades through several pits and terminates opposite a loaded enemy pit can clear a large section of their row in one turn.\n- **Namua entry point matters** — during the namua (reserve) phase, where you introduce a seed into your row determines which relay cascade follows; experienced players choose the entry pit to generate immediate captures rather than passive placements.\n- **Maintaining seed density in the back row provides flexibility** — back-row seeds are difficult to capture; building a seed reserve there allows sustained front-row attacks even after the front is depleted.\n- **Direction choice is a major weapon** — at key junctions where sowing direction is optional, choosing the direction that avoids completing a dead-end relay and instead feeds a capture chain is the primary high-level skill."}, {"id": 90, "type": "game", "source": "bao", "section": "Engines & current best play", "text": "Bao — Engines & current best play\n\n- **Strongest known program(s):** No widely-available dedicated Bao engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked publicly.\n- **Notes:** Bao has a strong human competitive tradition in East Africa (particularly Tanzania and Kenya); computational analysis lags far behind the human game's depth, and no game-theoretic value has been established."}, {"id": 91, "type": "game", "source": "bao", "section": "Complexity", "text": "Bao — Complexity\n\nNot well quantified in the solving literature, but clearly orders of magnitude\nbeyond Awari/Kalah owing to four rows and multi-lap sowing."}, {"id": 92, "type": "game", "source": "bao", "section": "References", "text": "Bao — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Bao_(game)) ([archive](http://web.archive.org/web/20260210162711/https://en.wikipedia.org/wiki/Bao_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 93, "type": "game", "source": "bao", "section": "See also", "text": "Bao — See also\n\n- [Awari (Oware)](awari.md) · [Kalah](kalah.md)\n- Lexicon: [state-space complexity](../lexicon/README.md#state-space-complexity)"}, {"id": 94, "type": "game", "source": "battleship", "section": "overview", "text": "Battleship\nThe childhood guessing game — an imperfect-information game; \"optimal play\"\nSolution status: Unsolved (as a game); search heuristics well studied. Game-theoretic value: Unknown — depends on opponent placement strategy. Players: 2. Type: Imperfect-information game (simultaneous hidden placement, then search)."}, {"id": 95, "type": "game", "source": "battleship", "section": "Description", "text": "Battleship — Description\n\nEach player secretly places a fleet of ships on a 10×10 grid. Players then\nalternate calling shots at coordinates on the opponent's grid; the opponent\nannounces hit or miss (and, in most rules, when a ship is sunk). The first player\nto sink the opponent's entire fleet wins. Because each player's board is hidden,\nthis is a game of **imperfect information**, unlike most entries in this archive."}, {"id": 96, "type": "game", "source": "battleship", "section": "Solution status", "text": "Battleship — Solution status\n\nBattleship is **not \"solved\"** in the combinatorial-game-theory sense, and the\nnotion barely applies: there is no single game-theoretic value because the\noutcome depends on each player's *placement* policy and *shooting* policy against\nthe other's. What *is* well studied is the **search problem** — given the\ninformation revealed so far, which square maximises the chance of a hit.\nProbability-density (\"parity\" plus hit-following) heuristics are known to be\nstrong and near-optimal for the shooting phase against a uniformly random\nplacement. The full game — including adversarial placement — is best modelled as\na two-player game with hidden information and has no published optimal-strategy\nsolution."}, {"id": 97, "type": "game", "source": "battleship", "section": "Consensus on optimal play", "text": "Battleship — Consensus on optimal play\n\n- **Use probability-density targeting during the hunting phase** — mentally (or computationally) track which squares can still contain an unsunk ship given all misses; always shoot at the square with the highest probability of being occupied.\n- **Exploit parity to reduce wasted shots** — ships occupy at least 2 consecutive squares; during the hunting phase, only fire at every other square in a checkerboard pattern to guarantee touching every possible 2-square ship with minimal shots.\n- **Follow hits in both directions** — when you score a hit, shoot the adjacent squares along a line until you find both ends of the ship before switching back to hunting; this sinks ships faster than scattering shots after a hit.\n- **Avoid placing ships at edges and corners** — against a probability-density hunter, ships near the edges are statistically easier to locate because fewer ship orientations fit there; interior placement forces the opponent to waste more shots.\n- **Separate your ships** — placing ships adjacent or near each other concentrates targets; a spread-out fleet makes each hit less informative about where neighbouring ships are.\n- **Vary your placement pattern against repeated opponents** — because Battleship has hidden information and depends on opponent strategy, the \"optimal\" placement is really a mixed strategy; avoid predictable patterns that a learning opponent can exploit."}, {"id": 98, "type": "game", "source": "battleship", "section": "Engines & current best play", "text": "Battleship — Engines & current best play\n\n- **Strongest known program(s):** Various AI agents implementing probability-density search (e.g., academic and hobbyist implementations); no single canonical named engine.\n- **Strength:** Optimal search-phase play is well approximated by probability-density algorithms; placement strategy against adversarial opponents remains heuristic.\n- **Where the proof / tablebase lives (if solved):** — (not solved as a full adversarial game)\n- **Notes:** The shooting sub-problem is a well-studied probability puzzle; the full two-player adversarial game including placement is an open problem in imperfect-information game theory."}, {"id": 99, "type": "game", "source": "battleship", "section": "Complexity", "text": "Battleship — Complexity\n\nThe placement space is large (hundreds of thousands of legal fleet layouts), and\nbecause information is hidden the game does not have a game-tree complexity in\nthe perfect-information sense."}, {"id": 100, "type": "game", "source": "battleship", "section": "References", "text": "Battleship — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Battleship_(game)) ([archive](http://web.archive.org/web/20260505154451/https://en.wikipedia.org/wiki/Battleship_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 101, "type": "game", "source": "battleship", "section": "See also", "text": "Battleship — See also\n\n- [Liar's dice](liars-dice.md) · [Mastermind](mastermind.md) · [Heads-up limit hold'em](heads-up-limit-holdem.md)\n- Lexicon: [perfect information](../lexicon/README.md#perfect-information) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 102, "type": "game", "source": "brandubh", "section": "overview", "text": "Brandubh\nThe small Irish tafl game on a 7×7 board — compact enough for endgame\nSolution status: Partially solved / analysed (no published full solution) **[verify]**. Game-theoretic value: Unknown (rule-set dependent) **[verify]**. Players: 2 (asymmetric: attackers vs. king's defenders). Type: Partisan asymmetric capture game."}, {"id": 103, "type": "game", "source": "brandubh", "section": "Description", "text": "Brandubh — Description\n\nThe smallest widely-played tafl variant: a 7×7 board with a central throne and\nfour corner squares. The **defender** has a king plus 4 soldiers; the\n**attacker** has 8 soldiers. All pieces move like a rook; captures are custodial\n(sandwiching an enemy between two friendly pieces). The king wins by reaching a\ncorner; the attackers win by capturing the king. As with all tafl games, the\nhistorical rules are incompletely recorded, so several modern rule sets coexist."}, {"id": 104, "type": "game", "source": "brandubh", "section": "Solution status", "text": "Brandubh — Solution status\n\nBrandubh is **not formally solved** in the literature, though its 7×7 board makes\nit the most tractable tafl variant — small enough that endgame retrograde\ndatabases and strong solvers are feasible, and several tafl-community analyses\nreport particular rule sets as decisive wins for one side. As with [Tablut](tablut.md),\n**rule ambiguity** is the central problem: there is no single canonical Brandubh\nwhose value could be quoted. Treat any specific value claim as **[verify]**\npending a peer-reviewed solution of a clearly stated rule set."}, {"id": 105, "type": "game", "source": "brandubh", "section": "Consensus on optimal play", "text": "Brandubh — Consensus on optimal play\n\n- **Defenders: break out early** — the king wins by reaching a corner, not by surviving indefinitely; early aggressive movement toward a corner puts the attacker on the back foot and is generally considered the stronger defensive strategy.\n- **Attackers: seal the diagonals** — the four corners are the king's only escape; attackers must cover the diagonal corridors leading to corners while simultaneously building a custodial capture threat around the king.\n- **Custodial forks decide close games** — a piece that simultaneously threatens to sandwich two different enemy pieces (a custodial fork) either wins material or forces the opponent into a losing repositioning.\n- **The throne is a double-edged asset for defenders** — the throne counts as a friendly piece for capturing the king when the king is adjacent to it (under most rule sets), but the king cannot re-enter it under some rules; know your rule set's throne interactions precisely.\n- **Balance is rule-set-sensitive** — the small board makes Brandubh highly sensitive to exact rules about shieldwall capture, corner entry, and throne mechanics; opening moves that are strong under one rule set may be losing under another."}, {"id": 106, "type": "game", "source": "brandubh", "section": "Engines & current best play", "text": "Brandubh — Engines & current best play\n\n- **Strongest known program(s):** No widely-distributed dedicated Brandubh engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)); the competitive tafl community uses custom programs for analysis.\n- **Strength:** Not benchmarked publicly.\n- **Notes:** Brandubh's 7×7 board makes full retrograde analysis computationally feasible; informal community analyses exist for specific rule sets, but no peer-reviewed solution of a canonical rule set has been published."}, {"id": 107, "type": "game", "source": "brandubh", "section": "Complexity", "text": "Brandubh — Complexity\n\nModerate — the 7×7 board and small piece count put full retrograde analysis of\nfixed rule sets within computational reach, even though none has been formally\npublished as a solution."}, {"id": 108, "type": "game", "source": "brandubh", "section": "References", "text": "Brandubh — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Brandubh) ([archive](http://web.archive.org/web/20210126053031/http://en.wikipedia.org/wiki/Brandubh))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 109, "type": "game", "source": "brandubh", "section": "See also", "text": "Brandubh — See also\n\n- [Tablut](tablut.md) · [Fox and Geese](fox-and-geese.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 110, "type": "game", "source": "brazilian-draughts", "section": "overview", "text": "Brazilian draughts\n8×8 international draughts — flying kings on a small board, unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan draughts."}, {"id": 111, "type": "game", "source": "brazilian-draughts", "section": "Description", "text": "Brazilian draughts — Description\n\nBrazilian draughts is essentially international draughts (Polish draughts)\nplayed on an 8×8 board: men capture backward, kings fly, and capture\nsequences must take the maximum number of pieces."}, {"id": 112, "type": "game", "source": "brazilian-draughts", "section": "Rules", "text": "Brazilian draughts — Rules\n\n1. Board: 8×8 with dark squares to the player's left. Each side has 12 men.\n2. Men move one square diagonally forward; men capture by jumping enemy\n   pieces forward or backward.\n3. Captures are mandatory and must take the **maximum number of pieces**\n   available.\n4. Men promoting on the back rank during a chain capture continue as kings if\n   they can still capture.\n5. Kings move and capture any distance along a diagonal (flying king); they\n   must land on the square immediately past the captured piece's row, but may\n   choose any empty square along the line.\n6. Loss conditions: no legal moves available."}, {"id": 113, "type": "game", "source": "brazilian-draughts", "section": "Solution status", "text": "Brazilian draughts — Solution status\n\nBrazilian draughts is **not solved**. Tablebase work covers small endgames;\nengines are strong but no full proof exists."}, {"id": 114, "type": "game", "source": "brazilian-draughts", "section": "Consensus on optimal play", "text": "Brazilian draughts — Consensus on optimal play\n\n- **Maximum-capture rule dominates tactics** — all legal captures are mandatory and you must take the maximum number of pieces; your entire tactical calculation must start by finding the longest capture chain available to each side before considering positional moves.\n- **Promote to flying king as fast as possible** — a king that can sweep diagonals is vastly more powerful than a man; advancing pieces toward the back rank while blocking opponent promotions is the primary strategic objective.\n- **Control the long diagonal** — as in international draughts, the long diagonal is a key highway for flying kings; anchoring a man or king on the central long diagonal squares restricts opponent king mobility.\n- **Maintain piece balance; avoid forced-exchange disadvantage** — the maximum-capture rule means exchanges can be forced; make sure your capture chains do not leave you with fewer or weaker pieces after the sequence resolves.\n- **Tempo matters in king endings** — king vs. king endings often hinge on who has the opposition (the right diagonal relationship); flying kings make triangulation manoeuvres important in pure king endgames."}, {"id": 115, "type": "game", "source": "brazilian-draughts", "section": "Engines & current best play", "text": "Brazilian draughts — Engines & current best play\n\n- **Strongest known program(s):** Various draughts engines adapted for Brazilian rules (e.g., Kingsrow or Cake variants); no single canonical public engine for this specific variant is prominently documented.\n- **Strength:** Super-human for endgame positions covered by tablebases; strong amateur to expert-level in midgame.\n- **Where the proof / tablebase lives (if solved):** Endgame tablebases for small piece counts exist; full game is unsolved.\n- **Notes:** Brazilian draughts sits between English draughts (solved) and international draughts (10×10, unsolved) in complexity; its flying-king rules make it considerably harder to solve than English draughts despite the same 8×8 board."}, {"id": 116, "type": "game", "source": "brazilian-draughts", "section": "Complexity", "text": "Brazilian draughts — Complexity\n\nSimilar to English draughts."}, {"id": 117, "type": "game", "source": "brazilian-draughts", "section": "References", "text": "Brazilian draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Brazilian_draughts) ([archive](http://web.archive.org/web/20260312045332/https://en.wikipedia.org/wiki/Brazilian_draughts))\n- [Schaeffer et al. (2007). *Checkers is Solved*.](../references.md#schaeffer2007) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 118, "type": "game", "source": "brazilian-draughts", "section": "See also", "text": "Brazilian draughts — See also\n\n- [International draughts](international-draughts.md) · [English draughts](checkers.md) · [Russian draughts](russian-draughts.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase)"}, {"id": 119, "type": "game", "source": "breakthrough", "section": "overview", "text": "Breakthrough\nA modern racing game with simple rules — weakly solved on small boards, but\nSolution status: Partially solved (small boards weakly solved; 8×8 unsolved). Game-theoretic value: Varies by board size; 8×8 unknown. Players: 2. Type: Partisan racing / capture game."}, {"id": 120, "type": "game", "source": "breakthrough", "section": "Description", "text": "Breakthrough — Description\n\nInvented by Dan Troyka in 2000 (it won a 2001 design competition). Played on a\nrectangular board — standard is 8×8. Each player has two rows of identical\npieces. A piece moves one square straight or diagonally forward; it may **capture\nonly diagonally** forward. There are no other captures and pieces never move\nbackward. The first player to reach the opponent's back rank — or to capture all\nenemy pieces — wins. Draws are impossible: material is monotonically depleted and\nsomeone must break through."}, {"id": 121, "type": "game", "source": "breakthrough", "section": "Solution status", "text": "Breakthrough — Solution status\n\nBreakthrough is **partially solved**. [Saffidine, Jouandeau & Cazenave (2012)](../references.md#saffidine-breakthrough2012)\nweakly solved several small boards (e.g. 3×n and other reduced sizes) using race\npatterns and job-level proof-number search; subsequent work extended weak\nsolutions to further small boards (such as 6×5). The **standard 8×8 game is\nunsolved** — its state space (~10^28) and branching factor remain out of reach of\na full weak solution, though it is a popular test bed for game-AI research and\nstrong programs play it well."}, {"id": 122, "type": "game", "source": "breakthrough", "section": "Consensus on optimal play", "text": "Breakthrough — Consensus on optimal play\n\n- **Advance on a broad front, not a single file** — pieces can only capture diagonally, so a piece advancing in a single column can be blocked by a lone defender directly ahead; spreading multiple pieces across files creates threats the opponent cannot all cover.\n- **Create a passed pawn equivalent** — a piece with no enemy piece that can diagonally intercept it on the way to the back rank is effectively won; creating such a \"passer\" while denying the opponent the same is the main strategic goal.\n- **Trade advantageously to open a lane** — a capture is always diagonal, never forward; use captures to remove pieces that would block or deflect your advance, choosing exchanges that leave your own pieces better positioned to race.\n- **Tempo is decisive in races** — both players simultaneously advance; counting how many moves each side needs to promote a piece (the \"race count\") tells you whether you can afford to spend a move on a capture or must push straight ahead.\n- **Use wing pieces to threaten diversionary attacks** — an advance on the flank forces the opponent to defend it, which can free up a path in the centre for your decisive breakthrough."}, {"id": 123, "type": "game", "source": "breakthrough", "section": "Engines & current best play", "text": "Breakthrough — Engines & current best play\n\n- **Strongest known program(s):** Various research bots used in the Computer Olympiad; no single canonical public engine is prominently distributed, but Breakthrough is a standard benchmark in Monte Carlo Tree Search research.\n- **Strength:** Super-human; strong MCTS engines consistently outperform human players.\n- **Where the proof / tablebase lives (if solved):** Small-board weak solutions (Saffidine, Jouandeau & Cazenave, 2012) — see [../references.md#saffidine-breakthrough2012](../references.md#saffidine-breakthrough2012); 8×8 unsolved.\n- **Notes:** Breakthrough is popular in game-AI research because its simple rules and no-draws property make it easy to implement and benchmark; proof-number search and MCTS have both been applied successfully on small boards."}, {"id": 124, "type": "game", "source": "breakthrough", "section": "Complexity", "text": "Breakthrough — Complexity\n\n8×8 state-space complexity is roughly 10^28; the branching factor is moderate\nbut the depth and width together keep a full solution out of reach."}, {"id": 125, "type": "game", "source": "breakthrough", "section": "References", "text": "Breakthrough — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Breakthrough_(board_game)) ([archive](http://web.archive.org/web/20251215011900/https://en.wikipedia.org/wiki/Breakthrough_(board_game)))\n- [Saffidine, Jouandeau & Cazenave (2012). *Solving Breakthrough with Race Patterns and Job-Level Proof Number Search*.](../references.md#saffidine-breakthrough2012)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 126, "type": "game", "source": "breakthrough", "section": "See also", "text": "Breakthrough — See also\n\n- [Hexapawn](hexapawn.md) · [Clobber](clobber.md) · [Chess](chess.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [proof-number search](../lexicon/README.md#proof-number-search)"}, {"id": 127, "type": "game", "source": "bridg-it", "section": "overview", "text": "Bridg-it\nA connection game with a complete, elegant solution — the first player wins,\nSolution status: Strongly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan connection game."}, {"id": 128, "type": "game", "source": "bridg-it", "section": "Description", "text": "Bridg-it — Description\n\nBridg-it (David Gale's game, marketed in the 1960s) is played on two\ninterleaved grids of dots — one for each player. Players alternately draw a\nshort edge connecting two adjacent dots of their own colour, never crossing an\nopponent's edge. One player tries to build a connected path between the top and\nbottom, the other between left and right. It is a specific instance of the\n[Shannon switching game](shannon-switching-game.md)."}, {"id": 129, "type": "game", "source": "bridg-it", "section": "Solution status", "text": "Bridg-it — Solution status\n\nBridg-it is **strongly solved**, and unusually it has *two* clean solutions:\n\n- **Pairing strategy.** Oliver Gross found an explicit\n  [pairing strategy](../lexicon/README.md#pairing-strategy) for the first player:\n  the board's edges can be paired up so that, after the first player's strong\n  opening move, every opponent move has a designated partner edge whose play\n  maintains the winning connection. This is a complete, easily executed winning\n  strategy.\n- **Matroid solution.** [Lehman (1964)](../references.md#lehman1964) solved the\n  general Shannon switching game (of which Bridg-it is a case) using matroid\n  theory, characterising exactly when the connecting player can win.\n\nEither way, the first player wins from the standard start, with a known\nstrategy — the definition of strongly solved."}, {"id": 130, "type": "game", "source": "bridg-it", "section": "Consensus on optimal play", "text": "Bridg-it — Consensus on optimal play\n\n- **First player makes one strong opening move, then mirrors using the pairing strategy** — after the opening, every edge on the board (except the one already played) can be paired with a partner edge; whenever the opponent plays one edge of a pair, the first player immediately plays its partner, guaranteeing a connected path regardless of what the opponent does.\n- **The pairing strategy is the complete answer** — there is no need for heuristic reasoning; the pairing strategy is a mathematically proven winning strategy and can be executed move-by-move without lookahead.\n- **Second player cannot win against the pairing strategy** — the game is decided; playing second in Bridg-it is a losing position with no recourse, making the swap (pie) rule essential for competitive fair play.\n- **The game has no draws** — by the parity of the board construction, one player must form a winning connection before the other; this is guaranteed by a topological argument (similar to Hex's no-draw proof)."}, {"id": 131, "type": "game", "source": "bridg-it", "section": "Engines & current best play", "text": "Bridg-it — Engines & current best play\n\n- **Strongest known program(s):** Any program that implements the pairing strategy plays perfectly. No specialised commercial engine needed.\n- **Strength:** Perfect play is achievable by a simple rule (pairing strategy) — no search required.\n- **Where the proof / tablebase lives (if solved):** Lehman (1964) via matroid theory; Gross pairing strategy described in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Bridg-it is one of the rare non-trivial two-player games with a clean, human-executable winning strategy rather than a lookup table or deep search tree."}, {"id": 132, "type": "game", "source": "bridg-it", "section": "Complexity", "text": "Bridg-it — Complexity\n\nGrows with board size, but the pairing strategy makes optimal play O(1) per move\nregardless."}, {"id": 133, "type": "game", "source": "bridg-it", "section": "References", "text": "Bridg-it — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shannon_switching_game) ([archive](http://web.archive.org/web/20260107194010/https://en.wikipedia.org/wiki/Shannon_switching_game))\n- [Lehman, A. (1964). *A solution of the Shannon switching game*.](../references.md#lehman1964)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 134, "type": "game", "source": "bridg-it", "section": "See also", "text": "Bridg-it — See also\n\n- [Shannon switching game](shannon-switching-game.md) · [Hex](hex.md) · [Y](y.md)\n- Lexicon: [pairing strategy](../lexicon/README.md#pairing-strategy) · [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 135, "type": "game", "source": "bridge", "section": "overview", "text": "Bridge\nThe classical trick-taking game — unsolved as a competitive game.\nSolution status: Unsolved as a full game; **double-dummy** (open hands) cases are computationally tractable. Game-theoretic value: Unknown. Players: 4 (two partnerships). Type: Trick-taking, imperfect information, partial-information."}, {"id": 136, "type": "game", "source": "bridge", "section": "Description", "text": "Bridge — Description\n\nBridge is the canonical 4-player partnership trick-taking game. Two phases —\n**bidding** (an information-revealing auction for the contract) and\n**play** (taking tricks against the contract) — combine into a game with\nhidden information, partner inference, and probabilistic squeeze play.\n\"Solving\" bridge in any robust sense remains an open problem; computer\ndouble-dummy solvers handle single-hand analysis where all cards are visible."}, {"id": 137, "type": "game", "source": "bridge", "section": "Rules", "text": "Bridge — Rules\n\n1. Players: 4, sitting as two opposing partnerships (N–S vs. E–W); a standard\n   52-card deck is dealt 13 cards to each.\n2. **Bidding**: starting with the dealer, each player makes a \"bid\" (a level\n   1–7 and a denomination of one of four suits or No-Trump), passes, doubles,\n   or redoubles. Bidding continues until three consecutive passes; the final\n   bid becomes the **contract**.\n3. The opener's left-hand opponent leads to the first trick; the partner of\n   the declaring player puts their hand face-up as **dummy**.\n4. **Play**: 13 tricks; standard trick-taking rules — must follow suit, may\n   trump if no suit card, highest card of led suit (or highest trump) wins.\n5. Scoring: the declaring side scores points based on contract level and\n   bonuses (game/slam/honours/vulnerability). The non-declaring side scores\n   for setting the contract."}, {"id": 138, "type": "game", "source": "bridge", "section": "Solution status", "text": "Bridge — Solution status\n\nBridge is **not solved**. The hidden-information and partnership-signalling\naspects mean optimality is best framed in equilibrium terms; the\ndouble-dummy (open-hands) restriction has been well analysed by\ncombinatorial solvers (e.g., Bo Haglund's `dds` library)."}, {"id": 139, "type": "game", "source": "bridge", "section": "Consensus on optimal play", "text": "Bridge — Consensus on optimal play\n\n- **Bid to the correct level based on combined partnership strength** — game contracts (3NT, 4♥, 4♠, 5♣/♦) score a large bonus; the key threshold is roughly 25+ high-card points for 3NT/4-of-a-major, 33+ for small slam; bidding short or past this threshold wastes or overspends the hand's value.\n- **Plan the play before playing to trick one** — the declarer should count winners and losers before leading from dummy; a plan that avoids blocking suit entries or prematurely releasing key controls prevents many avoidable defeats.\n- **Count the hand as a defender** — both defenders should track which high cards have appeared and infer from partner's signals (attitude, count, suit-preference leads) where the remaining honours sit; this inference replaces the missing full information.\n- **Signals and leads follow standard agreements** — standard fourth-best leads, attitude signals on partner's lead, and count signals when declarer draws trumps allow both defenders to build an accurate picture of the hidden cards; deviating from agreed signals hands declarer an inference advantage.\n- **Double-dummy analysis gives the absolute best play for each hand** — when all hands are known (e.g., post-deal analysis), the `dds` library computes the maximum tricks achievable for every possible contract; this is the gold standard for evaluating defensive or declaring errors."}, {"id": 140, "type": "game", "source": "bridge", "section": "Engines & current best play", "text": "Bridge — Engines & current best play\n\n- **Strongest known program(s):** Double-dummy solver: Bo Haglund's `dds` library; full-game AI: NooK (DeepMind/Google), BBO (Bridge Base Online) AI bots.\n- **Strength:** Double-dummy play is solved exactly; full-game bots are competitive with expert humans but not definitively stronger across all bidding and play situations.\n- **Where the proof / tablebase lives (if solved):** `dds` is open-source; full-game bridge remains unsolved.\n- **Notes:** Bridge is the hardest major card game to solve owing to its combination of hidden information, partnership communication (bidding system), and deep play; recent neural-network bots (NooK) have reached strong amateur level in full-game play."}, {"id": 141, "type": "game", "source": "bridge", "section": "Complexity", "text": "Bridge — Complexity\n\nEnormous in full form."}, {"id": 142, "type": "game", "source": "bridge", "section": "References", "text": "Bridge — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Contract_bridge) ([archive](http://web.archive.org/web/20260508033633/https://en.wikipedia.org/wiki/Contract_bridge))\n- [Thompson, K. (1997). *Bridge-card analysis history*.](../references.md#thompson-bridge) **[verify]**"}, {"id": 143, "type": "game", "source": "bridge", "section": "See also", "text": "Bridge — See also\n\n- [Skat](skat.md) · [Hanabi](hanabi.md)\n- Lexicon: [imperfect information](../lexicon/README.md#imperfect-information)"}, {"id": 144, "type": "game", "source": "brussels-sprouts", "section": "overview", "text": "Brussels Sprouts\nA \"game\" that looks strategic but whose winner is fixed before a single move\nSolution status: Strongly solved. Game-theoretic value: Completely determined by the number of starting crosses — no choice affects it. Players: 2. Type: Impartial combinatorial game (topological); a \"fake\" game."}, {"id": 145, "type": "game", "source": "brussels-sprouts", "section": "Description", "text": "Brussels Sprouts — Description\n\nA variant of [Sprouts](sprouts.md). Start with *n* crosses (+), each with four\nfree ends. On a turn a player draws a line connecting two free ends (not\ncrossing existing lines) and adds a small cross-bar in the middle, creating two\nnew free ends. The player unable to move loses."}, {"id": 146, "type": "game", "source": "brussels-sprouts", "section": "Solution status", "text": "Brussels Sprouts — Solution status\n\nBrussels Sprouts is **strongly solved** — trivially and completely. A simple\nEuler-characteristic argument shows that **every game starting from *n* crosses\nlasts exactly 5n − 2 moves**, no matter what either player does. Therefore the\noutcome depends only on the parity of 5n − 2: the first player wins iff that\nnumber is odd, i.e. iff *n* is odd.\n\nNo move ever matters. Brussels Sprouts is the canonical example of a\n**\"fake\" game** whose result is predetermined — a useful counterpoint to its\ngenuinely deep cousin [Sprouts](sprouts.md), and a reminder that \"strongly\nsolved\" can sometimes mean \"there was never anything to solve.\""}, {"id": 147, "type": "game", "source": "brussels-sprouts", "section": "Consensus on optimal play", "text": "Brussels Sprouts — Consensus on optimal play\n\n- **The winner is determined entirely by the starting count *n*** — every game from *n* crosses lasts exactly 5n − 2 moves; if that number is odd (i.e., *n* is odd), the first player wins; if even (*n* even), the second player wins. No move by either player can change this.\n- **No strategy is meaningful** — because the game length is fixed regardless of moves made, there is literally no \"better\" or \"worse\" play; any legal move has the same outcome as any other.\n- **Use this as a teaching example of predetermined games** — Brussels Sprouts illustrates that the appearance of decision-making does not imply strategic depth; the Euler-characteristic argument makes this one of the cleanest \"fake game\" examples in combinatorial game theory."}, {"id": 148, "type": "game", "source": "brussels-sprouts", "section": "Engines & current best play", "text": "Brussels Sprouts — Engines & current best play\n\n- **Strongest known program(s):** Any program that counts the starting crosses and returns the result in O(1) is \"optimal.\" No game-playing engine is meaningful here.\n- **Strength:** Perfect — trivially, since the outcome is computable without any game-tree search.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001) — Euler-characteristic argument for the fixed game length 5n − 2.\n- **Notes:** Brussels Sprouts is the archetypal \"predetermined game\" in CGT literature; it is useful pedagogically but has no strategic content whatsoever."}, {"id": 149, "type": "game", "source": "brussels-sprouts", "section": "Complexity", "text": "Brussels Sprouts — Complexity\n\nNot applicable — the game tree's depth is constant and the value is\nplay-independent."}, {"id": 150, "type": "game", "source": "brussels-sprouts", "section": "References", "text": "Brussels Sprouts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sprouts_(game)) ([archive](http://web.archive.org/web/20260511093257/https://en.wikipedia.org/wiki/Sprouts_(game)))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 151, "type": "game", "source": "brussels-sprouts", "section": "See also", "text": "Brussels Sprouts — See also\n\n- [Sprouts](sprouts.md) · [Conway's Soldiers](conways-soldiers.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 152, "type": "game", "source": "capablanca-chess", "section": "overview", "text": "Capablanca chess\n10×8 chess variant with added \"archbishop\" and \"chancellor\" pieces — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 153, "type": "game", "source": "capablanca-chess", "section": "Description", "text": "Capablanca chess — Description\n\nCapablanca chess (José Raúl Capablanca, ~1925) is played on a 10×8 board with\ntwo new pieces in addition to the standard chess army: the **archbishop**\n(bishop+knight) and the **chancellor** (rook+knight). Capablanca proposed it\nas a way to reduce draws he found inevitable in orthodox chess."}, {"id": 154, "type": "game", "source": "capablanca-chess", "section": "Rules", "text": "Capablanca chess — Rules\n\n1. Board: 10 files × 8 ranks. Each side has the usual chess pieces plus one\n   archbishop and one chancellor, placed in the back rank between minor pieces\n   and rooks (specific placement varies).\n2. **Archbishop**: moves either as a bishop or as a knight.\n3. **Chancellor**: moves either as a rook or as a knight.\n4. All other pieces move as in orthodox chess. Pawns promote on the 8th rank.\n5. Castling is defined to span the wider board; standard check/checkmate and\n   draw rules apply."}, {"id": 155, "type": "game", "source": "capablanca-chess", "section": "Solution status", "text": "Capablanca chess — Solution status\n\nCapablanca chess is **not solved**. The board and piece set make engine\nanalysis significantly more expensive than chess."}, {"id": 156, "type": "game", "source": "capablanca-chess", "section": "Consensus on optimal play", "text": "Capablanca chess — Consensus on optimal play\n\n- **The chancellor is roughly rook+knight in value (~8.5 pawns), the archbishop bishop+knight (~7 pawns)** — knowing these rough values (stronger than a queen) prevents naive piece trades that give up massive material; avoid exchanging a chancellor or archbishop for a queen without compensation.\n- **Control open files for your chancellor immediately** — the chancellor's rook component is dominant on open files; getting it active early (similar to a rook in orthodox chess) is a top priority.\n- **The archbishop is a long-range fork machine** — it can threaten squares a queen cannot reach; be alert to archbishop forks that simultaneously attack king and rook (or two pieces), since the knight component adds non-linear attack patterns to the bishop's long diagonals.\n- **Pawn structure principles from chess carry over** — doubled pawns, isolated pawns, and open-file weaknesses all function as in chess; the extra files (9th and 10th) simply provide more terrain for these structures to form.\n- **Opening development is less codified** — there is no large body of grandmaster-level theory; developing the powerful compound pieces quickly toward the centre and ensuring king safety (castling is available on the wide board) are the reliable fundamentals.\n- **Watch for back-rank threats from chancellors** — a chancellor on the 7th/8th rank is even more dangerous than a queen owing to its knight leap; the back-rank mate motifs from chess apply but with greater range."}, {"id": 157, "type": "game", "source": "capablanca-chess", "section": "Engines & current best play", "text": "Capablanca chess — Engines & current best play\n\n- **Strongest known program(s):** Fairy-Stockfish ([https://github.com/ianfab/Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish))) — Stockfish adapted for fairy chess variants including Capablanca chess.\n- **Strength:** Super-human (inherits Stockfish's deep search with fairy-piece extensions).\n- **Where the proof / tablebase lives (if solved):** — (unsolved; no endgame tablebase covering the full piece set)\n- **Notes:** Fairy-Stockfish is the standard analysis tool for Capablanca chess; no competitive human scene large enough to produce extensive opening theory exists, so engine analysis defines current best play."}, {"id": 158, "type": "game", "source": "capablanca-chess", "section": "Complexity", "text": "Capablanca chess — Complexity\n\nLarger than chess."}, {"id": 159, "type": "game", "source": "capablanca-chess", "section": "References", "text": "Capablanca chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Capablanca_chess) ([archive](http://web.archive.org/web/20260510041243/https://en.wikipedia.org/wiki/Capablanca_Chess))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 160, "type": "game", "source": "capablanca-chess", "section": "See also", "text": "Capablanca chess — See also\n\n- [Chess](chess.md) · [Courier chess](courier-chess.md) · [Chess960](chess960.md) · [Glinski hexagonal chess](glinski-hexagonal-chess.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 161, "type": "game", "source": "caro", "section": "overview", "text": "Caro\nThe Vietnamese five-in-a-row variant — like Gomoku but with a \"blocked\" rule\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan k-in-a-row game."}, {"id": 162, "type": "game", "source": "caro", "section": "Description", "text": "Caro — Description\n\nCaro is the Vietnamese form of five-in-a-row. It differs from\n[Gomoku](gomoku.md) in one key respect: a five-in-a-row is **not a win if it is\nblocked at both ends** by opposing stones. This single rule neutralises many of\nGomoku's standard \"double threat\" forcing sequences and gives Caro a more\ndefensive flavour."}, {"id": 163, "type": "game", "source": "caro", "section": "Rules", "text": "Caro — Rules\n\n1. A Go-style board (commonly 15×15 or 19×19), initially empty.\n2. Players alternate placing one stone of their colour on any empty\n   intersection.\n3. The first player to form an **uninterrupted line** of five or more stones\n   horizontally, vertically, or diagonally — **with at least one end of the\n   line not blocked by an opposing stone** — wins.\n4. If the board fills with no such line, the game is a draw (very rare)."}, {"id": 164, "type": "game", "source": "caro", "section": "Solution status", "text": "Caro — Solution status\n\nCaro is **not solved**. The \"blocked-at-both-ends doesn't count\" rule\nsubstantially weakens Gomoku's first-player advantage, but a formal value (and\nproof) is not published. Engines and competitive play exist and there is\npractical consensus that the first player is favoured but not decisively so."}, {"id": 165, "type": "game", "source": "caro", "section": "Consensus on optimal play", "text": "Caro — Consensus on optimal play\n\n- **Build unblocked fours and threes** — because a five-in-a-row blocked at both ends does not win, always orient your attack so that at least one end of your forming line is open; a \"four with two open ends\" (live four) is the most dangerous threat.\n- **Create double-open-three threats** — simultaneously threatening two different lines of three stones, each with both ends open (so both will become winning live fours), forces the opponent to defend both at once and is usually decisive.\n- **Block opponent threes at the earliest open end** — blocking a live three at one end converts it to a half-blocked three, severely reducing its threat value; waiting until it becomes a four is too late.\n- **Centre play opens the most attack lines** — central stones sit on more diagonals, horizontals, and verticals than edge stones; early central placement gives more directions from which to form a live five.\n- **Unlike Gomoku, direct first-player forcing wins are rarer** — the blocked rule prevents many classic Gomoku sequences; Caro rewards patient positional build-up over sharp tactical sequences, and second-player defensive resources are stronger."}, {"id": 166, "type": "game", "source": "caro", "section": "Engines & current best play", "text": "Caro — Engines & current best play\n\n- **Strongest known program(s):** Various Gomoku/five-in-a-row engines adapted for Caro rules (e.g., Yixin with Caro rule support); no single canonical public engine is prominently documented for Caro specifically.\n- **Strength:** Super-human for well-tuned engines; Vietnamese competitive community plays at high human level.\n- **Where the proof / tablebase lives (if solved):** — (unsolved)\n- **Notes:** Caro's blocked rule makes it significantly harder than Gomoku to solve; it has a large competitive following in Vietnam and online but limited academic-solving attention."}, {"id": 167, "type": "game", "source": "caro", "section": "Complexity", "text": "Caro — Complexity\n\nLarge — comparable to or harder than Gomoku because the blocked rule prevents\nmany cheap forcing wins, increasing search depth."}, {"id": 168, "type": "game", "source": "caro", "section": "References", "text": "Caro — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Gomoku) ([archive](http://web.archive.org/web/20260506025335/https://en.wikipedia.org/wiki/Gomoku))\n- [Allis, van den Herik & Huntjens (1996). *Go-Moku Solved by New Search Techniques*.](../references.md#allis-gomoku1996) (related)"}, {"id": 169, "type": "game", "source": "caro", "section": "See also", "text": "Caro — See also\n\n- [Gomoku](gomoku.md) · [Renju](renju.md) · [Pente](pente.md) · [Ninuki-renju](ninuki-renju.md)\n- Lexicon: [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 170, "type": "game", "source": "catch-the-hare", "section": "overview", "text": "Catch the Hare\nAsymmetric hunt game: hounds versus a single hare — partially analysed.\nSolution status: Solved for several boards **[verify]**. Game-theoretic value: Depends on board; small boards favour the hounds **[verify]**. Players: 2 (asymmetric). Type: Partisan asymmetric hunt game."}, {"id": 171, "type": "game", "source": "catch-the-hare", "section": "Description", "text": "Catch the Hare — Description\n\nCatch the Hare is the generic name for a family of asymmetric pursuit games\nin which a small number of hounds (typically 3) try to corner a single hare\non a small board. Versions appear across medieval Europe and the Americas."}, {"id": 172, "type": "game", "source": "catch-the-hare", "section": "Rules", "text": "Catch the Hare — Rules\n\n1. Board: small cross- or diamond-shaped grid (varies by tradition).\n2. One side controls 3 (sometimes 4) hounds; the other side controls 1 hare.\n3. Hounds may move one step **forward or sideways** but never backward;\n   the hare moves one step in any direction. No captures.\n4. The hounds win if the hare cannot move (cornered).\n5. The hare wins by **escaping past** the hounds — reaching the back row from\n   which the hounds started."}, {"id": 173, "type": "game", "source": "catch-the-hare", "section": "Solution status", "text": "Catch the Hare — Solution status\n\nSeveral specific Catch-the-Hare boards have been solved by exhaustive search;\nthe hounds typically win with perfect play if the board geometry favours\nthem, while the hare wins on more open boards. **[verify]** the canonical\nboard's value."}, {"id": 174, "type": "game", "source": "catch-the-hare", "section": "Consensus on optimal play", "text": "Catch the Hare — Consensus on optimal play\n\n- **Hounds must advance as an unbroken line** — because the hare can escape through any gap in the hound formation, the hounds must maintain a contiguous front with no jumpable spaces; a single-step gap between two adjacent hounds allows the hare to slip through and win.\n- **Hounds: never let a hound fall behind the others** — all three hounds should advance together at roughly the same rank; an isolated hound ahead of its companions is easily flanked and the formation breaks.\n- **Hare: immediately probe for and exploit gaps** — the hare's only winning strategy is to find or force a gap in the hound line and sprint through it; probing moves that threaten multiple gaps simultaneously are strongest.\n- **Hare: use lateral movement to stretch the hound line** — moving to the side forces hounds to spread their formation wider, increasing the risk of a gap; diagonal escapes to the corner regions are often the last resort.\n- **Geometry is decisive** — on narrow boards (few columns) the hounds can close all gaps easily and win reliably; on wide open boards the hare has room to manoeuvre around the line and escape. Know which regime your specific board falls into."}, {"id": 175, "type": "game", "source": "catch-the-hare", "section": "Engines & current best play", "text": "Catch the Hare — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Several specific board variants have been solved by exhaustive search in the academic literature; the state space is small enough that a complete minimax solve is straightforward for any fixed board geometry."}, {"id": 176, "type": "game", "source": "catch-the-hare", "section": "Complexity", "text": "Catch the Hare — Complexity\n\nSmall."}, {"id": 177, "type": "game", "source": "catch-the-hare", "section": "References", "text": "Catch the Hare — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hare_games) ([archive](http://web.archive.org/web/20260405215833/https://en.wikipedia.org/wiki/Hare_games))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 178, "type": "game", "source": "catch-the-hare", "section": "See also", "text": "Catch the Hare — See also\n\n- [Fox and Geese](fox-and-geese.md) · [Halatafl](halatafl.md) · [Tablut](tablut.md)\n- Lexicon: [hunt game](../lexicon/README.md#hunt-game)"}, {"id": 179, "type": "game", "source": "catchup", "section": "overview", "text": "Catchup\nHex-grid placement game where leading lets the trailing player play more — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement game with catch-up mechanic."}, {"id": 180, "type": "game", "source": "catchup", "section": "Description", "text": "Catchup — Description\n\nCatchup (Nick Bentley, 2010) is a connection-style game on a hexagonal grid\nwith an unusual move structure: the **size of the largest opposing group**\ndetermines how many pieces you place on your next turn, dynamically\n\"catching up\" the trailing player."}, {"id": 181, "type": "game", "source": "catchup", "section": "Rules", "text": "Catchup — Rules\n\n1. Board: hexagonal grid (commonly side 5 or 6, with 61 cells).\n2. The first player places one stone; thereafter the rule is:\n   - At the start of your turn, find the size *G* of the **largest connected\n     group of either colour** currently on the board.\n   - You then place **as many** stones as the lesser of *G* and the number of\n     empty cells, distributed one per cell.\n3. After the board is full, the player with the largest connected group wins;\n   ties are broken by next-largest group, then third-largest, etc.\n4. Stones once placed are never moved or removed."}, {"id": 182, "type": "game", "source": "catchup", "section": "Solution status", "text": "Catchup — Solution status\n\nCatchup is **not solved**. The dynamic move sizing makes the game tree\nunusual but no formal value has been computed."}, {"id": 183, "type": "game", "source": "catchup", "section": "Consensus on optimal play", "text": "Catchup — Consensus on optimal play\n\n- **Avoid creating large groups prematurely** — the catch-up rule means that growing the board's largest group hands your opponent more stones on their next turn; building many small scattered groups may be better than one large connected one until the late game.\n- **Cluster your stones before the endgame** — the winner is the player with the largest connected group at the end; stones must eventually connect, but the timing of merging clusters is key — merge just when the opponent cannot mount an equal-sized response.\n- **Exploit the opponent's catch-up moves against them** — when the opponent gets to place many stones (because your group is large), they are forced to spread across the board; use those forced placements to your advantage by ensuring they create only fragmented groups.\n- **Control the centre of the hexagonal board** — as in most hexagonal placement games, central stones are reachable from more directions and can join clusters on multiple axes; peripheral stones are easier to cut off.\n- **Count group sizes before each move** — precisely knowing the current largest group size tells you how many stones you will place next turn and how many your opponent will place; planning several moves ahead with these counts avoids being surprised by a sudden opponent surge."}, {"id": 184, "type": "game", "source": "catchup", "section": "Engines & current best play", "text": "Catchup — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked publicly.\n- **Notes:** Catchup's variable-move-count mechanic makes it an interesting research game for MCTS and planning algorithms; no dedicated competitive engine has been published."}, {"id": 185, "type": "game", "source": "catchup", "section": "Complexity", "text": "Catchup — Complexity\n\nModerate."}, {"id": 186, "type": "game", "source": "catchup", "section": "References", "text": "Catchup — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Catchup_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 187, "type": "game", "source": "catchup", "section": "See also", "text": "Catchup — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Hive](hive.md) · [Volo](volo.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 188, "type": "game", "source": "cathedral", "section": "overview", "text": "Cathedral\nA medieval-themed area-control game with polyominoes — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan polyomino placement game."}, {"id": 189, "type": "game", "source": "cathedral", "section": "Description", "text": "Cathedral — Description\n\nCathedral (Robert P. Moore, 1962) is a polyomino-placement game on a 10×10\nboard: each player places a fixed set of differently-shaped building pieces;\nsurrounding an opposing piece on all sides removes it from the board. The\nplayer with the fewer remaining un-played pieces wins."}, {"id": 190, "type": "game", "source": "cathedral", "section": "Rules", "text": "Cathedral — Rules\n\n1. Board: 10×10 grid, initially empty except for the neutral **Cathedral** piece\n   (a fixed polyomino).\n2. Each player has a fixed inventory of buildings, each a different polyomino\n   shape and orientation.\n3. The starting player places the Cathedral, then players alternate placing one\n   of their buildings on the board. Pieces may rotate but not overlap.\n4. Once each player has placed at least one piece, **enclosing rule**: if a\n   single small area is entirely bounded by one player's pieces (and/or the\n   board edge), any opposing pieces fully inside that area are removed and\n   returned to their owner.\n5. A player who cannot make a legal placement passes. When both players pass\n   consecutively, the player with the fewer points of unplayed pieces wins."}, {"id": 191, "type": "game", "source": "cathedral", "section": "Solution status", "text": "Cathedral — Solution status\n\nCathedral is **not solved**. The polyomino-shape inventory and 10×10 board make\nthe branching factor enormous; engines exist but no formal solving result."}, {"id": 192, "type": "game", "source": "cathedral", "section": "Consensus on optimal play", "text": "Cathedral — Consensus on optimal play\n\n- **Place the Cathedral near the centre to contest territory from both sides** — the Cathedral is neutral and placed first; a central placement denies both players optimal anchor squares while a corner placement largely wastes it as a shared border piece.\n- **Use large pieces early, small pieces to fill gaps late** — large polyominoes require contiguous open space; placing them when the board is open gives more placement options. Small pieces can fill awkward spaces later.\n- **Try to form enclosed regions quickly** — a closed region owned by you removes any opponent pieces inside it; regions that close with 4–6 squares can capture significant opponent pieces and simultaneously deny that space.\n- **Deny your opponent enclosing opportunities** — avoid clustering your pieces in a concave arrangement that the opponent can cap with a single piece to form a closed region around your buildings.\n- **Count unplayed piece-points, not placed pieces** — the winning condition is *fewer points of unplayed pieces remaining*, so deliberately placing high-value pieces (large buildings) early reduces your score even if you cannot create a great enclosure from them."}, {"id": 193, "type": "game", "source": "cathedral", "section": "Engines & current best play", "text": "Cathedral — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked publicly.\n- **Notes:** Cathedral has a dedicated hobbyist following but no published competitive engine or formal game-theoretic analysis; the high branching factor from polyomino orientations makes full solving computationally expensive."}, {"id": 194, "type": "game", "source": "cathedral", "section": "Complexity", "text": "Cathedral — Complexity\n\nBranching factor is the limiting issue; the state graph itself is moderate."}, {"id": 195, "type": "game", "source": "cathedral", "section": "References", "text": "Cathedral — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Cathedral_(board_game)) ([archive](http://web.archive.org/web/20260422123604/https://en.wikipedia.org/wiki/Cathedral_(board_game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 196, "type": "game", "source": "cathedral", "section": "See also", "text": "Cathedral — See also\n\n- [Quoridor](quoridor.md) · [Quarto](quarto.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 197, "type": "game", "source": "checkers", "section": "overview", "text": "Checkers (English draughts)\nThe most complex game ever weakly solved — eighteen years of computation\nSolution status: Weakly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan board game."}, {"id": 198, "type": "game", "source": "checkers", "section": "Description", "text": "Checkers (English draughts) — Description\n\nPlayed on the 32 dark squares of an 8×8 board, 12 pieces per side. Men move and\ncapture diagonally forward; captures are **compulsory**; reaching the far rank\nmakes a king, which moves and captures diagonally in any direction."}, {"id": 199, "type": "game", "source": "checkers", "section": "Solution status", "text": "Checkers (English draughts) — Solution status\n\nCheckers is **weakly solved**. [Schaeffer et al. (2007)](../references.md#schaeffer2007),\npublishing in *Science*, proved that **with perfect play the game is a draw**.\nThe proof combined two halves that met in the middle:\n\n- a complete endgame [tablebase](../lexicon/README.md#endgame-tablebase) for all\n  positions with ≤10 pieces (~3.9 × 10^13 positions), built by\n  [retrograde analysis](../lexicon/README.md#retrograde-analysis); and\n- a forward [proof-number search](../lexicon/README.md#proof-number-search) from\n  the opening that drove every relevant line into the tablebase.\n\nThe computation ran (with interruptions) from 1989 to 2007. At ~5 × 10^20 legal\npositions, checkers was — and for many years remained — the most complex game\nto be solved. The Chinook program had already become the first machine to win a\nhuman World Championship match (1994); the 2007 result proved it could never,\neven in principle, be beaten."}, {"id": 200, "type": "game", "source": "checkers", "section": "Consensus on optimal play", "text": "Checkers (English draughts) — Consensus on optimal play\n\n- **Hold the centre** — central squares (especially the \"dog hole\" squares d4/e5) control more diagonal lines than edge squares; a piece in the centre limits the opponent's manoeuvre options significantly.\n- **Maintain piece count parity; compulsory captures can be traps** — because all captures are mandatory, setting up a multi-jump sequence where you sacrifice one piece to capture two is a standard tactic; always check whether your intended move exposes you to a forced recapture chain.\n- **Promote kings without losing the back row** — a king is far stronger than a man; race to promote while keeping enough back-rank men to prevent opponent promotions; letting the opponent king up freely loses quickly.\n- **King mobility dominates endgames** — in king vs. king endings, the player whose king can reach the centre diagonal faster usually wins; triangulation (wasting moves to put the opponent in zugzwang) is a key technique.\n- **Avoid the \"single-corner\" trap** — a common tactical motif is forcing the opponent's king into a corner where it can only oscillate between two squares while your pieces tighten the net; recognise this pattern both to execute and to escape it.\n- **Draw technique: 3-2 or 2-1 king endings** — with correct play by the defender, many king-heavy endings are draws by repeated position; knowing the exact drawing moves in these endings avoids needless losses."}, {"id": 201, "type": "game", "source": "checkers", "section": "Engines & current best play", "text": "Checkers (English draughts) — Engines & current best play\n\n- **Strongest known program(s):** Chinook ([research page](https://webdocs.cs.ualberta.ca/~chinook/) ([archive](http://web.archive.org/web/20260424021226/https://webdocs.cs.ualberta.ca/~chinook/))) — alpha-beta with a complete endgame tablebase; weakly solved the game.\n- **Strength:** Perfect play available via the proof (draw from initial position); endgame tablebase covers all positions ≤10 pieces.\n- **Where the proof / tablebase lives (if solved):** [Schaeffer et al. (2007)](../references.md#schaeffer2007); tablebase at the University of Alberta Chinook project page.\n- **Notes:** Chinook became World Checkers Champion in 1994 and proved the draw result in 2007 after 18 years of computation; no human or program can beat the tablebase in the solved endgame positions."}, {"id": 202, "type": "game", "source": "checkers", "section": "Complexity", "text": "Checkers (English draughts) — Complexity\n\nState-space ~5 × 10^20; game-tree ~10^31\n([Schaeffer et al., 2007](../references.md#schaeffer2007))."}, {"id": 203, "type": "game", "source": "checkers", "section": "References", "text": "Checkers (English draughts) — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/English_draughts) ([archive](http://web.archive.org/web/20260310161143/https://en.wikipedia.org/wiki/English_draughts))\n- [Schaeffer, J. et al. (2007). *Checkers Is Solved*.](../references.md#schaeffer2007)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 204, "type": "game", "source": "checkers", "section": "See also", "text": "Checkers (English draughts) — See also\n\n- [International draughts](international-draughts.md) · [Fanorona](fanorona.md) · [Lasca](lasca.md) · [Othello](othello.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [proof-number search](../lexicon/README.md#proof-number-search)"}, {"id": 205, "type": "game", "source": "cherries", "section": "overview", "text": "Cherries\nA small partisan game on coloured stones — illustrates how a tiny rule\nSolution status: Partially analysed (specific positions have known CGT values). Game-theoretic value: Position-dependent. Players: 2. Type: Partisan combinatorial game."}, {"id": 206, "type": "game", "source": "cherries", "section": "Description", "text": "Cherries — Description\n\nCherries is one of several small partisan games developed in the CGT teaching\ntradition: a row of \"cherries\" coloured for Left or Right, with simple removal\nrules. Its purpose is to give a hands-on example of partisan value computation\nwithout the combinatorial weight of full board games."}, {"id": 207, "type": "game", "source": "cherries", "section": "Rules", "text": "Cherries — Rules\n\n1. A row (or set) of cherries, each coloured blue (Left) or red (Right). Specific\n   variants attach cherries in pairs or singletons. **[verify]** the canonical\n   rule set, which differs slightly between sources.\n2. **Left** moves: remove a blue cherry (and possibly its attached partner,\n   depending on the variant).\n3. **Right** moves: remove a red cherry similarly.\n4. The player unable to move loses (normal play)."}, {"id": 208, "type": "game", "source": "cherries", "section": "Solution status", "text": "Cherries — Solution status\n\nPartial. Specific Cherry positions have published CGT values, and the game is\nused to illustrate **mean value** and **temperature** in introductory CGT\nmaterial. A comprehensive theory for arbitrary configurations exists only via\ndirect recursive evaluation; we treat the game as partial pending a canonical\nrule set. **[verify]**"}, {"id": 209, "type": "game", "source": "cherries", "section": "Consensus on optimal play", "text": "Cherries — Consensus on optimal play\n\n- **Compute the CGT value of each isolated component first** — Cherries typically decomposes into independent sub-games (individual cherry pairs or singletons); compute the game value of each component separately, then sum them to find the overall value.\n- **Play the hottest component first** — in CGT terms the \"temperature\" of a component measures how much it is worth to move there next; always play in the highest-temperature component to maximise your advantage per move.\n- **A position with value > 0 is a Left (first player) win; < 0 a Right win; = 0 a second-player win** — reading the computed CGT sum directly gives the winner under optimal play; no heuristic reasoning is needed once the values are known.\n- **Use mean-value estimates to guide play when exact temperatures are complex** — the mean value of a game tells you roughly what each player will score from a position; comparing means across components identifies where your moves have the biggest impact.\n- **Cherries is a teaching example, not a competitive game** — its primary purpose is to illustrate partisan CGT concepts (switches, infinitesimals, temperature); in practice you compute the value analytically rather than searching a game tree."}, {"id": 210, "type": "game", "source": "cherries", "section": "Engines & current best play", "text": "Cherries — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. CGT values for specific positions are computed analytically (by hand or symbolic CGT tools).\n- **Strength:** Exact solutions for any specific position are achievable via direct CGT analysis; no game-playing engine is needed or meaningful.\n- **Notes:** Cherries is primarily a pedagogical vehicle for Berlekamp-Conway-Guy combinatorial game theory; optimal play for any well-defined configuration follows directly from the CGT value computation."}, {"id": 211, "type": "game", "source": "cherries", "section": "Complexity", "text": "Cherries — Complexity\n\nSmall."}, {"id": 212, "type": "game", "source": "cherries", "section": "References", "text": "Cherries — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Cherries_(game))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 213, "type": "game", "source": "cherries", "section": "See also", "text": "Cherries — See also\n\n- [Toppling Dominoes](toppling-dominoes.md) · [Toads and Frogs](toads-and-frogs.md) · [Domineering](domineering.md)\n- Lexicon: [temperature / hot game](../lexicon/README.md#temperature--hot-game) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 214, "type": "game", "source": "chess", "section": "overview", "text": "Chess\nThe most-studied game in history — superhuman engines, complete 7-piece\nSolution status: Unsolved (endgames strongly solved for ≤7 pieces). Game-theoretic value: Unknown (widely conjectured to be a draw). Players: 2. Type: Partisan board game."}, {"id": 215, "type": "game", "source": "chess", "section": "Description", "text": "Chess — Description\n\nThe standard 8×8 game with six piece types per side, castling, en passant, and\npromotion. Checkmate wins; stalemate and several rules (threefold repetition,\nfifty-move rule, insufficient material) produce draws."}, {"id": 216, "type": "game", "source": "chess", "section": "Solution status", "text": "Chess — Solution status\n\nChess is **unsolved**, and is expected to remain so indefinitely. The\ngame-tree complexity — Claude Shannon's classic estimate of about 10^120\n([Shannon, 1950](../references.md#shannon1950)) — is astronomically beyond any\nconceivable exhaustive search, and the legal-position count is on the order of\n10^44.\n\nWhat *is* solved is the **endgame**: complete\n[tablebases](../lexicon/README.md#endgame-tablebase) give the exact\ngame-theoretic value (and distance to mate) for **every position with 7 or\nfewer pieces** ([Lomonosov, 2012](../references.md#lomonosov2012)), built by\n[retrograde analysis](../lexicon/README.md#retrograde-analysis) in the tradition\nof [Thompson (1986)](../references.md#thompson1986). An 8-piece project is under\nway. These are genuine strong solutions — of sub-games, not of chess.\n\nModern engines (and self-play systems such as\n[AlphaZero](../references.md#silver-alphazero2018)) play far above the best\nhumans, but this is [strong play, not solving](../lexicon/README.md#solving-vs-strong-play):\nit proves nothing about the game-theoretic value of the initial position."}, {"id": 217, "type": "game", "source": "chess", "section": "Consensus on optimal play", "text": "Chess — Consensus on optimal play\n\nThe overwhelming expert and engine consensus is that chess is a **draw** with\nbest play — but this is a belief supported by evidence, not a proof."}, {"id": 218, "type": "game", "source": "chess", "section": "Engines & current best play", "text": "Chess — Engines & current best play\n\n- **Strongest known programs:** [Stockfish](https://stockfishchess.org/) ([archive](http://web.archive.org/web/20260512103017/https://stockfishchess.org/)) (alpha-beta + NNUE evaluation, open source) and [Leela Chess Zero](https://lczero.org/) ([archive](http://web.archive.org/web/20260501191219/https://lczero.org/)) (MCTS + deep residual policy/value network, AlphaZero-style, open source); historically [AlphaZero](../references.md#silver-alphazero2018) (DeepMind, closed) demonstrated the self-play approach.\n- **Strength:** Vastly super-human. Top engines rate ~3600 Elo on [CCRL](https://www.computerchess.org.uk/ccrl/) ([archive](http://web.archive.org/web/20251204164819/https://computerchess.org.uk/ccrl/)) lists vs. ~2830 for the strongest human; engine-vs-engine matches (TCEC, CCC) settle a large majority of games as draws under balanced openings.\n- **Endgame tablebases (genuine strong solutions of sub-games):** [Syzygy](https://syzygy-tables.info/) ([archive](http://web.archive.org/web/20260512112537/https://syzygy-tables.info/)) (≤7 pieces, distance-to-zeroing) and [Lomonosov 7-piece](../references.md#lomonosov2012) (distance-to-mate). An 8-piece project is under way.\n- **Notes:** No engine \"solves\" chess — they prove only that *they* draw or beat their opponents, not the game-theoretic value of the start position. See [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)."}, {"id": 219, "type": "game", "source": "chess", "section": "Complexity", "text": "Chess — Complexity\n\nState-space ~10^44; game-tree ~10^120\n([Shannon, 1950](../references.md#shannon1950);\n[van den Herik et al., 2002](../references.md#vandenherik2002)). Generalised\n(n×n) chess is EXPTIME-complete."}, {"id": 220, "type": "game", "source": "chess", "section": "References", "text": "Chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Chess) ([archive](http://web.archive.org/web/20260514055155/https://en.wikipedia.org/wiki/Chess))\n- [Shannon, C. E. (1950). *Programming a Computer for Playing Chess*.](../references.md#shannon1950)\n- [Thompson, K. (1986). *Retrograde Analysis of Certain Endgames*.](../references.md#thompson1986)\n- [Lomonosov 7-piece endgame tablebases (2012).](../references.md#lomonosov2012)\n- [Silver et al. (2018). *AlphaZero*.](../references.md#silver-alphazero2018)\n- [Zermelo, E. (1913).](../references.md#zermelo1913)"}, {"id": 221, "type": "game", "source": "chess", "section": "See also", "text": "Chess — See also\n\n- [Losing chess](losing-chess.md) · [Minichess](minichess.md) · [Shogi](shogi.md) · [Xiangqi](xiangqi.md) · [Shatranj](shatranj.md) · [Go](go.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play) · [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 222, "type": "game", "source": "chess960", "section": "overview", "text": "Chess960\nChess with randomised back-rank setup (960 starting positions) — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 223, "type": "game", "source": "chess960", "section": "Description", "text": "Chess960 — Description\n\nChess960 (Bobby Fischer, 1996) keeps every rule of chess intact but randomises\nthe back-rank pieces among the **960** legal starting positions (bishops on\nopposite colours, king between the rooks). The goal is to defeat memorised\nopening theory."}, {"id": 224, "type": "game", "source": "chess960", "section": "Rules", "text": "Chess960 — Rules\n\n1. Before play begins one of the 960 valid starting positions is chosen (the\n   pawns always occupy the 2nd/7th ranks, the king lies between the rooks, and\n   the two bishops sit on opposite colours).\n2. Both players start with mirrored back ranks. After setup, play proceeds\n   exactly as in orthodox chess.\n3. **Castling** is generalised: the king and chosen rook end on the standard\n   castled squares (c1/g1 or c8/g8 for king; d1/f1 or d8/f8 for rook),\n   irrespective of their starting files; all squares the king passes through\n   must be safe and unoccupied save for the castling rook.\n4. All other rules — pawn double-step, en passant, promotion, stalemate, 50-move\n   rule — are unchanged."}, {"id": 225, "type": "game", "source": "chess960", "section": "Solution status", "text": "Chess960 — Solution status\n\nChess960 is **not solved**. Each of 960 starting positions is its own opening\nproblem and engines play the game very strongly."}, {"id": 226, "type": "game", "source": "chess960", "section": "Consensus on optimal play", "text": "Chess960 — Consensus on optimal play\n\n- **Standard chess middlegame and endgame principles apply fully** — piece activity, king safety, pawn structure, and endgame technique carry over unchanged from orthodox chess; Chess960 only randomises the opening, not the underlying strategy.\n- **Develop toward the centre quickly regardless of starting position** — without memorised opening theory to lean on, moving central pawns and developing minor pieces to active squares is even more important; reactive, passive development is more easily punished.\n- **Understand castling rights before committing king or rook** — Chess960 castling rules can be counter-intuitive (e.g., a rook starting on g1 still castles to f1); confirm which pieces must stay in place before making commitments that forfeit castling.\n- **Use engine analysis per starting position** — Stockfish, Leela Chess Zero, and Fairy-Stockfish all handle Chess960 natively; analysing your specific starting position beforehand gives opening guidance that substitutes for memorised theory.\n- **Symmetric starts tend toward equality; asymmetric starts may offer sharper imbalances** — engine evaluation of the 960 positions shows most are roughly balanced but a few starting configurations give one side structurally superior piece placement; knowing which regime you are in calibrates risk tolerance."}, {"id": 227, "type": "game", "source": "chess960", "section": "Engines & current best play", "text": "Chess960 — Engines & current best play\n\n- **Strongest known program(s):** Stockfish ([https://stockfishchess.org/](https://stockfishchess.org/) ([archive](http://web.archive.org/web/20260512103017/https://stockfishchess.org/))) and Leela Chess Zero ([https://lczero.org/](https://lczero.org/) ([archive](http://web.archive.org/web/20260501191219/https://lczero.org/))) — both natively support Chess960; Lichess offers Chess960 play at [https://lichess.org/variant/chess960](https://lichess.org/variant/chess960) ([archive](http://web.archive.org/web/20260429091725/https://lichess.org/variant/chess960)).\n- **Strength:** Super-human (same engines as for orthodox chess).\n- **Where the proof / tablebase lives (if solved):** — (unsolved; no full tablebase; standard chess endgame tablebases apply for late-game positions)\n- **Notes:** Chess960's main purpose is to eliminate opening preparation advantages; in practice, professional events (Chess960 World Championship) use it and top players treat it as equivalent in depth to classical chess from move 10 onward."}, {"id": 228, "type": "game", "source": "chess960", "section": "Complexity", "text": "Chess960 — Complexity\n\nSimilar to chess."}, {"id": 229, "type": "game", "source": "chess960", "section": "References", "text": "Chess960 — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Chess960) ([archive](http://web.archive.org/web/20260501000739/https://en.wikipedia.org/wiki/Chess960))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 230, "type": "game", "source": "chess960", "section": "See also", "text": "Chess960 — See also\n\n- [Chess](chess.md) · [Capablanca chess](capablanca-chess.md) · [Courier chess](courier-chess.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 231, "type": "game", "source": "chomp", "section": "overview", "text": "Chomp\nA game where we *know* the first player wins — but, remarkably, no one knows\nSolution status: Ultra-weakly solved. Game-theoretic value: First-player win (all non-trivial boards). Players: 2. Type: Impartial combinatorial game."}, {"id": 232, "type": "game", "source": "chomp", "section": "Description", "text": "Chomp — Description\n\nPlayed on a rectangular grid of cells, thought of as a chocolate bar; the\ntop-left cell is \"poisoned.\" On a turn a player picks a cell and \"chomps\" it\ntogether with every cell below and to the right of it. The player forced to eat\nthe poisoned top-left cell loses."}, {"id": 233, "type": "game", "source": "chomp", "section": "Solution status", "text": "Chomp — Solution status\n\nChomp is **ultra-weakly solved**, and it is the textbook illustration of why\nthat category exists. A [strategy-stealing argument](../lexicon/README.md#strategy-stealing)\nshows the first player wins on any board larger than 1×1: if the second player\nhad a winning reply to the first player's \"chomp only the bottom-right cell,\"\nthe first player could have played that winning move themselves to begin with.\n\nThis proves a winning strategy **exists** — but the argument is\nnon-constructive. For general *m*×*n* boards **no explicit winning strategy is\nknown**, and finding one is a well-known open problem. (Explicit strategies are\nknown only for special cases such as square boards, 2×*n* boards, and 3×*n*\nboards.) Chomp is thus solved in value but not in strategy: the canonical\nultra-weak solution."}, {"id": 234, "type": "game", "source": "chomp", "section": "Consensus on optimal play", "text": "Chomp — Consensus on optimal play\n\n- **First player wins on any non-trivial rectangular board** — this is proven (via strategy stealing), so if you are the first player you are in a won position; the challenge is finding the actual winning move.\n- **On a 2×n board, the winning first move is to take from the bottom row leaving a 2×1 column** — explicit solutions exist for 2×n boards; memorise the winning pattern for small cases (it involves leaving the opponent with an L-shaped position they cannot handle).\n- **On square boards, first move: take the bottom-right corner only** — this leaves a symmetric non-square position; the winning strategy on square n×n boards is known: maintain a specific symmetry until the opponent is forced to eat the poison.\n- **On large general boards, no known efficient strategy exists** — the ultra-weak solution only proves a winner exists; finding the winning move in a given position requires game-tree search, and no polynomial algorithm is known for general *m*×*n*.\n- **As second player on a general board, your only hope is opponent error** — against a perfect opponent, second player loses; focus on positions (narrow boards, specific sizes) where explicit first-player strategies are known if you want to improve."}, {"id": 235, "type": "game", "source": "chomp", "section": "Engines & current best play", "text": "Chomp — Engines & current best play\n\n- **Strongest known program(s):** No dedicated public Chomp engine known to the cataloguer; custom minimax programs solve small boards.\n- **Strength:** Perfect play on small boards (2×n, 3×n, square boards with known strategies); no known efficient algorithm for general large boards.\n- **Where the proof / tablebase lives (if solved):** Ultra-weak solution: Gale (1974) strategy-stealing argument; explicit strategies for special cases in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Chomp is the canonical example of a game whose value is proven without a constructive strategy; finding a general polynomial winning strategy remains an open problem in combinatorial game theory."}, {"id": 236, "type": "game", "source": "chomp", "section": "Complexity", "text": "Chomp — Complexity\n\nThe number of positions on an *m*×*n* board equals the number of monotone\nstaircase shapes, which grows quickly; no polynomial strategy is known."}, {"id": 237, "type": "game", "source": "chomp", "section": "References", "text": "Chomp — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Chomp) ([archive](http://web.archive.org/web/20260509224037/https://en.wikipedia.org/wiki/Chomp))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- D. Gale (1974). *A curious Nim-type game*. American Mathematical Monthly 81:876–879. **[verify]**"}, {"id": 238, "type": "game", "source": "chomp", "section": "See also", "text": "Chomp — See also\n\n- [Hex](hex.md) (also ultra-weakly solved by strategy stealing) · [Nim](nim.md) · [Sprouts](sprouts.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing) · [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved)"}, {"id": 239, "type": "game", "source": "clobber", "section": "overview", "text": "Clobber\nA young partisan game (2001) of capturing adjacent enemy stones; rich in CGT\nSolution status: Partially solved (small boards; CGT theory of components). Game-theoretic value: Known for small boards and many component shapes. Players: 2. Type: Partisan combinatorial game."}, {"id": 240, "type": "game", "source": "clobber", "section": "Description", "text": "Clobber — Description\n\nPlayed on a grid initially filled in a checkerboard pattern of black and white\nstones. On a turn a player moves one of **their own** stones onto an\n**orthogonally adjacent enemy** stone, removing (\"clobbering\") that enemy stone.\nA player unable to move loses ([normal play](../lexicon/README.md#normal-play-convention))."}, {"id": 241, "type": "game", "source": "clobber", "section": "Solution status", "text": "Clobber — Solution status\n\nClobber is **partially solved**. It was designed (2001) as a testbed for\ncombinatorial game theory, and positions split into independent components whose\nCGT values can be computed and summed. Values, including\n[infinitesimal](../lexicon/README.md#temperature--hot-game) and `*`-type values,\nare tabulated for many small components. Specific small boards have been solved\nby exhaustive search, and Clobber has been a regular event in computer-games\nolympiads. But the standard playing boards are not solved, and there is no\ngeneral theory giving every position's value."}, {"id": 242, "type": "game", "source": "clobber", "section": "Consensus on optimal play", "text": "Clobber — Consensus on optimal play\n\n- **Preserve mobility while reducing the opponent's** — each capture removes an enemy stone and moves one of yours; the endgame is a race to leave the opponent with no adjacent enemy to move onto; prioritise captures that give you future move options while stranding opponent clusters.\n- **Identify components early** — the board typically fragments into regions of alternating stones; each isolated component has a CGT value that can be computed independently; the game value is the sum of component values, so compute these before deciding where to play.\n- **Play in the hottest component first** — CGT temperature tells you how much it is worth to move in a given component; always play in the highest-temperature component to maximise your gain.\n- **Value zero-temperature components as \"free moves\" for the opponent** — a component with value 0 is a second-player win in isolation; leaving such a component undisturbed while playing elsewhere often lets you steer the sum toward a winning value.\n- **Avoid creating isolated singleton stones** — a stone with no adjacent enemy cannot be moved; creating such orphans prematurely reduces your move count and risks losing by immobility."}, {"id": 243, "type": "game", "source": "clobber", "section": "Engines & current best play", "text": "Clobber — Engines & current best play\n\n- **Strongest known program(s):** Various research bots built for Computer Olympiad play; no single widely-distributed public engine known to the cataloguer.\n- **Strength:** Strong on small boards using CGT evaluation; competitive on standard tournament boards.\n- **Where the proof / tablebase lives (if solved):** CGT component values tabulated in Albert, Grossman, Nowakowski, Wolfe (2005); see also *Winning Ways* [../references.md#bcg2001](../references.md#bcg2001).\n- **Notes:** Clobber was explicitly designed to test CGT methods on a natural board game; it remains a primary research vehicle for infinitesimal game analysis and computer-olympiad benchmarking."}, {"id": 244, "type": "game", "source": "clobber", "section": "Complexity", "text": "Clobber — Complexity\n\nGrows quickly with board size; CGT decomposition helps but does not tame the\nlargest boards."}, {"id": 245, "type": "game", "source": "clobber", "section": "References", "text": "Clobber — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Clobber) ([archive](http://web.archive.org/web/20251017003913/https://en.wikipedia.org/wiki/Clobber))\n- M. H. Albert, J. P. Grossman, R. J. Nowakowski, D. Wolfe (2005). *An introduction to Clobber*. INTEGERS / Games of No Chance. **[verify]**\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 246, "type": "game", "source": "clobber", "section": "See also", "text": "Clobber — See also\n\n- [Domineering](domineering.md) · [Toads and Frogs](toads-and-frogs.md) · [Amazons](amazons.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [combinatorial game theory](../lexicon/README.md#combinatorial-game-theory)"}, {"id": 247, "type": "game", "source": "col", "section": "overview", "text": "Col\nA map-colouring partisan game, fully solved by combinatorial game theory: its\nSolution status: Strongly solved (as a theory). Game-theoretic value: Every position has an exact CGT value (a number or number-plus-star). Players: 2. Type: Partisan combinatorial game."}, {"id": 248, "type": "game", "source": "col", "section": "Description", "text": "Col — Description\n\nPlayed on a map (a graph of regions). One player colours regions **blue**, the\nother **red**, with the rule that **adjacent regions may not share a colour**\n(as in proper map colouring). A player who cannot legally colour a region loses\n([normal play](../lexicon/README.md#normal-play-convention))."}, {"id": 249, "type": "game", "source": "col", "section": "Solution status", "text": "Col — Solution status\n\nCol is **strongly solved as a theory**. [Conway (1976)](../references.md#conway1976)\nand [*Winning Ways*](../references.md#bcg2001) showed that every Col position\nhas an exact combinatorial-game value, and — a notable structural fact — those\nvalues are always either a [number](../lexicon/README.md#surreal-number) or a\nnumber plus the infinitesimal `*` (star). A region already adjacent to both\ncolours is \"dead\"; a clever device of *tinting* with a region adjacent to a\ncolour-of-itself lets the whole position be evaluated by the standard CGT\ndisjunctive-sum calculus.\n\nSo although a particular large map still takes computation to evaluate, the\n*game* is solved: there is a complete, exact method to value any position and\nplay optimally."}, {"id": 250, "type": "game", "source": "col", "section": "Consensus on optimal play", "text": "Col — Consensus on optimal play\n\n- **Compute the CGT value of each component, then sum** — a Col position decomposes into independent sub-maps; each has an exact value (a number or number + *); the overall position value is the sum, which determines who wins and by how much.\n- **A position with value > 0 is a Left win, < 0 a Right win, = 0 is a second-player win** — reading the numeric CGT value directly gives the game result; no tree search is needed once values are computed.\n- **Mark \"dead\" regions immediately** — a region adjacent to both blue and red is unavailable to either player; identifying and discarding dead regions simplifies the position and avoids wasted computation.\n- **Prefer moves that maximise the remaining position's value (for Left) or minimise it (for Right)** — in CGT parlance, Left always wants to leave a position as positive as possible; at each step choose the move from the component that shifts the total sum furthest in your favour.\n- **Tinting technique eliminates constrained regions** — a region already adjacent to one colour can be replaced by a simpler representation (its \"tint\"); applying this before summing reduces the position to its canonical form."}, {"id": 251, "type": "game", "source": "col", "section": "Engines & current best play", "text": "Col — Engines & current best play\n\n- **Strongest known program(s):** No dedicated Col engine known to the cataloguer; symbolic CGT tools (e.g., CGSuite) can evaluate arbitrary Col positions analytically.\n- **Strength:** Perfect — any position is solvable exactly by the CGT value computation method; no game-tree search required.\n- **Where the proof / tablebase lives (if solved):** [Conway (1976)](../references.md#conway1976) and [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Col is solved as a *theory*: the evaluation method is polynomial for maps with bounded treewidth; for general maps it may be computationally expensive but the method is exact and complete."}, {"id": 252, "type": "game", "source": "col", "section": "Complexity", "text": "Col — Complexity\n\nDepends on the map; the CGT decomposition keeps it tractable for modest maps."}, {"id": 253, "type": "game", "source": "col", "section": "References", "text": "Col — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Col_(game)) ([archive](http://web.archive.org/web/20260503203320/https://en.wikipedia.org/wiki/Col_(game)))\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 254, "type": "game", "source": "col", "section": "See also", "text": "Col — See also\n\n- [Snort](snort.md) (its \"kissing\" companion game) · [Hackenbush](hackenbush.md) · [Domineering](domineering.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [combinatorial game theory](../lexicon/README.md#combinatorial-game-theory)"}, {"id": 255, "type": "game", "source": "conhex", "section": "overview", "text": "ConHex\nA hybrid of Hex and Othello — connection on a board of \"tiles\" that flip when\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 256, "type": "game", "source": "conhex", "section": "Description", "text": "ConHex — Description\n\nConHex (Michail Antonow, 2003) combines [Hex](hex.md)-style connection goals\nwith [Othello](othello.md)-style capture: each \"tile\" of the board is a small\npolygon containing several corner-points, and a player who has placed pieces in\na majority of a tile's corners **claims** the tile for themselves. Connection\nof *claimed tiles* between opposite sides wins."}, {"id": 257, "type": "game", "source": "conhex", "section": "Rules", "text": "ConHex — Rules\n\n1. The board is a non-uniform tiling: most cells are pentagons containing 5\n   corner-points; some are smaller cells with 3 or 4 corners.\n2. Players alternate placing one peg of their colour on any empty corner-point.\n3. When a tile has more pegs of one colour than the other in its corner-points,\n   that tile is **claimed** by that colour (and re-evaluated whenever a corner\n   is filled).\n4. The first player to form a connected chain of *claimed tiles* linking their\n   two opposite sides wins.\n5. Like Hex, draws are not possible."}, {"id": 258, "type": "game", "source": "conhex", "section": "Solution status", "text": "ConHex — Solution status\n\nConHex is **unsolved**. The tile-claiming rule introduces an aggregation step\nthat complicates retrograde analysis; the game has a small but active\ncommunity. No published solution exists."}, {"id": 259, "type": "game", "source": "conhex", "section": "Consensus on optimal play", "text": "ConHex — Consensus on optimal play\n\n- **Claim tiles, not corner-points** — placing a peg matters only insofar as it shifts a tile toward your colour majority; a corner-point shared by multiple tiles is especially valuable as it can influence several tile outcomes simultaneously.\n- **Contest high-valency corner-points first** — points that are shared by 2–3 tiles are \"multi-tile\" pegs; securing them forces the opponent to overinvest in defense of multiple tiles while you build your chain efficiently.\n- **Think in tile-connectivity chains, not peg lines** — the winning path is made of *claimed tiles*, not individual pegs; visualise which sequence of tiles you need to claim to connect your two sides and invest in those tiles' corner-points.\n- **Apply virtual connection reasoning from Hex at the tile level** — two groups of claimed tiles that cannot both be disconnected (they share two disjoint connecting tile-paths) are virtually connected; recognise these structures to play confidently without fully resolving each tile.\n- **The strategy-stealing argument applies** — an extra peg is never a liability, so first player has at least a theoretical draw; in practice first player appears to have an advantage, and the swap rule is appropriate for fair play."}, {"id": 260, "type": "game", "source": "conhex", "section": "Engines & current best play", "text": "ConHex — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** ConHex has a small dedicated community but no published competitive engine or formal game-theoretic analysis; heuristics from Hex analysis offer the closest applicable guidance."}, {"id": 261, "type": "game", "source": "conhex", "section": "Complexity", "text": "ConHex — Complexity\n\nLarger than plain Hex on a comparable board, due to the additional tile-state."}, {"id": 262, "type": "game", "source": "conhex", "section": "References", "text": "ConHex — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/ConHex)\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework)"}, {"id": 263, "type": "game", "source": "conhex", "section": "See also", "text": "ConHex — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Crossway](crossway.md) · [Havannah](havannah.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 264, "type": "game", "source": "connect-four", "section": "overview", "text": "Connect Four\nThe classic dropping-disc game — weakly solved in 1988 as a first-player win.\nSolution status: Weakly solved (standard 7×6); strongly solved in practice via full databases. Game-theoretic value: First-player win (with the correct first move in the centre column). Players: 2. Type: Partisan positional (k-in-a-row, with gravity) game."}, {"id": 265, "type": "game", "source": "connect-four", "section": "Description", "text": "Connect Four — Description\n\nPlayed on a vertical 7-column × 6-row grid. Players alternately drop a disc into\na column; it falls to the lowest empty cell. The winner is the first to get four\nof their discs in a line — horizontally, vertically, or diagonally."}, {"id": 266, "type": "game", "source": "connect-four", "section": "Solution status", "text": "Connect Four — Solution status\n\nConnect Four is **weakly solved**. [Allis (1988)](../references.md#allis1988)\nsolved the standard 7×6 board in his M.Sc. thesis using a knowledge-based\napproach with nine strategic rules plus search; **James Dow Allen** and\n**[John Tromp](../references.md#tromp-connectfour)** reached the same result\nindependently around the same time. The verdict: with perfect play the\n**first player wins**, and must do so by playing the **centre column** first —\nany other opening move at least throws away the win (the centre is the unique\nwinning first move).\n\nModern solvers carry full databases and effectively play perfectly from every\nposition, so Connect Four is now solved in the strongest practical sense; Tromp\nalso solved many non-standard board sizes."}, {"id": 267, "type": "game", "source": "connect-four", "section": "Consensus on optimal play", "text": "Connect Four — Consensus on optimal play\n\n- **Always open column 4 (the centre)** — the centre column is the unique winning first move; every other first-column choice either loses or draws with perfect opponent play; this is the most famous single-move result in solved game theory.\n- **Control the centre columns (3–5) throughout the game** — pieces in the centre participate in more potential four-in-a-row lines (horizontal, diagonal, vertical) than edge columns; central presence denies the opponent threats from multiple directions.\n- **Create odd-row threats** — in Connect Four, who wins depends partly on the parity of the threat row; first player benefits from threats on odd rows (1, 3, 5 from the bottom) because the disc sequence means first player tends to fill odd rows; second player should aim for even-row threats.\n- **Set up zugzwang with a \"double threat\"** — threatening four-in-a-row in two different places simultaneously forces the opponent to block only one; building positions where your opponent must fill a column on their turn that activates your second threat is the primary winning technique.\n- **Avoid filling columns under an opponent threat** — dropping into a column can hand the opponent a free four-in-a-row if they are waiting for a disc in that column on the next row up; count ahead which columns trigger threats before committing."}, {"id": 268, "type": "game", "source": "connect-four", "section": "Engines & current best play", "text": "Connect Four — Engines & current best play\n\n- **Strongest known program(s):** Various implementations of Tromp's and Allis's solver; Pascal Pons's open-source Connect-4 solver is a clean modern reference.\n- **Strength:** Perfect — full databases allow optimal play from any position.\n- **Where the proof / tablebase lives (if solved):** [Allis (1988)](../references.md#allis1988); Tromp's analysis at [../references.md#tromp-connectfour](../references.md#tromp-connectfour); online solvers widely available.\n- **Notes:** Connect Four was one of the first commercially popular games to be solved; its solution is widely cited as a milestone in game AI and is routinely used to teach alpha-beta pruning and game-tree search."}, {"id": 269, "type": "game", "source": "connect-four", "section": "Complexity", "text": "Connect Four — Complexity\n\n~4.5 × 10^12 legal positions; game-tree complexity ~10^21\n([van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 270, "type": "game", "source": "connect-four", "section": "References", "text": "Connect Four — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Connect_Four) ([archive](http://web.archive.org/web/20260503215600/https://en.wikipedia.org/wiki/Connect_Four))\n- [Allis, V. (1988). *A Knowledge-based Approach of Connect-Four*.](../references.md#allis1988)\n- [Allis, V. (1994). *Searching for Solutions in Games and Artificial Intelligence*.](../references.md#allis1994)\n- [Tromp, J. *John's Connect Four Playground*.](../references.md#tromp-connectfour)"}, {"id": 271, "type": "game", "source": "connect-four", "section": "See also", "text": "Connect Four — See also\n\n- [Score Four](score-four.md) · [Qubic](qubic.md) · [Gomoku](gomoku.md) · [Connect6](connect6.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 272, "type": "game", "source": "connect6", "section": "overview", "text": "Connect6\nA k-in-a-row game deliberately designed for fairness — two stones per turn —\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan positional (k-in-a-row) game."}, {"id": 273, "type": "game", "source": "connect6", "section": "Description", "text": "Connect6 — Description\n\nIntroduced by Professor I-Chen Wu in 2003. Played on a Go board (or larger).\nThe first player places **one** stone; thereafter **each player places two\nstones per turn**. The winner is the first to get **six** (or more) of their\nstones in a row, horizontally, vertically, or diagonally."}, {"id": 274, "type": "game", "source": "connect6", "section": "Solution status", "text": "Connect6 — Solution status\n\nConnect6 is **unsolved**. The two-stones-per-turn rule was chosen specifically\nto neutralise the overwhelming first-player advantage that plagues\n[Gomoku](gomoku.md): after the opening single stone, each side always has the\nsame total number of stones on the board, which makes the game empirically very\nbalanced — and that balance, plus the large board, makes it hard to solve.\nConnect6 has an active competitive and computer-games community, but no proof of\nthe game-theoretic value of the standard opening exists."}, {"id": 275, "type": "game", "source": "connect6", "section": "Consensus on optimal play", "text": "Connect6 — Consensus on optimal play\n\n- **Build \"live fours\" (four-in-a-row with both ends open) with two-stone turns** — placing two stones in one turn means you can simultaneously advance two different threats; a live four is nearly unblockable because the opponent cannot cover both ends in a single two-stone turn while also advancing their own attack.\n- **Create double live-four threats (\"double four\")** — having two independent live fours on the board at once is an immediate win, since the opponent's two stones cannot block both; the entire strategy of expert Connect6 play converges on creating this double-four situation.\n- **Respond to opponent threats first, then build** — with two stones per turn, you can usually block one serious threat and create a new one in the same move; falling behind in threat count is typically fatal.\n- **Spread your stones over multiple lines, not one** — concentrating all stones in a single row telegraphs your intention; mixing line directions (horizontal, vertical, two diagonals) makes it harder for the opponent to pre-emptively block.\n- **The opening single stone confers no lasting advantage** — unlike Gomoku, the one-stone start is quickly equalised; do not play as though you have a persistent first-move edge; play for balanced development."}, {"id": 276, "type": "game", "source": "connect6", "section": "Engines & current best play", "text": "Connect6 — Engines & current best play\n\n- **Strongest known program(s):** NCTU6-Lite and related programs from I-Chen Wu's group at NCTU; these use MCTS combined with pattern-matching threat-space search.\n- **Strength:** Super-human; top engines consistently defeat the best human players.\n- **Where the proof / tablebase lives (if solved):** — (unsolved; no tablebase)\n- **Notes:** Connect6 has been a Computer Olympiad event since 2006; the NCTU family of programs has dominated competition, and the game is widely regarded as one of the better-balanced unsolved k-in-a-row games."}, {"id": 277, "type": "game", "source": "connect6", "section": "Complexity", "text": "Connect6 — Complexity\n\nPlayed on boards as large as 19×19 or 59×59; state and game-tree complexity are\non the order of Go's or larger."}, {"id": 278, "type": "game", "source": "connect6", "section": "References", "text": "Connect6 — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Connect6)\n- I-C. Wu, D.-Y. Huang (2005). *A New Family of k-in-a-row Games*. Advances in Computer Games. **[verify]**\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 279, "type": "game", "source": "connect6", "section": "See also", "text": "Connect6 — See also\n\n- [Gomoku](gomoku.md) · [Renju](renju.md) · [Pente](pente.md) · [Connect Four](connect-four.md)\n- Lexicon: [first-player advantage](../lexicon/README.md#first-player-advantage) · [maker-breaker game](../lexicon/README.md#maker-breaker-game)"}, {"id": 280, "type": "game", "source": "conways-soldiers", "section": "overview", "text": "Conway's Soldiers\nA one-player peg puzzle with a beautiful impossibility proof: you can reach\nSolution status: Strongly solved. Game-theoretic value: N/A — proven: rows 1–4 reachable, row 5 unreachable. Players: 1 (solitaire puzzle). Type: One-player combinatorial puzzle."}, {"id": 281, "type": "game", "source": "conways-soldiers", "section": "Description", "text": "Conway's Soldiers — Description\n\nOn an infinite checkerboard, a horizontal line divides it into a \"lower\" half\n(filled with as many soldiers as you like) and an empty \"upper\" half. Soldiers\nmove only by peg-solitaire jumps — horizontally or vertically over an adjacent\nsoldier into an empty cell, removing the jumped soldier. The goal is to advance\na soldier as far up into the empty half as possible."}, {"id": 282, "type": "game", "source": "conways-soldiers", "section": "Solution status", "text": "Conway's Soldiers — Solution status\n\nConway's Soldiers is **strongly solved** by a famous theorem. Using a\n*potential-function* argument (assign each cell a weight that is a power of\n1/φ where φ is the golden ratio, chosen so a jump never increases total\npotential), [Conway (1976)](../references.md#conway1976) proved:\n\n- a soldier can be brought to **row 1, 2, 3, or 4** above the line with finitely\n  many soldiers, but\n- **row 5 can never be reached** with any finite starting army.\n\nThe impossibility is exact, not empirical. (Variants relax the result:\ndiagonal moves, or an infinite limiting process, change which rows are\nreachable — but the standard puzzle is completely settled.)"}, {"id": 283, "type": "game", "source": "conways-soldiers", "section": "Consensus on optimal play", "text": "Conway's Soldiers — Consensus on optimal play\n\n- **Row 4 is the maximum reachable row; attempting row 5 is provably futile** — no matter how many soldiers you start with or which sequence of jumps you make, row 5 above the line is unreachable; do not waste time searching for a clever configuration.\n- **Use the potential-function argument to check any candidate construction** — assign each cell weight (1/φ)^(row distance above line); any valid jump sequence cannot increase total potential; if your starting configuration's potential is less than (1/φ)^target, reaching that target is impossible regardless of strategy.\n- **For row 4: the minimum known construction requires exactly 20 soldiers in a specific triangular formation** — this is the established record; the efficient 20-soldier pattern is the \"optimal\" solution for row 4.\n- **For rows 1–3: constructions are much smaller and straightforward** — small symmetric formations bring a soldier up 1, 2, or 3 rows respectively; these are tractable and well-documented in the references.\n- **Diagonal-jump variants change the reachability threshold** — if diagonal jumps are allowed, different rows become reachable; the classical result applies only to the standard orthogonal-jump rule."}, {"id": 284, "type": "game", "source": "conways-soldiers", "section": "Engines & current best play", "text": "Conway's Soldiers — Engines & current best play\n\n- **Strongest known program(s):** No game-playing engine is relevant — this is a one-player solitaire puzzle with a proven impossibility result.\n- **Strength:** Perfect — the complete characterisation is a closed theorem, not a search result.\n- **Where the proof / tablebase lives (if solved):** [Conway (1976)](../references.md#conway1976); see also [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Conway's Soldiers is one of the most elegant impossibility results in recreational mathematics; the golden-ratio potential function is the key insight and is non-obvious despite the simple rules."}, {"id": 285, "type": "game", "source": "conways-soldiers", "section": "Complexity", "text": "Conway's Soldiers — Complexity\n\nNot a search problem — the answer is a theorem with a short proof."}, {"id": 286, "type": "game", "source": "conways-soldiers", "section": "References", "text": "Conway's Soldiers — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Conway%27s_Soldiers) ([archive](http://web.archive.org/web/20260418214622/https://en.wikipedia.org/wiki/Conway%27s_Soldiers))\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 287, "type": "game", "source": "conways-soldiers", "section": "See also", "text": "Conway's Soldiers — See also\n\n- [Peg solitaire](pegs-solitaire.md) · [Brussels Sprouts](brussels-sprouts.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 288, "type": "game", "source": "courier-chess", "section": "overview", "text": "Courier chess\nMedieval 12×8 chess variant — historically extinct, unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 289, "type": "game", "source": "courier-chess", "section": "Description", "text": "Courier chess — Description\n\nCourier chess (medieval Germany, ~1200 CE; documented in Selenus 1616) is a\n12×8 board variant that introduces extra pieces — most famously the\n**Courier** (a long-range bishop) — and is considered a precursor of modern\nchess. It vanished from play by the 19th century."}, {"id": 290, "type": "game", "source": "courier-chess", "section": "Rules", "text": "Courier chess — Rules\n\n1. Board: 12 files × 8 ranks. Each side has 24 pieces — the usual chess\n   pieces plus two Couriers, a Sage (man/king-mover that can be captured), a\n   Schleich (a 1-step piece) and additional pawns.\n2. The **Courier** moves as a modern bishop (long diagonal).\n3. The medieval pieces — Ferz, Alfil and Mann — retain their pre-modern moves\n   (one step diagonally, two-step diagonal jump and one step in any direction\n   respectively). **[verify]** exact starting placement and minor-piece rules.\n4. Win condition: checkmate the king. Pawn promotion rules follow the\n   medieval convention (promotion to Ferz on the 8th rank)."}, {"id": 291, "type": "game", "source": "courier-chess", "section": "Solution status", "text": "Courier chess — Solution status\n\nCourier chess is **not solved**. As an extinct game its engine analysis is\nrudimentary."}, {"id": 292, "type": "game", "source": "courier-chess", "section": "Consensus on optimal play", "text": "Courier chess — Consensus on optimal play\n\n- **Activate the Couriers early** — the Courier (long-range bishop) is the most powerful piece in the game; an open long diagonal lets it dominate the wide 12-file board in a way the medieval Ferz or Alfil cannot match.\n- **Modern bishops from chess analogy apply to the Courier** — standard chess principles for bishops (open diagonals, outposts, bishop pairs) carry over directly to the Couriers; fighting for both open diagonal colours early is a strong positional goal.\n- **Be patient with the slow medieval pieces** — the Ferz (one-step diagonal), Alfil (two-step diagonal jump), and Mann (one-step any direction) have very limited range; use them for defensive duties and pawn support rather than long-range attack.\n- **The wide board rewards piece coordination over tactics** — on a 12×8 board, pieces are often too far apart for quick tactical combinations; manoeuvring to improve piece placement and controlling key diagonals/files is more reliable than seeking immediate tactical wins.\n- **Pawn promotion is to Ferz, not queen** — unlike modern chess, promoting a pawn yields a weak Ferz (one diagonal step) rather than a queen; pawn majorities are less decisive than in modern chess, so piece play dominates the endgame more than pawn races."}, {"id": 293, "type": "game", "source": "courier-chess", "section": "Engines & current best play", "text": "Courier chess — Engines & current best play\n\n- **Strongest known program(s):** Fairy-Stockfish ([https://github.com/ianfab/Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish))) — supports historical and fairy chess variants including medieval piece types.\n- **Strength:** Weak to moderate; Courier chess is rarely played and engine evaluation is not tuned for it.\n- **Where the proof / tablebase lives (if solved):** — (unsolved; historical interest only)\n- **Notes:** Courier chess is primarily of historical significance as a precursor to modern chess; active competitive play is essentially non-existent, making engine tuning and formal analysis very limited."}, {"id": 294, "type": "game", "source": "courier-chess", "section": "Complexity", "text": "Courier chess — Complexity\n\nLarger than chess."}, {"id": 295, "type": "game", "source": "courier-chess", "section": "References", "text": "Courier chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Courier_chess) ([archive](http://web.archive.org/web/20260106104507/https://en.wikipedia.org/wiki/Courier_chess))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 296, "type": "game", "source": "courier-chess", "section": "See also", "text": "Courier chess — See also\n\n- [Chess](chess.md) · [Capablanca chess](capablanca-chess.md) · [Shatranj](shatranj.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 297, "type": "game", "source": "cram", "section": "overview", "text": "Cram\nThe impartial version of Domineering: both players place dominoes in either\nSolution status: Partially solved (parity arguments + computed nim-values for many boards). Game-theoretic value: Depends on board; many boards settled by symmetry parity. Players: 2. Type: Impartial combinatorial game."}, {"id": 298, "type": "game", "source": "cram", "section": "Description", "text": "Cram — Description\n\nPlayed on a rectangular grid. **Both** players place 1×2 dominoes in **either**\norientation (horizontal or vertical). Since both have the same moves, Cram is\n[impartial](../lexicon/README.md#impartial-game). The player unable to move loses."}, {"id": 299, "type": "game", "source": "cram", "section": "Solution status", "text": "Cram — Solution status\n\nCram is **partially solved**. Two ingredients give a lot for free:\n\n- **Symmetry strategy.** On boards with both dimensions even, the second player\n  wins by central-point reflection; on boards with exactly one even dimension,\n  the first player wins by playing the centre domino and then reflecting. So\n  *all even-related boards are settled by a one-line parity argument.*\n- For the remaining (odd×odd) boards, [nim-values](../lexicon/README.md#nim-value)\n  have been computed by machine for many sizes, settling them individually.\n\nThere is no general closed form for all boards, so the game as a whole is not\nsolved — but a large fraction of board sizes are, and the easy ones fall to pure\nsymmetry."}, {"id": 300, "type": "game", "source": "cram", "section": "Consensus on optimal play", "text": "Cram — Consensus on optimal play\n\n- **On even×even boards, second player wins by central-reflection** — after any first-player domino placement, the second player places the rotationally symmetric (180°) domino; this strategy guarantees the second player always has a legal move as long as the first player does.\n- **On odd×even boards (one dimension odd, one even), first player wins by playing the centre domino first, then reflecting** — placing the domino on the central axis removes the symmetry centre and lets first player adopt the same reflection strategy for the remainder.\n- **For odd×odd boards, consult computed nim-value tables** — parity arguments do not directly settle these; the winner is determined by the nim-value of the position, which has been computed by machine for many specific sizes.\n- **Blocking open spaces is as important as placing efficiently** — a domino that creates two isolated single squares (unable to fit a 1×2 domino) is often a strong move because it reduces the opponent's future options significantly.\n- **In late-game tight positions, count remaining placements** — when only a few regions remain, count exactly how many dominoes can fit in each; the player who leaves the opponent with an even total of remaining moves wins."}, {"id": 301, "type": "game", "source": "cram", "section": "Engines & current best play", "text": "Cram — Engines & current best play\n\n- **Strongest known program(s):** No widely-distributed dedicated Cram engine known to the cataloguer; nim-value computations for specific boards appear in academic research.\n- **Strength:** Perfect on symmetry-settled boards (by the mirroring strategy); computed-optimal for boards whose nim-values have been tabulated.\n- **Where the proof / tablebase lives (if solved):** Symmetry strategy and partial results in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Cram nicely illustrates how symmetry arguments can solve a large class of positions without any search; the remaining odd×odd cases are an ongoing area of combinatorial game theory research."}, {"id": 302, "type": "game", "source": "cram", "section": "Complexity", "text": "Cram — Complexity\n\nEven-related boards: trivial. Odd×odd boards: nim-value computation grows quickly\nwith size."}, {"id": 303, "type": "game", "source": "cram", "section": "References", "text": "Cram — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Cram_(game))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 304, "type": "game", "source": "cram", "section": "See also", "text": "Cram — See also\n\n- [Domineering](domineering.md) (partisan sibling) · [Nim](nim.md)\n- Lexicon: [impartial game](../lexicon/README.md#impartial-game) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 305, "type": "game", "source": "crazyhouse", "section": "overview", "text": "Crazyhouse\nChess variant where captured pieces become reusable — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant with piece-drop."}, {"id": 306, "type": "game", "source": "crazyhouse", "section": "Description", "text": "Crazyhouse — Description\n\nCrazyhouse is a chess variant inspired by Bughouse: when you capture an enemy\npiece you keep it in reserve, and on a later turn you may **drop** it onto any\nempty square (with restrictions on pawns) instead of moving a piece on the\nboard. The added reserve state explodes the position space, and the game is\nfar from solved."}, {"id": 307, "type": "game", "source": "crazyhouse", "section": "Rules", "text": "Crazyhouse — Rules\n\n1. Same setup, movement, and check/checkmate rules as orthodox chess.\n2. When you capture an opposing piece, it is placed in your **reserve** as one\n   of your colour (promoted pawns revert to pawns when captured).\n3. On any turn instead of moving a piece, you may **drop** a reserve piece on\n   any empty square subject to: pawns may not be dropped on the 1st or 8th\n   rank; a dropped pawn cannot promote on the drop move.\n4. A drop that delivers checkmate is legal (no \"no drop-mate\" rule).\n5. Win by checkmate; draws by stalemate, threefold, and the 50-move rule apply."}, {"id": 308, "type": "game", "source": "crazyhouse", "section": "Solution status", "text": "Crazyhouse — Solution status\n\nCrazyhouse is **not solved**. The reserve adds combinatorial state on top of\nchess, and engine strength has only recently caught up with top humans."}, {"id": 309, "type": "game", "source": "crazyhouse", "section": "Consensus on optimal play", "text": "Crazyhouse — Consensus on optimal play\n\nHeuristics from strong human and engine play:\n\n- **Pieces in hand are worth more than pieces on the board** — top players will sacrifice material on the board to gain a piece in reserve that can be dropped with tempo.\n- **Knights are king** — knights are disproportionately valuable because drop-checks with knights are uncontested by interposition and can fork the king and queen.\n- **Pawn breaks beat pawn structure** — opening up files to drop pieces matters more than long-term structural weaknesses.\n- **The king must run** — castling is common but the king often walks to safety along the back rank because dropped pieces can fork castled kings easily.\n- **First-move advantage is large** — practical statistics and engine self-play show a clear White edge, though no proof exists."}, {"id": 310, "type": "game", "source": "crazyhouse", "section": "Engines & current best play", "text": "Crazyhouse — Engines & current best play\n\n- **Strongest known programs:** [Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish)) (open source); the now-retired [JannLee](https://lichess.org/@/JannLee) ([archive](http://web.archive.org/web/20251104133530/https://lichess.org/@/JannLee)) (custom search + handcrafted eval) was a long-dominant Lichess bot. [Sjeng-Crazyhouse](https://www.sjeng.org/indexold.html) ([archive](http://web.archive.org/web/20260315205258/https://www.sjeng.org/indexold.html)) was an early classic.\n- **Strength:** Super-human; engines surpassed top humans around 2017.\n- **Notes:** No tablebases — the unbounded reserve state precludes them."}, {"id": 311, "type": "game", "source": "crazyhouse", "section": "Complexity", "text": "Crazyhouse — Complexity\n\nLarger than chess."}, {"id": 312, "type": "game", "source": "crazyhouse", "section": "References", "text": "Crazyhouse — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Crazyhouse) ([archive](http://web.archive.org/web/20260511100435/https://en.wikipedia.org/wiki/Crazyhouse))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 313, "type": "game", "source": "crazyhouse", "section": "See also", "text": "Crazyhouse — See also\n\n- [Chess](chess.md) · [Shogi](shogi.md) · [Atomic chess](atomic-chess.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 314, "type": "game", "source": "crossway", "section": "overview", "text": "Crossway\nA drawless connection game where a \"no checkerboard pattern\" rule rules out\nSolution status: Ultra-weakly solved **[verify]**. Game-theoretic value: First-player win. Players: 2. Type: Partisan connection game."}, {"id": 315, "type": "game", "source": "crossway", "section": "Description", "text": "Crossway — Description\n\nCrossway (Mark Steere, 2007) is a connection game on a square grid that uses a\nclever rule to **prevent draws**: a player may not create a 2×2 \"checkerboard\"\npattern. Combined with a Hex-style connection win condition, this guarantees\nexactly one player connects."}, {"id": 316, "type": "game", "source": "crossway", "section": "Rules", "text": "Crossway — Rules\n\n1. Square board (commonly 19×19, but any size works). Each player owns two\n   opposite sides.\n2. Players alternate placing one stone of their colour on any empty cell, with\n   one restriction: the move **must not create any 2×2 block of cells with a\n   checkerboard pattern** of the two colours.\n3. The first player to form a connected chain of their stones linking their two\n   sides wins. (Orthogonal *and* diagonal connection both count.)\n4. Draws are impossible."}, {"id": 317, "type": "game", "source": "crossway", "section": "Solution status", "text": "Crossway — Solution status\n\nUltra-weakly solved by strategy-stealing: drawless plus symmetric makes it a\n**first-player win**. **[verify]** the formal statement — Crossway's strategy-\nstealing argument is community-folklore rather than a formal paper. As with\n[Hex](hex.md) and [Y](y.md), the proof is non-constructive."}, {"id": 318, "type": "game", "source": "crossway", "section": "Consensus on optimal play", "text": "Crossway — Consensus on optimal play\n\n- **First player wins with optimal play (strategy stealing)** — the game is provably a first-player win; use the swap (pie) rule in competitive play to restore fairness.\n- **Treat the no-checkerboard restriction as a dual-use tool** — the restriction prevents you from creating a 2×2 checkerboard pattern, but it equally prevents your opponent; spots where your opponent is blocked from playing (because placing there would create a checkerboard) are free real estate — probe those areas.\n- **Diagonal connections count equally with orthogonal** — unlike many connection games, both diagonal and orthogonal adjacency form a path; this makes connection easier to achieve and means your threat detection must account for diagonal chains.\n- **Build broad chains rather than narrow lines** — a chain two or more cells wide is harder to cut than a single-cell corridor; investing extra stones to widen your connection makes it more robust against the checkerboard restriction interfering with a thin path.\n- **Virtual connection reasoning from Hex applies at the path level** — groups with two disjoint connecting paths to the goal are virtually connected; once established, these cannot both be cut simultaneously."}, {"id": 319, "type": "game", "source": "crossway", "section": "Engines & current best play", "text": "Crossway — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Crossway has a small community of connection-game enthusiasts but no dedicated published engine; its ultra-weak solution (first-player win) rests on a strategy-stealing argument that is community folklore rather than a peer-reviewed proof."}, {"id": 320, "type": "game", "source": "crossway", "section": "Complexity", "text": "Crossway — Complexity\n\nComparable to Hex on the same board size."}, {"id": 321, "type": "game", "source": "crossway", "section": "References", "text": "Crossway — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Crossway_(game))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework for drawless connection games)"}, {"id": 322, "type": "game", "source": "crossway", "section": "See also", "text": "Crossway — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Gonnect](gonnect.md) · [Havannah](havannah.md)\n- Lexicon: [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved) · [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 323, "type": "game", "source": "dao", "section": "overview", "text": "Dao\n4×4 line-and-corner movement game — solved by exhaustive search.\nSolution status: Strongly solved **[verify]**. Game-theoretic value: First-player win **[verify]**. Players: 2. Type: Partisan movement game."}, {"id": 324, "type": "game", "source": "dao", "section": "Description", "text": "Dao — Description\n\nDao (Ben van Buskirk, 1999) is a 4×4 abstract game in which each player has\n4 stones and tries to achieve any of three winning patterns. The board is\nsmall enough to be fully analysed."}, {"id": 325, "type": "game", "source": "dao", "section": "Rules", "text": "Dao — Rules\n\n1. Board: 4×4 grid; each player has 4 stones placed in opposite corners\n   (one player's at a1, b2, a3, b4 — the other in a mirrored arrangement;\n   **[verify]** the canonical starting layout).\n2. On a turn the player picks one of their stones and **slides** it in any\n   of the eight directions as far as it can go before hitting an edge or\n   another piece (cannot stop short).\n3. Stones never capture; the board content shifts but never decreases.\n4. A player wins immediately by achieving any of:\n   - All 4 stones in one row, column, or diagonal.\n   - All 4 stones occupying the four corners.\n   - All 4 stones occupying a 2×2 square.\n   - All 4 stones surrounding a single opposing stone (a 2×2 enclosure rule\n     — varies by variant)."}, {"id": 326, "type": "game", "source": "dao", "section": "Solution status", "text": "Dao — Solution status\n\nDao has been **strongly solved** by exhaustive analysis: results circulated\non the abstract-games mailing list around 2002 suggest a first-player win.\n**[verify]** the canonical attribution and published date."}, {"id": 327, "type": "game", "source": "dao", "section": "Consensus on optimal play", "text": "Dao — Consensus on optimal play\n\n- **Work toward multiple winning threats simultaneously** — Dao has three different winning configurations (line/diagonal, four corners, 2×2 square); threatening two different configurations at once forces the opponent to defend both, which is usually impossible on the tiny 4×4 board.\n- **Stones slide to the edge — plan the endpoint, not the path** — a stone in an unobstructed line always slides to the board edge; before moving, trace exactly where each stone will land and which patterns become threatened or blocked by that landing square.\n- **The 2×2 winning cluster is easiest to threaten covertly** — a 2×2 square can form in any of nine positions on the board; grouping your stones centrally gives the most potential 2×2 formations and makes your intent hardest to read.\n- **Opponent blocking is mutual** — your own stones block your opponent's slides and vice versa; placing a stone as a blocker that simultaneously threatens a pattern of yours is the most efficient use of a turn.\n- **The four-corners pattern is hardest to block** — the opponent must keep pieces off all four corners to prevent this; threatening corners forces the opponent to occupy them, which constrains their own pattern formation."}, {"id": 328, "type": "game", "source": "dao", "section": "Engines & current best play", "text": "Dao — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer; the exhaustive-search solution from ~2002 serves as the perfect-play oracle.\n- **Strength:** Perfect play achievable via the solved tables; not benchmarked against humans.\n- **Where the proof / tablebase lives (if solved):** Community analysis from the abstract-games mailing list, ~2002; see [../references.md#herik-dao2002](../references.md#herik-dao2002) [verify].\n- **Notes:** Dao's small state space makes it trivially solvable by exhaustive search; the result (first-player win) is widely reported but lacks a peer-reviewed published citation."}, {"id": 329, "type": "game", "source": "dao", "section": "Complexity", "text": "Dao — Complexity\n\nSmall."}, {"id": 330, "type": "game", "source": "dao", "section": "References", "text": "Dao — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Dao_(game)) ([archive](http://web.archive.org/web/20251125063343/https://en.wikipedia.org/wiki/Dao_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [van den Herik *Dao* analysis (2002).](../references.md#herik-dao2002) **[verify]**"}, {"id": 331, "type": "game", "source": "dao", "section": "See also", "text": "Dao — See also\n\n- [Quarto](quarto.md) · [Quixo](quixo.md) · [Volo](volo.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 332, "type": "game", "source": "dara", "section": "overview", "text": "Dara\nA West African three-in-a-row game with a much larger board than the morris\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement+movement game."}, {"id": 333, "type": "game", "source": "dara", "section": "Description", "text": "Dara — Description\n\nDara is a traditional Nigerian/Sahelian abstract strategy game played in sand\nor on a wooden board. Although superficially related to\n[Nine Men's Morris](nine-mens-morris.md) — placement followed by movement,\nwinning by making a \"line\" — the larger board (commonly 5×6 with 24 pieces\ntotal) and the **capture-on-three-in-a-row** rule give it a quite different\ncharacter."}, {"id": 334, "type": "game", "source": "dara", "section": "Rules", "text": "Dara — Rules\n\n1. Board: 5 × 6 grid (some variants 6×6) of intersections.\n2. Each player has **12 stones**.\n3. **Placement phase**: players alternate placing stones on empty\n   intersections. Three-in-a-row is **not** allowed during placement.\n4. **Movement phase**: players alternate sliding one of their stones one square\n   orthogonally to an empty adjacent intersection.\n5. A player who makes a row of exactly three stones in a straight line (along\n   the grid, not diagonal) **captures** one opponent stone of their choice.\n   Only horizontal and vertical lines count; rows of four or more do not score.\n6. The first player reduced to fewer than 3 stones (and so unable ever to make\n   a three-row) loses."}, {"id": 335, "type": "game", "source": "dara", "section": "Solution status", "text": "Dara — Solution status\n\nDara is **not solved**. The state space — a 5×6 board with up to 24 placed\nstones — is moderate but no formal solving result has been published. There are\npractical Dara-playing programs and a competitive human tradition, but no proven\ngame-theoretic value."}, {"id": 336, "type": "game", "source": "dara", "section": "Consensus on optimal play", "text": "Dara — Consensus on optimal play\n\n- **In the placement phase, build near-rows without completing them** — three-in-a-row is prohibited during placement; place stones that create two-in-a-row configurations that will become immediate capture threats the moment the movement phase begins, without triggering the placement ban.\n- **Capture pieces that support opponent three-row threats** — when you score a three-in-a-row and can remove an opponent stone, prioritise removing the stone that is most integral to their next potential three-in-a-row; this both denies them a capture and may reduce them below the three-stone losing threshold faster.\n- **Control the centre columns/rows** — central intersections participate in more potential three-in-a-row lines (horizontal and vertical) than edge intersections; a piece in the centre can contribute to multiple future rows simultaneously.\n- **Avoid rows of four or more** — only exactly three-in-a-row scores; a line of four does not, so extending a three into a four on your own initiative wastes the capture and blocks the scoring line. Know when to stop.\n- **Manage piece count carefully** — the game is won by reducing the opponent below 3 pieces; count captures and plan whether you are on a winning attrition track or need to slow down to preserve your own material."}, {"id": 337, "type": "game", "source": "dara", "section": "Engines & current best play", "text": "Dara — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Dara has a living competitive tradition in Nigeria and the Sahel but limited computational analysis; the state space is comparable to Nine Men's Morris (which has been strongly solved), suggesting a formal solution is feasible but has not been published."}, {"id": 338, "type": "game", "source": "dara", "section": "Complexity", "text": "Dara — Complexity\n\nModerate — comparable in size to Nine Men's Morris, plausibly within reach of\nmodern solvers but not yet attempted."}, {"id": 339, "type": "game", "source": "dara", "section": "References", "text": "Dara — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Dara_(game)) ([archive](http://web.archive.org/web/20260111160444/https://en.wikipedia.org/wiki/Dara_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 340, "type": "game", "source": "dara", "section": "See also", "text": "Dara — See also\n\n- [Nine Men's Morris](nine-mens-morris.md) · [Twelve Men's Morris](twelve-mens-morris.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [state-space complexity](../lexicon/README.md#state-space-complexity)"}, {"id": 341, "type": "game", "source": "dawsons-chess", "section": "overview", "text": "Dawson's chess\nA chess-derived impartial game that reduces to the octal game 0.137 and is\nSolution status: Strongly solved. Game-theoretic value: Depends on the row; determined by nim-values. Players: 2. Type: Impartial combinatorial game (octal game)."}, {"id": 342, "type": "game", "source": "dawsons-chess", "section": "Description", "text": "Dawson's chess — Description\n\nDawson's chess was posed by T. R. Dawson (1935) as a chess problem: pawns of\nopposite colours face off on a 3×*n* board, captures are *compulsory*, and —\nanalysed under the [misère](../lexicon/README.md#misère-play) convention as\nDawson intended — it asks who is forced to make the last capture. The position\nabstracts to a take-and-break game on a row, and under\n[normal play](../lexicon/README.md#normal-play-convention) it is the octal game\n**0.137** (remove a run of three, or related moves, splitting the row)."}, {"id": 343, "type": "game", "source": "dawsons-chess", "section": "Solution status", "text": "Dawson's chess — Solution status\n\nDawson's chess is **strongly solved**. [Guy & Smith (1956)](../references.md#guy-smith1956)\ncomputed its [nim-values](../lexicon/README.md#nim-value) and proved the\nsingle-row sequence is **eventually periodic with period 34** (with finitely\nmany exceptions). Multi-component positions then resolve by\n[nim-sum](../lexicon/README.md#nim-sum), so every position has a known value and\noptimal move.\n\nHistorically Dawson intended the *misère* version; the misère analysis is\nsubstantially more delicate, but the normal-play octal game is the clean,\nfully-solved object usually meant by \"Dawson's chess\" in CGT."}, {"id": 344, "type": "game", "source": "dawsons-chess", "section": "Consensus on optimal play", "text": "Dawson's chess — Consensus on optimal play\n\n- **Look up the nim-value from the period-34 table** — for normal play, the nim-value (Grundy value) of a single row of length *n* follows a period-34 pattern (after a short non-periodic prefix); the table is the complete strategy — no other reasoning is needed.\n- **Combine multi-component positions with nim-sum** — if the position has multiple independent rows/components, compute each component's nim-value separately, then XOR (nim-sum) them; a position with nim-sum 0 is a second-player win, any non-zero nim-sum is a first-player win.\n- **To win from a non-zero nim-sum position, move to make the nim-sum 0** — find the component whose nim-value, when replaced by a reachable nim-value, makes the total nim-sum zero; that is your optimal move.\n- **For misère play, apply misère-quotient theory** — the normal-play strategy almost always works for misère too, with the exception: when all components have nim-value 0 or 1, invert the normal-play winning condition (move to leave an *odd* number of 1s instead of an even number).\n- **Period-34 means the game is essentially mechanical** — after looking up the table, optimal play requires no insight beyond nim-sum arithmetic; the game has no room for creative play."}, {"id": 345, "type": "game", "source": "dawsons-chess", "section": "Engines & current best play", "text": "Dawson's chess — Engines & current best play\n\n- **Strongest known program(s):** Any program that implements the period-34 nim-value table and nim-sum calculation plays perfectly. No game-specific competitive engine needed.\n- **Strength:** Perfect — the nim-value table and nim-sum give the exact optimal move in constant time per component.\n- **Where the proof / tablebase lives (if solved):** [Guy & Smith (1956)](../references.md#guy-smith1956); see also [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Dawson's chess is a canonical example of an octal game solved by eventual periodicity of Grundy values; it illustrates how CGT reduces a combinatorial game to pure arithmetic."}, {"id": 346, "type": "game", "source": "dawsons-chess", "section": "Complexity", "text": "Dawson's chess — Complexity\n\nPeriod-34 nim-value table; per-position analysis is linear in the number of\ncomponents."}, {"id": 347, "type": "game", "source": "dawsons-chess", "section": "References", "text": "Dawson's chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Dawson%27s_chess) ([archive](http://web.archive.org/web/20251016032223/https://en.wikipedia.org/wiki/Dawson%27s_chess))\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 348, "type": "game", "source": "dawsons-chess", "section": "See also", "text": "Dawson's chess — See also\n\n- [Kayles](kayles.md) · [Treblecross](treblecross.md) · [Mock Turtles](mock-turtles.md) · [Grundy's game](grundys-game.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [misère play](../lexicon/README.md#misère-play)"}, {"id": 349, "type": "game", "source": "dobutsu-shogi", "section": "overview", "text": "Dōbutsu shōgi\n\"Animal chess\" — a 3×4 shogi for children that has been completely solved.\nSolution status: Weakly solved (in practice strongly solved — full search). Game-theoretic value: Second-player win. Players: 2. Type: Partisan board game (shogi variant)."}, {"id": 350, "type": "game", "source": "dobutsu-shogi", "section": "Description", "text": "Dōbutsu shōgi — Description\n\nPlayed on a 3-column × 4-row board. Each side has four pieces — Lion (the king),\nGiraffe, Elephant, and Chick — with simple movement. As in [shogi](shogi.md),\ncaptured pieces are kept in hand and may be **dropped** back into play; the\nChick promotes to a Hen. You win by capturing the enemy Lion or by marching your\nown Lion safely to the far rank."}, {"id": 351, "type": "game", "source": "dobutsu-shogi", "section": "Solution status", "text": "Dōbutsu shōgi — Solution status\n\nDōbutsu shōgi is **completely solved**. The reachable state space is only about\n1.6 billion positions — small enough to enumerate exhaustively — and a full\nsolution was computed in 2009. The result: **with perfect play the second\nplayer (Gote) wins**; equivalently, the game is a loss for the first player. The\nsolution is, in effect, a strong one: a value is known for every reachable\nposition.\n\nIt is the standard example of a genuine shogi-family game (drops and all) being\nsolved, precisely because shrinking the board to 3×4 collapses the otherwise\nexplosive complexity of [shogi](shogi.md).\n\n> **[verify]** — The 2009 computational solution and the second-player-win\n> verdict are widely reported; this archive should pin the primary publication\n> (and confirm the exact reachable-position count) in\n> [references.md](../references.md)."}, {"id": 352, "type": "game", "source": "dobutsu-shogi", "section": "Consensus on optimal play", "text": "Dōbutsu shōgi — Consensus on optimal play\n\n- **Second player (Gote) wins with perfect play** — first player is in a losing position from move 1; as Sente, your only hope is opponent error; as Gote, follow the solved database and you cannot lose.\n- **Lion advancement is the decisive threat** — the game ends when a Lion reaches the far rank safely OR when a Lion is captured; controlling whether YOUR Lion can advance safely to the goal rank while preventing the opponent's is the central strategic calculation on the tiny 3×4 board.\n- **Use drops to create immediate threats** — captured pieces can be dropped anywhere on your turn; a well-timed Giraffe or Elephant drop that attacks the enemy Lion immediately forces a defensive response and is often more powerful than advancing a piece already on the board.\n- **The Chick → Hen promotion doubles its value** — getting a Chick to promote on the far rank converts it from a single-step-forward piece to a multi-direction Hen; promoting while also threatening the Lion is a strong combined goal.\n- **On a 3-wide board, flanking is impossible** — the Lion has nowhere to hide; every piece threatens the central file; defensive play often means keeping your Lion near the back rank while advancing supported threats."}, {"id": 353, "type": "game", "source": "dobutsu-shogi", "section": "Engines & current best play", "text": "Dōbutsu shōgi — Engines & current best play\n\n- **Strongest known program(s):** 2009 computational solution — effectively a complete minimax database for all ~1.6 billion positions; any program querying this database plays perfectly.\n- **Strength:** Perfect — the complete database gives the exact result and optimal move for every reachable position.\n- **Where the proof / tablebase lives (if solved):** 2009 exhaustive search (primary publication to be confirmed, see [verify] note above); widely reproduced in the abstract-game community.\n- **Notes:** Dōbutsu shōgi is the canonical example of a shogi-family game made tractable by board reduction; it demonstrates that shogi's drop mechanic does not inherently prevent solving — the board size is the bottleneck."}, {"id": 354, "type": "game", "source": "dobutsu-shogi", "section": "Complexity", "text": "Dōbutsu shōgi — Complexity\n\n~1.6 × 10^9 reachable positions — fully enumerated."}, {"id": 355, "type": "game", "source": "dobutsu-shogi", "section": "References", "text": "Dōbutsu shōgi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/D%C5%8Dbutsu_sh%C5%8Dgi)\n- 2009 computational solution of Dōbutsu shōgi (primary publication to be confirmed). **[verify]**\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 356, "type": "game", "source": "dobutsu-shogi", "section": "See also", "text": "Dōbutsu shōgi — See also\n\n- [Shogi](shogi.md) · [Chess](chess.md) · [Minichess](minichess.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 357, "type": "game", "source": "domineering", "section": "overview", "text": "Domineering\nA partisan tiling game — one player places vertical dominoes, the other\nSolution status: Partially solved (many rectangular boards weakly solved). Game-theoretic value: Depends on board; e.g. the 11×11 board is a first-player win. Players: 2. Type: Partisan combinatorial game."}, {"id": 358, "type": "game", "source": "domineering", "section": "Description", "text": "Domineering — Description\n\nPlayed on a rectangular grid. One player (\"Vertical\") places 1×2 dominoes\nvertically, the other (\"Horizontal\") places them horizontally; dominoes may not\noverlap. A player unable to place a domino loses\n([normal play](../lexicon/README.md#normal-play-convention))."}, {"id": 359, "type": "game", "source": "domineering", "section": "Solution status", "text": "Domineering — Solution status\n\nDomineering is **partially solved**: it is fully amenable to combinatorial game\ntheory (positions decompose into disjunctive sums with exact CGT values), and\nmany specific boards have been **weakly solved** by a combination of CGT and\nsearch. Known results include all small rectangles and a continuing series of\nsquare boards. [Uiterwijk (2016)](../references.md#uiterwijk-domineering2016)\nweakly solved the **11×11** board (a first-player win), extending earlier\nsolutions for 8×8, 9×9, 10×10 and many *m*×*n* rectangles.\n\nThe standard game has no single canonical board size, so \"Domineering\" as a\nwhole is not \"solved\" — but a large and growing table of board sizes is."}, {"id": 360, "type": "game", "source": "domineering", "section": "Consensus on optimal play", "text": "Domineering — Consensus on optimal play\n\n- **Vertical wants tall open corridors; Horizontal wants wide open corridors** — each player benefits from regions shaped for their orientation; early play should occupy and block regions that suit the opponent's orientation while exploiting those that suit yours.\n- **Decompose the board into independent regions and evaluate each** — as the game progresses, the board breaks into disconnected areas; each area has an exact CGT value; the game result is determined by the sum of these values, so evaluate each region separately.\n- **A region with CGT value 0 favours the second player; > 0 favours Vertical; < 0 favours Horizontal** — reading off the CGT sum tells you directly who wins and roughly by how much without playing out all moves.\n- **Play in \"hot\" regions first** — temperature measures how urgently you should move in a region; always play in the highest-temperature region to maximise your advantage per move.\n- **Square boards tend to favour first player by slight margins** — solved square boards (up through 11×11) generally give the first player the win; the advantage arises from asymmetric region formation that slightly favours the first player's orientation."}, {"id": 361, "type": "game", "source": "domineering", "section": "Engines & current best play", "text": "Domineering — Engines & current best play\n\n- **Strongest known program(s):** Custom research solvers (Breuker, Uiterwijk, van den Herik group) combined with CGT tools (e.g., CGSuite).\n- **Strength:** Perfect for solved board sizes; CGT-guided play is strong on unsolved larger boards.\n- **Where the proof / tablebase lives (if solved):** [Uiterwijk (2016)](../references.md#uiterwijk-domineering2016) for 11×11; earlier results in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Domineering is one of the most theoretically rich partisan games in CGT; it serves as a primary test case for hot-game theory, temperature, and thermographic analysis."}, {"id": 362, "type": "game", "source": "domineering", "section": "Complexity", "text": "Domineering — Complexity\n\nGrows quickly with board size; CGT decomposition is what makes boards up to\n~11×11 tractable."}, {"id": 363, "type": "game", "source": "domineering", "section": "References", "text": "Domineering — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Domineering) ([archive](http://web.archive.org/web/20251112162954/https://en.wikipedia.org/wiki/Domineering))\n- [Uiterwijk, J. W. H. M. (2016). *11×11 Domineering Is Solved: The First Player Wins*.](../references.md#uiterwijk-domineering2016)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)"}, {"id": 364, "type": "game", "source": "domineering", "section": "See also", "text": "Domineering — See also\n\n- [Cram](cram.md) (impartial sibling) · [Col](col.md) · [Snort](snort.md) · [Hackenbush](hackenbush.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [weakly solved](../lexicon/README.md#weakly-solved)"}, {"id": 365, "type": "game", "source": "dots-and-boxes", "section": "overview", "text": "Dots and Boxes\nA childhood classic with surprisingly deep theory — solved for small grids,\nSolution status: Partially solved (small boards solved exactly). Game-theoretic value: Known for small boards; e.g. analysed exhaustively up to roughly the 4×5-box grid. Players: 2. Type: Scoring game (last-move structure with capture)."}, {"id": 366, "type": "game", "source": "dots-and-boxes", "section": "Description", "text": "Dots and Boxes — Description\n\nOn a grid of dots, players alternately draw one unit edge between adjacent dots.\nCompleting the fourth side of a 1×1 box scores that box **and grants another\nmove**. When all boxes are claimed, the player with more boxes wins."}, {"id": 367, "type": "game", "source": "dots-and-boxes", "section": "Solution status", "text": "Dots and Boxes — Solution status\n\nDots and Boxes is **partially solved**. The standard competition boards (e.g.\n5×5 boxes) are **not** solved. However:\n\n- **Small boards are solved exactly.** Exhaustive computer analysis (notably\n  David Wilson's) has determined the optimal result for grids up to around the\n  4×5-box size, and selected larger cases.\n- **A deep partial theory exists.** [Berlekamp's](../references.md#berlekamp-dotsandboxes2000)\n  analysis recasts the endgame as a *Nimber* / loony-endgame theory built on the\n  **long-chain rule** and the strategic device of the *double-cross*. This does\n  not solve the game, but it gives strong, often provably optimal, endgame play\n  and explains why parity of long chains dominates expert play.\n\nSo Dots and Boxes sits between \"solved\" and \"unsolved\": rigorous for small\nboards and for endgames, open for full-size competition play."}, {"id": 368, "type": "game", "source": "dots-and-boxes", "section": "Consensus on optimal play", "text": "Dots and Boxes — Consensus on optimal play\n\n- **Avoid completing the third side of any box until forced** — drawing the third side \"opens\" a chain for the opponent to sweep; expert play delays entering chains as long as possible and aims to force the opponent to open them.\n- **Count long chains and control their parity** — a \"long chain\" is a sequence of 3+ boxes that, once opened, can all be captured in one turn; Berlekamp's long-chain rule: if the number of long chains is odd, the first player wins (in normal play); use this parity count to guide your moves.\n- **Use the double-cross sacrifice to control chain parity** — when forced to give up a chain, you can sacrifice two boxes by leaving a \"double-cross\" (cross instead of completing the end of the chain); this hands the opponent two boxes but lets you close the chain yourself and take the rest, while changing the chain-parity count in your favour.\n- **Never take a 3-chain without considering the sacrifice** — automatically sweeping a 3-box chain may give your opponent the winning parity; the sacrifice (give 2, take the rest) is often the correct play to maintain favourable chain parity.\n- **In the opening, create loops not chains** — closed loops are harder to exploit offensively than open chains; preferring loop-forming moves over chain-creating moves in the midgame gives more endgame flexibility."}, {"id": 369, "type": "game", "source": "dots-and-boxes", "section": "Engines & current best play", "text": "Dots and Boxes — Engines & current best play\n\n- **Strongest known program(s):** Various research programs implementing Berlekamp's chain-parity theory with alpha-beta search; David Wilson's solver for small boards.\n- **Strength:** Perfect on small boards (up to ~4×5); strong (but not provably optimal) on larger boards using chain-parity heuristics.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp (2000)](../references.md#berlekamp-dotsandboxes2000); small-board exact solutions by David Wilson (unpublished/circulated).\n- **Notes:** Dots and Boxes is deceptively deep; Berlekamp's chain-parity theory shows that expert endgame play is essentially nim-like, but the opening and midgame of large boards remain open research questions."}, {"id": 370, "type": "game", "source": "dots-and-boxes", "section": "Complexity", "text": "Dots and Boxes — Complexity\n\nState space grows roughly as 3^(number of edges); full-size boards are far\nbeyond exhaustive search."}, {"id": 371, "type": "game", "source": "dots-and-boxes", "section": "References", "text": "Dots and Boxes — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Dots_and_Boxes) ([archive](http://web.archive.org/web/20260503203320/https://en.wikipedia.org/wiki/Dots_and_boxes))\n- [Berlekamp, E. R. (2000). *The Dots and Boxes Game: Sophisticated Child's Play*.](../references.md#berlekamp-dotsandboxes2000)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 372, "type": "game", "source": "dots-and-boxes", "section": "See also", "text": "Dots and Boxes — See also\n\n- [Nimber theory via Nim](nim.md) · [Sprouts](sprouts.md)\n- Lexicon: [temperature / hot game](../lexicon/README.md#temperature--hot-game) · [partially solved](../lexicon/README.md#solved-game)"}, {"id": 373, "type": "game", "source": "dvonn", "section": "overview", "text": "DVONN\nA stacking game with three \"DVONN\" pieces that anchor the board — unsolved\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan stacking game."}, {"id": 374, "type": "game", "source": "dvonn", "section": "Description", "text": "DVONN — Description\n\nDVONN (Kris Burm, 2001) is the third GIPF-project game. Two players have\nsingle-coloured pieces (black and white); a small set of red **DVONN pieces**\nanchor groups of pieces. The board shrinks as orphaned stacks (groups not\nconnected via a DVONN piece) fall off."}, {"id": 375, "type": "game", "source": "dvonn", "section": "Rules", "text": "DVONN — Rules\n\n1. Board: 49-cell hexagonal grid.\n2. **Placement phase**: starting with the DVONN pieces (red), then alternating\n   their own colours, players place all pieces onto empty cells until the board\n   is filled.\n3. **Movement phase**: players alternate moving a stack they control on top of\n   (= same colour topping the stack). A stack moves exactly the number of\n   cells equal to its height, in any of six directions, and must land on\n   another stack (cannot move to an empty cell).\n4. After every move, any stack not connected (via stacks) to a DVONN piece\n   falls off the board.\n5. A player who cannot move passes; once both pass, the game ends and the\n   player with the **taller controlled total height** wins."}, {"id": 376, "type": "game", "source": "dvonn", "section": "Solution status", "text": "DVONN — Solution status\n\nDVONN is **not solved**. It is a common research target for Monte-Carlo and\nneural-network game programs, and engines play it strongly, but no formal\nsolution exists."}, {"id": 377, "type": "game", "source": "dvonn", "section": "Consensus on optimal play", "text": "DVONN — Consensus on optimal play\n\n- **Control DVONN piece proximity** — every stack on the board must stay connected to a DVONN piece; placing your stacks to maintain and threaten connection while forcing opponent stacks to become isolated is the dominant strategic theme.\n- **In the placement phase, surround DVONN pieces with your colour** — stacks near a DVONN piece are harder to isolate; placing your pieces in a loose ring around the DVONN pieces in the placement phase gives you reliable anchors for the movement phase.\n- **Build tall stacks before moving, not during** — a tall stack is harder to isolate (it moves a long distance and lands on many targets), but building height requires sacrificing short stacks by merging early; identify which merges build height efficiently without leaving isolated pieces.\n- **Cut opponent connection lines** — moving a stack between an opponent stack and the nearest DVONN piece severs that stack's lifeline; isolation threats force the opponent into defensive moves that waste their tempo.\n- **Prefer to have the last move in tight endgames** — when most stacks are large, the player who can make the last controlling move to absorb the remaining DVONN-connected stacks often wins; parity counting matters in the late game."}, {"id": 378, "type": "game", "source": "dvonn", "section": "Engines & current best play", "text": "DVONN — Engines & current best play\n\n- **Strongest known program(s):** Various MCTS and neural-network programs from Computer Olympiad entries; no single widely-distributed canonical engine known to the cataloguer.\n- **Strength:** Super-human; top engines consistently outperform top human players.\n- **Where the proof / tablebase lives (if solved):** — (unsolved)\n- **Notes:** DVONN has been a Computer Olympiad event; MCTS with domain-specific heuristics has been the dominant engine architecture, though neural-network approaches have also been applied successfully."}, {"id": 379, "type": "game", "source": "dvonn", "section": "Complexity", "text": "DVONN — Complexity\n\nLarge."}, {"id": 380, "type": "game", "source": "dvonn", "section": "References", "text": "DVONN — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/DVONN) ([archive](http://web.archive.org/web/20260130165830/https://en.wikipedia.org/wiki/DVONN))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 381, "type": "game", "source": "dvonn", "section": "See also", "text": "DVONN — See also\n\n- [GIPF](gipf.md) · [ZÈRTZ](zertz.md) · [YINSH](yinsh.md) · [LYNGK](lyngk.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 382, "type": "game", "source": "einstein-wurfelt-nicht", "section": "overview", "text": "EinStein würfelt nicht!\nA small modern dice-driven abstract — endgames and reduced boards are solved,\nSolution status: Partially solved (small boards / endgames; standard 5×5 not formally solved). Game-theoretic value: Unknown (a chance game — measured as expected win probability). Players: 2. Type: Stochastic abstract game."}, {"id": 383, "type": "game", "source": "einstein-wurfelt-nicht", "section": "Description", "text": "EinStein würfelt nicht! — Description\n\nDesigned by Ingo Althöfer (2004). On a 5×5 board each player has six numbered\ncubes (1–6) in one corner. Each turn a die is rolled; the player must move the\ncube of that number — or, if it has been captured, the nearest numbered cube\nabove or below. Moves go toward the opposite corner; landing on any cube (yours\nor the opponent's) removes it. A player wins by getting a cube to the opposite\ncorner or by capturing all the opponent's cubes. The die roll means the game has\n**chance but no hidden information**."}, {"id": 384, "type": "game", "source": "einstein-wurfelt-nicht", "section": "Solution status", "text": "EinStein würfelt nicht! — Solution status\n\nEinStein würfelt nicht! is **partially solved**. Because it has chance, \"solving\"\nmeans computing exact **expected win probabilities** rather than a win/draw/loss\nvalue. The board is small (5×5, six pieces a side), so **endgame positions and\nreduced material configurations have been solved exactly by retrograde-style\nexpectimax analysis**, and the game is a regular AI-competition test bed with\nstrong programs. But a published exact solution of the **full standard game**\nfrom the initial setup — accounting for the players' free initial cube placement\n— is not, to this archive's knowledge, established; treat the overall value as\nopen."}, {"id": 385, "type": "game", "source": "einstein-wurfelt-nicht", "section": "Consensus on optimal play", "text": "EinStein würfelt nicht! — Consensus on optimal play\n\n- **Place your low-numbered cubes (1, 2) centrally at setup** — the placement phase before the first roll is free; low numbers are rolled more frequently per valid move opportunity, so placing small-numbered cubes on the direct path to the goal corner maximises their expected contribution.\n- **Move diagonally toward the goal whenever the die allows** — diagonal moves (decreasing both row and column simultaneously) bring your cube closest to the opposite corner per step; prefer diagonal over straight moves when both are legal.\n- **Use cube 1 and 2 aggressively; they are the fastest attackers** — when rolled, low cubes that are well-placed can reach the goal in fewer turns on average; protecting them by not exposing them to capture is worth more than shielding large-numbered cubes.\n- **Capturing is often better than advancing** — removing an opponent cube denies them a potential fast mover and also forces their remaining cubes to cover that number's roll; a capture that also advances you toward the goal is nearly always correct.\n- **Stay off the edges when possible** — cubes on the board's edge have fewer valid diagonal moves; central or near-central positions maximise your options when a specific number is rolled."}, {"id": 386, "type": "game", "source": "einstein-wurfelt-nicht", "section": "Engines & current best play", "text": "EinStein würfelt nicht! — Engines & current best play\n\n- **Strongest known program(s):** Various Computer Olympiad bots implementing expectimax search with endgame evaluation; no single canonical public engine widely distributed.\n- **Strength:** Super-human for endgame positions with known exact values; strong for the full game via deep expectimax search.\n- **Where the proof / tablebase lives (if solved):** Endgame expectation tables computed by researchers for reduced material; no published full-game solution.\n- **Notes:** EWN is a regular Computer Olympiad event; the combination of dice chance and no hidden information makes it a useful benchmark for probabilistic game-tree search (expectimax / expectation-maximisation)."}, {"id": 387, "type": "game", "source": "einstein-wurfelt-nicht", "section": "Complexity", "text": "EinStein würfelt nicht! — Complexity\n\nModerate: the 5×5 board with six pieces per side gives a state space well within\nreach of exact analysis for endgames, and within reach of strong expectimax\nsearch for the whole game — the gap to a \"solution\" is one of formal publication\nand the placement phase rather than raw intractability."}, {"id": 388, "type": "game", "source": "einstein-wurfelt-nicht", "section": "References", "text": "EinStein würfelt nicht! — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/EinStein_w%C3%BCrfelt_nicht!) ([archive](http://web.archive.org/web/20251204160654/https://en.wikipedia.org/wiki/EinStein_w%C3%BCrfelt_nicht!))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 389, "type": "game", "source": "einstein-wurfelt-nicht", "section": "See also", "text": "EinStein würfelt nicht! — See also\n\n- [Backgammon](backgammon.md) · [Yahtzee](yahtzee.md)\n- Lexicon: [chance element](../lexicon/README.md#chance-element) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 390, "type": "game", "source": "eleven-mens-morris", "section": "overview", "text": "Eleven Men's Morris\nA larger morris variant — not formally solved.\nSolution status: Unsolved **[verify]**. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement+movement game."}, {"id": 391, "type": "game", "source": "eleven-mens-morris", "section": "Description", "text": "Eleven Men's Morris — Description\n\nA morris variant intermediate between [Nine Men's Morris](nine-mens-morris.md)\nand [Twelve Men's Morris](twelve-mens-morris.md): an extended board with 24\npoints (or 32, depending on the regional variant) and **11 stones per player**.\nIt is documented historically in several European traditions but is much less\nplayed and analysed than the 9- or 12-stone games. **[verify]** — the exact\nboard geometry varies by source."}, {"id": 392, "type": "game", "source": "eleven-mens-morris", "section": "Rules", "text": "Eleven Men's Morris — Rules\n\n1. Board: typically the standard 24-point morris board, possibly with extra\n   connections (diagonals) added to enable mills along extra lines. **[verify]**\n2. Each player has **11 stones**.\n3. **Placement phase**: players alternate placing stones on empty points,\n   removing an opponent's stone for each mill (three-in-a-row) formed.\n4. **Movement phase**: when all 22 stones are placed, players alternate sliding\n   stones to adjacent empty points; mills again remove opponent stones.\n5. A player reduced below 3 stones, or unable to move, loses."}, {"id": 393, "type": "game", "source": "eleven-mens-morris", "section": "Solution status", "text": "Eleven Men's Morris — Solution status\n\nTo this archive's knowledge, no peer-reviewed solving result exists for Eleven\nMen's Morris. The state space is larger than Nine Men's Morris, but well within\nmodern retrograde-analysis reach — the gap is a matter of effort and rule\ncanonicalisation rather than feasibility. Treat as **unsolved** and **[verify]**."}, {"id": 394, "type": "game", "source": "eleven-mens-morris", "section": "Consensus on optimal play", "text": "Eleven Men's Morris — Consensus on optimal play\n\n- **Form mills while denying opponent mills during placement** — the placement phase is decisive; placing a stone that creates your mill (removing an opponent stone) while blocking a near-mill of theirs simultaneously is the highest-value move type.\n- **Remove opponent stones that support multiple potential mills** — when you make a mill, target the opponent stone that participates in the most of their potential mill lines; this cripples their future attack options most efficiently.\n- **Create double-mill \"hammers\"** — a piece that can slide back and forth between two mill-forming positions creates a mill on every other turn; setting up this oscillating structure with 11 stones is the key winning technique.\n- **Maintain at least 4 stones to avoid the \"flying\" endgame** — in most morris variants, a player with 3 stones gains the right to jump anywhere; if you have exactly 3 stones you lose this deterrent; keep well above 3 while grinding the opponent down toward the 3-stone threshold.\n- **Nine Men's Morris strategy carries over** — principles proven for the solved 9-stone game (e.g., Gasser 1996) apply here: controlling corner and T-junction points, building crossed mill threats, and forcing zugzwang in the movement phase are all valid strategic goals."}, {"id": 395, "type": "game", "source": "eleven-mens-morris", "section": "Engines & current best play", "text": "Eleven Men's Morris — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. General morris engines (e.g., those for Nine Men's Morris) can approximate strong play with minor rule adaptation.\n- **Strength:** Not benchmarked.\n- **Notes:** Eleven Men's Morris has a small competitive community and no published peer-reviewed solving result; the state space is larger than Nine Men's Morris (~10^10 positions) but likely within reach of modern retrograde analysis if attempted."}, {"id": 396, "type": "game", "source": "eleven-mens-morris", "section": "Complexity", "text": "Eleven Men's Morris — Complexity\n\nLarger than Nine Men's Morris's ~10^10 positions, but presumably tractable."}, {"id": 397, "type": "game", "source": "eleven-mens-morris", "section": "References", "text": "Eleven Men's Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nine_men%27s_morris) ([archive](http://web.archive.org/web/20260328233608/https://en.wikipedia.org/wiki/Nine_Men%27s_Morris))\n- [Gasser (1996). *Solving Nine Men's Morris*.](../references.md#gasser1996) (general framework)"}, {"id": 398, "type": "game", "source": "eleven-mens-morris", "section": "See also", "text": "Eleven Men's Morris — See also\n\n- [Nine Men's Morris](nine-mens-morris.md) · [Twelve Men's Morris](twelve-mens-morris.md) · [Lasker Morris](lasker-morris.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 399, "type": "game", "source": "euclids-game", "section": "overview", "text": "Euclid's game\nA two-pile subtraction game whose winning positions are governed by the\nSolution status: Strongly solved. Game-theoretic value: First-player win iff (max/min) ≥ φ (golden ratio). Players: 2. Type: Impartial subtraction game."}, {"id": 400, "type": "game", "source": "euclids-game", "section": "Description", "text": "Euclid's game — Description\n\nA small impartial game with a number-theoretic flavour: it strips down the\nEuclidean algorithm into a two-player contest. Despite its tiny rules, the\nlosing-position structure is exactly the [Wythoff](wythoffs-game.md)-style\ngolden-ratio one."}, {"id": 401, "type": "game", "source": "euclids-game", "section": "Rules", "text": "Euclid's game — Rules\n\n1. The position is an ordered pair of positive integers (a, b).\n2. On your turn, you must subtract a positive multiple of the smaller from the\n   larger — i.e. replace (a, b) with (a, b − ka) for some k ≥ 1 with b − ka ≥ 0.\n3. The player who reduces one pile to zero (or who cannot move, equivalently)\n   *wins* under the standard convention (alt: the player who makes the last\n   legal move wins — same outcomes)."}, {"id": 402, "type": "game", "source": "euclids-game", "section": "Solution status", "text": "Euclid's game — Solution status\n\nStrongly solved. [Cole & Davie](../references.md#vandenherik2002) **[verify]**\nproved that the position (a, b) with a ≤ b is a **first-player win** iff\n`b/a ≥ φ`, where φ = (1+√5)/2 ≈ 1.618 — and in that case the winning move is\nthe unique one that *crosses* the golden-ratio boundary. From positions with\n`b/a < φ` only one move is legal up to flipping order, and it lands in a\nP-position."}, {"id": 403, "type": "game", "source": "euclids-game", "section": "Consensus on optimal play", "text": "Euclid's game — Consensus on optimal play\n\n- **Check whether b/a ≥ φ (≈ 1.618) — if yes, you win; if no, you are in a P-position** — this single inequality is the complete decision rule; from a winning (N-position) you can always move to a P-position; from a P-position any move leads to an N-position for your opponent.\n- **From an N-position, your winning move crosses the golden-ratio boundary** — find k such that b − ka satisfies a/(b − ka) ≥ φ (i.e., leave the ratio below φ); there is exactly one such valid k in any N-position (or occasionally two, if b is an exact multiple of a).\n- **When b/a < φ, you are in a P-position — only one move is legal** — there is only one integer multiple k available (k = floor(b/a) = 1), so you have no choice; you will hand your opponent an N-position. There is nothing to optimise here.\n- **The game terminates quickly** — because each move strips the larger pile by at least the smaller pile, the position shrinks like the Euclidean algorithm: at most O(log(max(a,b))) moves total. A game starting from (8, 13) or any Fibonacci pair is a P-position.\n- **Fibonacci pairs are the canonical P-positions** — (1,1), (1,2), (2,3), (3,5), (5,8)… (consecutive Fibonacci numbers) all satisfy b/a → φ from below and are second-player wins; recognise these to avoid playing into them."}, {"id": 404, "type": "game", "source": "euclids-game", "section": "Engines & current best play", "text": "Euclid's game — Engines & current best play\n\n- **Strongest known program(s):** Any program that checks b/a ≥ φ and computes the unique crossing move plays perfectly. No dedicated engine needed.\n- **Strength:** Perfect — O(1) per move.\n- **Where the proof / tablebase lives (if solved):** Cole & Davie (1969); also in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Euclid's game is a gem of elementary game theory: rules that mirror the Euclidean algorithm, a complete solution in one inequality, and P-positions at exactly the Fibonacci/golden-ratio pairs."}, {"id": 405, "type": "game", "source": "euclids-game", "section": "Complexity", "text": "Euclid's game — Complexity\n\nTrivial: each game lasts at most O(log min(a,b)) moves, and the decision rule\nis O(1)."}, {"id": 406, "type": "game", "source": "euclids-game", "section": "References", "text": "Euclid's game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Euclid_game)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 407, "type": "game", "source": "euclids-game", "section": "See also", "text": "Euclid's game — See also\n\n- [Wythoff's game](wythoffs-game.md) · [Nim](nim.md) · [Fibonacci Nim](fibonacci-nim.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 408, "type": "game", "source": "fanorona", "section": "overview", "text": "Fanorona\nMadagascar's national board game, with capture by approach and withdrawal —\nSolution status: Weakly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan board game."}, {"id": 409, "type": "game", "source": "fanorona", "section": "Description", "text": "Fanorona — Description\n\nThe standard board (*Fanorona-Tsivy*) is a 9×5 grid of points connected by\nlines. Pieces capture by **approach** (moving toward an adjacent enemy line) or\n**withdrawal** (moving away from one), removing the whole line of enemy pieces\nbeyond; capturing is compulsory when possible and capture-chains can continue.\nA player wins by removing all enemy pieces."}, {"id": 410, "type": "game", "source": "fanorona", "section": "Solution status", "text": "Fanorona — Solution status\n\nFanorona is **weakly solved**. [Schadd et al. (2008)](../references.md#schadd-fanorona2008)\nproved that **with perfect play the standard game is a draw**, using\n[proof-number search](../lexicon/README.md#proof-number-search) together with\nendgame [databases](../lexicon/README.md#endgame-tablebase) built by\n[retrograde analysis](../lexicon/README.md#retrograde-analysis) — the same\nmeet-in-the-middle methodology used for checkers and Nine Men's Morris. At\n~10^21 positions, Fanorona is of comparable scale to English\n[checkers](checkers.md)."}, {"id": 411, "type": "game", "source": "fanorona", "section": "Consensus on optimal play", "text": "Fanorona — Consensus on optimal play\n\n- **Prioritise capture chains over single captures** — each capture allows an additional move in the same turn (using a different direction and piece); a multi-capture sequence that removes 4–6 enemy pieces in one turn is often decisive; plan the chain before committing to the first capture.\n- **Distinguish approach from withdrawal before each capture** — approach captures the line of pieces in front of you (the ones you move toward), withdrawal captures the line behind you (the ones you move away from); choosing the correct direction often doubles or triples the number of pieces removed.\n- **Use the central points for maximum capture reach** — the Fanorona board has both orthogonal and diagonal lines; central points intersect more lines than edge points and give pieces more potential capture directions in a chain.\n- **The \"passing\" capture restriction prevents infinite loops** — a piece cannot revisit a position it has already occupied in the current capture chain; keep track of where you have been to avoid cutting off your own chain mid-sequence.\n- **With perfect play the game is a draw** — neither side should expect to win against a strong opponent; the correct defensive goal is to maintain sufficient piece density to answer all capture chains, not to race for a material advantage."}, {"id": 412, "type": "game", "source": "fanorona", "section": "Engines & current best play", "text": "Fanorona — Engines & current best play\n\n- **Strongest known program(s):** Schadd et al.'s 2008 proof-search engine combined with endgame tablebases; the program that proved the draw result.\n- **Strength:** Perfect from the proven weak solution; the endgame databases give exact play for all covered positions.\n- **Where the proof / tablebase lives (if solved):** [Schadd et al. (2008)](../references.md#schadd-fanorona2008); endgame databases computed at Maastricht University.\n- **Notes:** Fanorona's solution required combining proof-number search from the opening with retrograde-analysis endgame databases in a meet-in-the-middle approach, mirroring the technique used for checkers."}, {"id": 413, "type": "game", "source": "fanorona", "section": "Complexity", "text": "Fanorona — Complexity\n\nState-space ~10^21; game-tree ~10^46\n([Schadd et al., 2008](../references.md#schadd-fanorona2008))."}, {"id": 414, "type": "game", "source": "fanorona", "section": "References", "text": "Fanorona — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Fanorona) ([archive](http://web.archive.org/web/20260129053911/https://en.wikipedia.org/wiki/Fanorona))\n- [Schadd, M. P. D. et al. (2008). *Best Play in Fanorona Leads to Draw*.](../references.md#schadd-fanorona2008)"}, {"id": 415, "type": "game", "source": "fanorona", "section": "See also", "text": "Fanorona — See also\n\n- [Checkers (English draughts)](checkers.md) · [International draughts](international-draughts.md) · [Awari](awari.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [proof-number search](../lexicon/README.md#proof-number-search)"}, {"id": 416, "type": "game", "source": "fibonacci-nim", "section": "overview", "text": "Fibonacci Nim\nA one-heap take-away game with a moving limit, solved via the Zeckendorf\nSolution status: Strongly solved. Game-theoretic value: First-player win unless the heap size is a Fibonacci number. Players: 2. Type: Impartial combinatorial game."}, {"id": 417, "type": "game", "source": "fibonacci-nim", "section": "Description", "text": "Fibonacci Nim — Description\n\nA single heap of *n* objects. The first player may remove any positive number\nof objects but **not the entire heap**. Thereafter a player may remove at most\n*twice* the number their opponent just removed (and at least one). The player\ntaking the last object wins."}, {"id": 418, "type": "game", "source": "fibonacci-nim", "section": "Solution status", "text": "Fibonacci Nim — Solution status\n\nFibonacci Nim is **strongly solved** ([Whinihan, 1963](../references.md#guy-smith1956)\nrecords the result; the analysis is commonly credited to R. E. Gaskell). The\nsecond player wins if and only if *n* is a Fibonacci number. Otherwise the first\nplayer wins, and the winning strategy uses the **Zeckendorf representation** —\nthe unique way to write *n* as a sum of non-consecutive Fibonacci numbers: take\nthe *smallest* Fibonacci number in that representation, and continue to respond\nanalogously.\n\n> Note: the citation anchor above is a placeholder; the standard reference is\n> M. J. Whinihan, \"Fibonacci Nim,\" *Fibonacci Quarterly* 1(4):9–13, 1963.\n> **[verify]**"}, {"id": 419, "type": "game", "source": "fibonacci-nim", "section": "Consensus on optimal play", "text": "Fibonacci Nim — Consensus on optimal play\n\n- **If *n* is a Fibonacci number, you are in a P-position (second player wins)** — being first to move from any Fibonacci heap size is losing with optimal opponent play; your only strategy is to hope for opponent error.\n- **Otherwise, write *n* in its Zeckendorf representation and take the smallest Fibonacci summand** — the Zeckendorf representation of *n* is its unique sum of non-consecutive Fibonacci numbers; removing that smallest Fibonacci piece leaves the opponent in a P-position (a Fibonacci number remainder), and every response they make allows you to apply the same rule again.\n- **Never take so many that your move number doubles to the opponent's desired response** — the doubling-limit rule means your opponent can respond with up to twice your removal; after your Zeckendorf move, the remaining pile is a Fibonacci number, and any removal from a Fibonacci number leads to a non-Fibonacci position — which is an N-position for the next player. The strategy self-reinforces.\n- **On the first move, take only the smallest Fibonacci summand of *n*** — the restriction that you cannot take the entire heap on the first move is the only special constraint; the Zeckendorf rule already handles this because the smallest Fibonacci summand of any non-Fibonacci *n* is always less than *n*.\n- **Fibonacci pairs in Wythoff's game are the analogous safe positions** — the golden-ratio structure here (Fibonacci P-positions) mirrors Wythoff's game; players familiar with one can read across to the other."}, {"id": 420, "type": "game", "source": "fibonacci-nim", "section": "Engines & current best play", "text": "Fibonacci Nim — Engines & current best play\n\n- **Strongest known program(s):** Any program implementing Zeckendorf decomposition plays perfectly. No dedicated engine is needed.\n- **Strength:** Perfect — O(log n) computation via Zeckendorf representation.\n- **Where the proof / tablebase lives (if solved):** Whinihan (1963), *Fibonacci Quarterly* 1(4):9–13; also in [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Fibonacci Nim is the canonical example of a take-away game whose P-positions form a well-known mathematical sequence; its complete solution via Zeckendorf representations makes it a pedagogically valuable counterpart to standard Nim."}, {"id": 421, "type": "game", "source": "fibonacci-nim", "section": "Complexity", "text": "Fibonacci Nim — Complexity\n\nA family of positions; optimal moves computable directly from the Zeckendorf\nrepresentation of the heap size."}, {"id": 422, "type": "game", "source": "fibonacci-nim", "section": "References", "text": "Fibonacci Nim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Fibonacci_nim) ([archive](http://web.archive.org/web/20251219020355/https://en.wikipedia.org/wiki/Fibonacci_nim))\n- M. J. Whinihan (1963). *Fibonacci Nim*. Fibonacci Quarterly 1(4):9–13. **[verify]**\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 423, "type": "game", "source": "fibonacci-nim", "section": "See also", "text": "Fibonacci Nim — See also\n\n- [Nim](nim.md) · [Wythoff's game](wythoffs-game.md) · [Subtract-a-square](subtract-a-square.md)\n- Lexicon: [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 424, "type": "game", "source": "fifteen-puzzle", "section": "overview", "text": "15 puzzle\nThe classic sliding-tile puzzle — its solvability is fully characterised and\nSolution status: Strongly solved (solvability characterised; diameter = 80). Game-theoretic value: N/A (puzzle) — exactly half of all start states are solvable. Players: 1 (puzzle). Type: Single-player sliding-tile puzzle."}, {"id": 425, "type": "game", "source": "fifteen-puzzle", "section": "Description", "text": "15 puzzle — Description\n\nFifteen numbered tiles in a 4×4 frame with one empty space; a tile orthogonally\nadjacent to the gap can slide into it. The goal is to reach the ordered\nconfiguration. As a puzzle it has no game-theoretic value; the meaningful results\nare **which** scrambles are solvable and **how far** the hardest solvable one is\nfrom solved."}, {"id": 426, "type": "game", "source": "fifteen-puzzle", "section": "Solution status", "text": "15 puzzle — Solution status\n\nThe 15 puzzle is **strongly solved** as a puzzle:\n\n- **Solvability** was settled by Johnson & Story in 1879: a configuration is\n  solvable if and only if the permutation parity of the tiles matches the parity\n  of the blank's taxicab distance from its home square. Exactly **half** of the\n  16! arrangements are reachable — the famous \"14-15\" swap is one of the\n  unsolvable half.\n- **Diameter.** Brute-force breadth-first search over all ~10^13 solvable states\n  established that the hardest positions require **80** moves to solve optimally\n  — the 15-puzzle's \"God's Number\" — with 17 positions at that maximum distance.\n  **[verify]** the exact attribution and date of the diameter computation."}, {"id": 427, "type": "game", "source": "fifteen-puzzle", "section": "Consensus on optimal play", "text": "15 puzzle — Consensus on optimal play\n\n- **First, check solvability with the parity test** — count the number of inversions in the tile sequence, then add the row number of the blank (counting from the bottom); if that sum is even, the puzzle is solvable; if odd, it is unsolvable (famously, a 14-15 swap produces an unsolvable configuration).\n- **For humans: solve row by row, top to bottom, then column by column** — the standard human method (fill rows 1 and 2, then columns, then solve the last 2×4 or 2×3 block with known sequences) is far from optimal but tractable to learn and apply.\n- **For computers: use IDA* with a 6-6-3 or 5-5-5 pattern database heuristic** — iterative deepening A* with a precomputed lower-bound heuristic (summing taxicab distances of disjoint tile subsets) finds optimal solutions efficiently; this is the standard benchmark algorithm for the 15-puzzle and sliding-tile puzzles generally.\n- **The maximum optimal solution is 80 moves** — any solvable position can be solved in at most 80 single-tile slides; a solver that exceeds 80 moves is suboptimal.\n- **Memorise a few \"commutator\" sequences for 2×2 and 2×3 blocks** — the hardest part of the puzzle for humans is the final 3-tile block; several short sequences (typically 8–12 moves) cycle tiles without disturbing the rest of the board and can be combined to solve any configuration."}, {"id": 428, "type": "game", "source": "fifteen-puzzle", "section": "Engines & current best play", "text": "15 puzzle — Engines & current best play\n\n- **Strongest known program(s):** IDA* with pattern databases — the standard optimal solver. No single canonical named program, but the algorithm is widely implemented (e.g., in academic search-algorithm libraries).\n- **Strength:** Perfect — IDA* with the 6-6-3 pattern database finds an optimal (shortest) solution for any solvable 15-puzzle position.\n- **Where the proof / tablebase lives (if solved):** Solvability: Johnson & Story (1879); diameter (80 moves): computational result attributed to various groups, ~2011 [verify]; see also [Ratner & Warmuth (1990)](../references.md#ratner-warmuth1990).\n- **Notes:** The 15-puzzle is a canonical AI search benchmark; solving it optimally with IDA* and pattern databases is a standard textbook exercise in heuristic search. The n×n generalisation is NP-hard to solve optimally."}, {"id": 429, "type": "game", "source": "fifteen-puzzle", "section": "Complexity", "text": "15 puzzle — Complexity\n\n16!/2 ≈ 1.0 × 10^13 reachable states; configuration-graph diameter 80 in the\nsingle-move metric. The n×n generalisation is NP-hard to solve optimally."}, {"id": 430, "type": "game", "source": "fifteen-puzzle", "section": "References", "text": "15 puzzle — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/15_puzzle) ([archive](http://web.archive.org/web/20260403031423/https://en.wikipedia.org/wiki/15_puzzle))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)\n- [Ratner & Warmuth (1990). *The (n²−1)-puzzle and related relocation problems*.](../references.md#ratner-warmuth1990)"}, {"id": 431, "type": "game", "source": "fifteen-puzzle", "section": "See also", "text": "15 puzzle — See also\n\n- [Rubik's Cube](rubiks-cube.md) · [Peg solitaire](pegs-solitaire.md)\n- Lexicon: [God's number](../lexicon/README.md#gods-number) · [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 432, "type": "game", "source": "four-d-tic-tac-toe", "section": "overview", "text": "4-D tic-tac-toe\nTic-tac-toe on a 4-D 4×4×4×4 board — a known first-player win by\nSolution status: Partially solved (small dimensions); 4×4×4×4 is a first-player win. Game-theoretic value: First-player win (for sufficiently large dimensions / boards). Players: 2. Type: Partisan k-in-a-row / placement game."}, {"id": 433, "type": "game", "source": "four-d-tic-tac-toe", "section": "Description", "text": "4-D tic-tac-toe — Description\n\nThe natural higher-dimensional generalisation of tic-tac-toe: a 4×4×4×4\nhypercube of 256 cells. Players alternately place X or O; the first to align\nfour in a row along any of the many available \"lines\" (axis-, plane-, or\nhyperplane-aligned) wins. This is the canonical example of how raising the\ndimension dwarfs the winning sets and shifts the game decisively toward the\nfirst player."}, {"id": 434, "type": "game", "source": "four-d-tic-tac-toe", "section": "Rules", "text": "4-D tic-tac-toe — Rules\n\n1. Board: 4×4×4×4 hypercube of cells, with the 4^4 = 256 cells indexed by\n   (i, j, k, l) with each coordinate in {1, 2, 3, 4}.\n2. Players alternate placing X or O on any empty cell.\n3. The first player to **align four of their marks along any straight line** in\n   the hypercube wins. (Many more lines than in 3-D Qubic; total counts derive\n   from the geometry of the 4-D hypercube.)\n4. If the board fills with no four-in-a-line, the game is a draw (in this\n   particular setting, draws are not achievable under optimal play)."}, {"id": 435, "type": "game", "source": "four-d-tic-tac-toe", "section": "Solution status", "text": "4-D tic-tac-toe — Solution status\n\nThe **Hales–Jewett theorem (1963)** is a non-constructive existence result:\nfor any fixed line length k and number of players, a sufficiently\nhigh-dimensional cube tic-tac-toe cannot draw. For 4×4×4×4 specifically, the\nresult is the **standard \"n×n×n×n\" Hales-Jewett bound** plus a small computer\nanalysis: it is a **first-player win**, and explicit Maker strategies have been\nfound. **[verify]** the precise solver attribution."}, {"id": 436, "type": "game", "source": "four-d-tic-tac-toe", "section": "Consensus on optimal play", "text": "4-D tic-tac-toe — Consensus on optimal play\n\n- **The first player wins — any solid opening in the centre region is correct** — unlike standard tic-tac-toe, draws are not achievable in the 4×4×4×4 setting; the first player's task is to find any of the many winning threat sequences, not to avoid traps.\n- **Exploit the huge number of winning lines** — the 4-D hypercube contains many more collinear 4-tuples than 3-D Qubic; first-player advantage is overwhelming because simultaneous threats along multiple dimensions (axis lines, plane diagonals, space diagonals, hyperplane diagonals) are impossible to block all at once.\n- **Create threats along multiple dimensional axes simultaneously** — placing a piece at a cell that lies on 2-D, 3-D, and 4-D diagonals simultaneously creates more threats per stone than any edge or axis-only placement.\n- **Winning is achieved through a \"threat tree\"** — the practical first-player strategy involves building a tree of forcing threats (create-threat, force-block, create-another-threat) until the opponent cannot cover all branches simultaneously; this is how the computer analysis confirmed the win.\n- **The Hales–Jewett theorem guarantees no draw is possible** — for any position that fills completely with no winner, that would contradict the theorem; there is no need to play for a draw."}, {"id": 437, "type": "game", "source": "four-d-tic-tac-toe", "section": "Engines & current best play", "text": "4-D tic-tac-toe — Engines & current best play\n\n- **Strongest known program(s):** Computer threat-tree analysis has established the first-player win; no specific public interactive engine for 4-D tic-tac-toe is known to the cataloguer.\n- **Strength:** First-player win established computationally; no benchmarked interactive engine.\n- **Where the proof / tablebase lives (if solved):** Hales & Jewett (1963) for the general theorem; specific 4×4×4×4 analysis in game-AI literature [verify exact citation].\n- **Notes:** The 4×4×4×4 game is primarily of theoretical interest, illustrating how the Hales–Jewett theorem applies to concrete hypercube tic-tac-toe; it is rarely played competitively."}, {"id": 438, "type": "game", "source": "four-d-tic-tac-toe", "section": "Complexity", "text": "4-D tic-tac-toe — Complexity\n\nLarge — 256 cells means a state space of 3^256 in the trivial bound, well\nbeyond exhaustive search, though the first-player win has been demonstrated by\nthreat-tree analysis."}, {"id": 439, "type": "game", "source": "four-d-tic-tac-toe", "section": "References", "text": "4-D tic-tac-toe — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tic-tac-toe_variants)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Patashnik (1980). *Qubic: 4×4×4 Tic-Tac-Toe*.](../references.md#patashnik1980) (3-D analogue)"}, {"id": 440, "type": "game", "source": "four-d-tic-tac-toe", "section": "See also", "text": "4-D tic-tac-toe — See also\n\n- [Qubic](qubic.md) · [Tic-tac-toe](tic-tac-toe.md) · [m,n,k-games](mnk-games.md)\n- Lexicon: [pairing strategy](../lexicon/README.md#pairing-strategy) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 441, "type": "game", "source": "fox-and-geese", "section": "overview", "text": "Fox and Geese\nA classic asymmetric hunt game on a cross-shaped board — solved in the\nSolution status: Weakly solved. Game-theoretic value: Geese win with correct play. Players: 2 (asymmetric: one fox vs. several geese). Type: Partisan hunt game."}, {"id": 442, "type": "game", "source": "fox-and-geese", "section": "Description", "text": "Fox and Geese — Description\n\nPlayed on a cross-shaped (plus-shaped) subset of a checkers board. One player\ncontrols a single **fox**, the other controls a group of **geese** (commonly 13,\nthough counts vary by tradition). The fox moves like a checkers king and may, in\nsome rule sets, capture geese by jumping; the geese move forward/sideways only\nand never capture. The geese win by hemming the fox in so it cannot move; the\nfox wins by breaking through the goose formation (or reducing the geese below a\nthreshold where they can no longer trap it)."}, {"id": 443, "type": "game", "source": "fox-and-geese", "section": "Solution status", "text": "Fox and Geese — Solution status\n\nFox and Geese is **weakly solved**. The game and its optimal strategy are\nanalysed in [*Winning Ways*](../references.md#bcg2001): with correct play the\n**geese win** — they can advance in a solid phalanx that never offers the fox a\ngap, eventually walling it into a corner. The result is sensitive to the exact\nrule set and number of geese; the headline \"geese win\" applies to the standard\n13-geese, no-goose-capture version analysed in the CGT literature. Because the\nstate space is tiny, the game is also trivially solvable by exhaustive search."}, {"id": 444, "type": "game", "source": "fox-and-geese", "section": "Consensus on optimal play", "text": "Fox and Geese — Consensus on optimal play\n\n- **Maintain an unbroken front** — the geese must advance as a solid, gapless line; any hole lets the fox slip through and the game is lost.\n- **Never retreat a goose** — geese can only move forward or sideways, so a goose committed to the wrong square can create permanent weaknesses; plan each step.\n- **Advance the centre geese first** — building a convex front that pushes the fox toward the corners before compressing it minimises the chance of a flank break.\n- **Fox: probe the flanks** — the fox's only winning chance is an edge or corner escape; head for the sides and look for the slightest gap in the formation.\n- **Fox: force pace changes** — threatening a rush can bait geese into out-of-sync moves; the fox should use the entire board to disrupt the rhythm of the advancing wall.\n- **Endgame: geese aim to corner, not just stop** — hemming the fox against a wall without a second row of geese behind can allow a diagonal escape; the trap needs depth."}, {"id": 445, "type": "game", "source": "fox-and-geese", "section": "Engines & current best play", "text": "Fox and Geese — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** Analysed in *Winning Ways* (Berlekamp, Conway & Guy, 1982/2001); no separate online tablebase.\n- **Notes:** The state space is small enough that exhaustive search is trivial; any competent tree-search implementation finds optimal play instantly."}, {"id": 446, "type": "game", "source": "fox-and-geese", "section": "Complexity", "text": "Fox and Geese — Complexity\n\nThe board has only 33 squares and one fox, so the state space is small and the\ngame is fully tractable to search; its interest is strategic and historical, not\ncomputational."}, {"id": 447, "type": "game", "source": "fox-and-geese", "section": "References", "text": "Fox and Geese — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Fox_games) ([archive](http://web.archive.org/web/20260311131615/https://en.wikipedia.org/wiki/Fox_games))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 448, "type": "game", "source": "fox-and-geese", "section": "See also", "text": "Fox and Geese — See also\n\n- [Hare and Hounds](hare-and-hounds.md) · [Tablut](tablut.md) · [Brandubh](brandubh.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 449, "type": "game", "source": "frisian-draughts", "section": "overview", "text": "Frisian draughts\nDutch-Frisian draughts on 10×10 with orthogonal capture by kings — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan draughts."}, {"id": 450, "type": "game", "source": "frisian-draughts", "section": "Description", "text": "Frisian draughts — Description\n\nFrisian draughts is a 10×10 draughts variant from Friesland in which pieces\nmay also capture **orthogonally** in addition to diagonally. The orthogonal\ncaptures make the tactical play distinctively sharp."}, {"id": 451, "type": "game", "source": "frisian-draughts", "section": "Rules", "text": "Frisian draughts — Rules\n\n1. Board: 10×10, same starting positions as international draughts (20 men\n   per side on dark squares of the first four ranks).\n2. Men move and capture diagonally forward as in international draughts.\n3. **Orthogonal capture**: in addition to the usual diagonal jump, men may\n   capture by jumping an adjacent enemy piece orthogonally (sideways or\n   forward) to the next empty cell. Kings may capture orthogonally at any\n   distance.\n4. The maximum-capture rule applies: a player must choose a capture sequence\n   taking the largest number of pieces.\n5. Promotion happens on the back rank; flying kings move any distance along\n   diagonals (and capture diagonally or orthogonally).\n6. A player with no legal move loses."}, {"id": 452, "type": "game", "source": "frisian-draughts", "section": "Solution status", "text": "Frisian draughts — Solution status\n\nFrisian draughts is **not solved**. Engines exist (e.g., Damage) but the\northogonal capture rule makes the state-graph distinct from international\ndraughts and tablebases are correspondingly smaller."}, {"id": 453, "type": "game", "source": "frisian-draughts", "section": "Consensus on optimal play", "text": "Frisian draughts — Consensus on optimal play\n\n- **Orthogonal threats multiply danger** — a king that threatens both diagonal and orthogonal captures can fork positions impossible in standard draughts; always scan both axes when calculating.\n- **Centralise kings early** — Frisian kings are exceptionally powerful because of orthogonal range; getting kings to central files lets them dominate entire ranks and files simultaneously.\n- **Maximum-capture obligations can be exploited** — sacrificing a piece that forces the opponent into a long capture sequence can leave their pieces out of position; look for \"shot\" combinations that reverse material count.\n- **Guard against long-range orthogonal sweeps** — a king on an open rank or file can hoover multiple pieces; keep your piece clusters diagonal to the opponent's kings, not aligned orthogonally.\n- **Piece count matters more than in regular draughts** — the extra capture power means a man-down position is often immediately decisive; avoid loose pieces.\n- **Endgames: king vs. two men is often won for the king** — the orthogonal range allows the king to catch men that would escape in international draughts."}, {"id": 454, "type": "game", "source": "frisian-draughts", "section": "Engines & current best play", "text": "Frisian draughts — Engines & current best play\n\n- **Strongest known program(s):** Damage — dedicated Frisian draughts engine by Bert Tuyt.\n- **Strength:** Super-human at competitive level.\n- **Where the proof / tablebase lives (if solved):** Not solved; no public complete tablebase.\n- **Notes:** Competitive scene centred in the Netherlands; Damage is the reference engine used in online play on Toernooibase and lidraughts."}, {"id": 455, "type": "game", "source": "frisian-draughts", "section": "Complexity", "text": "Frisian draughts — Complexity\n\nSimilar to international draughts."}, {"id": 456, "type": "game", "source": "frisian-draughts", "section": "References", "text": "Frisian draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Frisian_draughts) ([archive](http://web.archive.org/web/20251004011548/https://en.wikipedia.org/wiki/Frisian_draughts))\n- [Schaeffer et al. (2007). *Checkers is Solved*.](../references.md#schaeffer2007) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 457, "type": "game", "source": "frisian-draughts", "section": "See also", "text": "Frisian draughts — See also\n\n- [International draughts](international-draughts.md) · [Turkish draughts](turkish-draughts.md) · [Russian draughts](russian-draughts.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 458, "type": "game", "source": "geography", "section": "overview", "text": "Generalized Geography\nThe textbook PSPACE-complete game — a directed-graph reachability game whose\nSolution status: Strongly solved as a theory (decision problem PSPACE-complete). Game-theoretic value: Position-dependent. Players: 2. Type: Impartial graph game."}, {"id": 459, "type": "game", "source": "geography", "section": "Description", "text": "Generalized Geography — Description\n\nThe motivating example is the children's game in which players alternate\nnaming places, each starting with the last letter of the previous one; \"places\nalready used\" are forbidden. Generalised on a directed graph, this becomes the\nstandard model for hardness reductions in combinatorial game theory."}, {"id": 460, "type": "game", "source": "geography", "section": "Rules", "text": "Generalized Geography — Rules\n\n1. A directed graph G and a starting vertex v are given.\n2. A token starts at v. Players alternate moving the token along a directed\n   edge to an unvisited vertex.\n3. The player unable to move loses (normal play).\n\nVariants:\n\n- **Vertex Geography** — once a vertex is visited, it cannot be re-entered.\n- **Edge Geography** — once an edge is traversed, it cannot be re-used."}, {"id": 461, "type": "game", "source": "geography", "section": "Solution status", "text": "Generalized Geography — Solution status\n\nSolved as a theory. [Schaefer (1978)](../references.md#schaefer1978) proved\n**Generalized Geography is PSPACE-complete** — given a graph and start vertex,\ndeciding who wins is hard for polynomial space, and it is the canonical reduction\ntarget used to prove other games PSPACE-hard (Hex, Othello, Amazons, many\nothers). The undirected variant differs sharply: see\n[Undirected Vertex Geography](undirected-vertex-geography.md)."}, {"id": 462, "type": "game", "source": "geography", "section": "Consensus on optimal play", "text": "Generalized Geography — Consensus on optimal play\n\n- **Move to vertices with the fewest outgoing edges** — restricting the opponent's future options is the core heuristic; a vertex with degree 1 is essentially a trap to push the opponent toward.\n- **Force the opponent into a dead-end path** — count the length of reachable chains; if you can steer into a path of odd length, the opponent faces the last move and loses.\n- **Bipartite structure is decisive** — on bipartite directed graphs, the first player loses if and only if the starting vertex is matched in every maximum matching; checking this is the efficient algorithm for those cases.\n- **Cut vertices are key resources** — moving through a cut vertex seals off a subgraph; identify which player benefits from that subgraph being isolated before committing.\n- **For small instances, retrograde analysis is the exact solver** — work backwards from positions with no moves (losses for the mover) to classify every reachable position as W or L."}, {"id": 463, "type": "game", "source": "geography", "section": "Engines & current best play", "text": "Generalized Geography — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Any retrograde-analysis tool on directed graphs solves specific instances.\n- **Strength:** Not benchmarked (PSPACE-complete in general; polynomial for bipartite graphs).\n- **Where the proof / tablebase lives (if solved):** [Schaefer (1978)](../references.md#schaefer1978) — proof of PSPACE-completeness; no universal tablebase possible.\n- **Notes:** Geography is primarily a complexity-theory benchmark; for any fixed small graph, position evaluation is fast via retrograde analysis."}, {"id": 464, "type": "game", "source": "geography", "section": "Complexity", "text": "Generalized Geography — Complexity\n\nPSPACE-complete in the size of the input graph."}, {"id": 465, "type": "game", "source": "geography", "section": "References", "text": "Generalized Geography — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Generalized_geography) ([archive](http://web.archive.org/web/20251116130356/https://en.wikipedia.org/wiki/Generalized_geography))\n- [Schaefer (1978). *On the complexity of some two-person perfect-information games*.](../references.md#schaefer1978)\n- [Lichtenstein & Sipser (1980). *GO is polynomial-space hard*.](../references.md#lichtenstein-sipser1980)"}, {"id": 466, "type": "game", "source": "geography", "section": "See also", "text": "Generalized Geography — See also\n\n- [Undirected Vertex Geography](undirected-vertex-geography.md) · [Node Kayles](node-kayles.md) · [Shannon switching game](shannon-switching-game.md)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete)"}, {"id": 467, "type": "game", "source": "gipf", "section": "overview", "text": "GIPF\nKris Burm's hex-board flagship of the GIPF project — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan abstract strategy game."}, {"id": 468, "type": "game", "source": "gipf", "section": "Description", "text": "GIPF — Description\n\nGIPF (Kris Burm, 1997) is the eponymous first title in the GIPF Project — a\nseries of six interconnected abstract games (see [ZÈRTZ](zertz.md),\n[DVONN](dvonn.md), [YINSH](yinsh.md), [PÜNCT](punct.md), [TZAAR](tzaar.md),\n[TAMSK](tamsk.md), [LYNGK](lyngk.md)). It is played on a small hexagonal\nboard: pieces are pushed onto the board from the edges and captured in lines of\nfour."}, {"id": 469, "type": "game", "source": "gipf", "section": "Rules", "text": "GIPF — Rules\n\n1. Board: a hexagonal star of intersections (a small hex grid with the corners\n   trimmed).\n2. Each player has a reserve of pieces (15 in the standard setup) and a stock\n   of \"GIPF\" double-pieces in the basic variant.\n3. On a turn, a player **pushes** one of their pieces onto the board from one\n   of the edge-entry points along a line, sliding all pieces in that line by\n   one cell.\n4. Whenever a row of four same-colour pieces is formed, those pieces are\n   **captured** by their owner; any pieces of the *other* colour at either end\n   of the run are captured by the player who formed the row.\n5. The first player who cannot push a piece in onto the board loses (typically\n   because their reserve is empty).\n6. **Tournament variant**: each player starts with a fixed number of \"GIPF\n   pieces\" (double-stacked pieces); losing all your GIPF pieces is also a loss\n   condition."}, {"id": 470, "type": "game", "source": "gipf", "section": "Solution status", "text": "GIPF — Solution status\n\nGIPF is **not solved**. The combination of the small board with the sliding /\nrow-of-four-captures dynamics gives a rich, non-monotone tactical structure.\nThere is a small competitive community and some computer players, but no\npublished solution."}, {"id": 471, "type": "game", "source": "gipf", "section": "Consensus on optimal play", "text": "GIPF — Consensus on optimal play\n\n- **Control the centre intersections** — pushes from the edge affect lines that pass through the centre; owning the central cells makes it easier to threaten row-of-four from multiple directions.\n- **Create two simultaneous threats** — one row-of-four threat can be blocked by a single push from the opponent; two threats at once (a \"fork\") forces a concession.\n- **Recapture efficiently with GIPF pieces** — GIPF double-pieces are immune to capture and return captured pieces to your reserve; placing and keeping at least one GIPF piece in play is vital in the tournament variant.\n- **Deplete the opponent's reserve, not just the board** — the game is lost when you cannot push a piece; forcing captures that return pieces to your reserve while denying the opponent the same is the path to victory.\n- **Avoid clustering same-colour pieces on the same line** — four in a row is a reward, but three already-positioned pieces on a line invite an opponent's blocking push that spoils your plan; vary threat directions.\n- **Endgame tempo** — when reserves are low, each push is a countdown move; calculate who runs out of pieces first and steer toward positions where that counter favours you."}, {"id": 472, "type": "game", "source": "gipf", "section": "Engines & current best play", "text": "GIPF — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** Not solved; no tablebase.\n- **Notes:** Small but active competitive community; human expert theory is more developed than published computer analysis."}, {"id": 473, "type": "game", "source": "gipf", "section": "Complexity", "text": "GIPF — Complexity\n\nLarge enough to be out of reach of full search; small enough that strong\nprograms are feasible."}, {"id": 474, "type": "game", "source": "gipf", "section": "References", "text": "GIPF — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/GIPF_(game)) ([archive](http://web.archive.org/web/20251004213101/https://en.wikipedia.org/wiki/GIPF_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 475, "type": "game", "source": "gipf", "section": "See also", "text": "GIPF — See also\n\n- [ZÈRTZ](zertz.md) · [DVONN](dvonn.md) · [YINSH](yinsh.md) · [TZAAR](tzaar.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 476, "type": "game", "source": "glinski-hexagonal-chess", "section": "overview", "text": "Glinski hexagonal chess\nChess on a hexagonal board with three pawn directions — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 477, "type": "game", "source": "glinski-hexagonal-chess", "section": "Description", "text": "Glinski hexagonal chess — Description\n\nGlinski hexagonal chess (Władysław Gliński, 1936) maps chess onto a 91-cell\nhexagonal board. The classical chess pieces are redefined for hex geometry:\nbishops move along three diagonal directions, rooks along three orthogonal\nones, and there are three bishops per side (one per cell colour)."}, {"id": 478, "type": "game", "source": "glinski-hexagonal-chess", "section": "Rules", "text": "Glinski hexagonal chess — Rules\n\n1. Board: 91 hexagons arranged as a regular hexagon of side 6.\n2. Each side has 9 pawns, 2 rooks, 2 knights, 3 bishops (one per cell colour),\n   1 queen, 1 king.\n3. Pieces move along the natural hex generalisations: rooks along the three\n   orthogonal axes, bishops along the three diagonal axes, queen as both;\n   knights have a defined leap; pawns move straight forward and capture\n   diagonally forward (three directions).\n4. Pawns promote on the far edge; there is no castling and no en-passant in\n   Glinski's original rules.\n5. Win is by checkmate; standard stalemate/draw rules apply with Glinski's\n   adjustments."}, {"id": 479, "type": "game", "source": "glinski-hexagonal-chess", "section": "Solution status", "text": "Glinski hexagonal chess — Solution status\n\nGlinski hexagonal chess is **not solved**. Engine play exists but is far less\ndeveloped than orthodox chess."}, {"id": 480, "type": "game", "source": "glinski-hexagonal-chess", "section": "Consensus on optimal play", "text": "Glinski hexagonal chess — Consensus on optimal play\n\n- **The third bishop matters** — with three bishops of different cell colours, each player can attack every hex; keeping all three active prevents colour-blind defensive setups that work in orthodox chess.\n- **Centre control has six axes** — the hex board has three orthogonal and three diagonal directions; centralised pieces threaten more of the board than in square chess, making central occupation even more valuable.\n- **Pawns are weaker than in orthodox chess** — three forward capture directions make pawn chains harder to form and easier to disrupt; avoid pawn-heavy positional play and favour piece activity.\n- **Knights are relatively stronger** — knights leap, bypassing the six-directional flow; their fixed jump pattern is harder to anticipate on a hex board, making them excellent for surprise attacks.\n- **King safety requires guarding six directions** — the hex king is approached from six rather than eight squares; ensure at least four of those approaches are covered by your own pieces."}, {"id": 481, "type": "game", "source": "glinski-hexagonal-chess", "section": "Engines & current best play", "text": "Glinski hexagonal chess — Engines & current best play\n\n- **Strongest known program(s):** No widely-known public engine for Glinski's variant; Fairy-Stockfish ([https://github.com/ianfab/Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish))) supports some hex-chess variants but Glinski's specific rules may not be fully implemented.\n- **Strength:** Weak to moderate; far below the strength of orthodox chess engines.\n- **Where the proof / tablebase lives (if solved):** Not solved; no tablebase.\n- **Notes:** Small competitive scene in continental Europe; human theory is the primary reference."}, {"id": 482, "type": "game", "source": "glinski-hexagonal-chess", "section": "Complexity", "text": "Glinski hexagonal chess — Complexity\n\nLarger than chess."}, {"id": 483, "type": "game", "source": "glinski-hexagonal-chess", "section": "References", "text": "Glinski hexagonal chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hexagonal_chess) ([archive](http://web.archive.org/web/20260513200804/https://en.wikipedia.org/wiki/Hexagonal_chess))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 484, "type": "game", "source": "glinski-hexagonal-chess", "section": "See also", "text": "Glinski hexagonal chess — See also\n\n- [Chess](chess.md) · [Capablanca chess](capablanca-chess.md) · [Chess960](chess960.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 485, "type": "game", "source": "go", "section": "overview", "text": "Go\nThe largest classical board game — superhuman AI exists, small boards are\nSolution status: Unsolved (19×19); small boards solved — 5×5 weakly solved. Game-theoretic value: Unknown (19×19). 5×5: first player (Black) wins by 25. Players: 2. Type: Partisan territory game."}, {"id": 486, "type": "game", "source": "go", "section": "Description", "text": "Go — Description\n\nPlayed on the intersections of a 19×19 grid. Players alternately place stones;\nstones with no liberties are captured; the goal is to control more territory\n(plus captures) than the opponent. The *ko* rule forbids immediate board\nrepetition."}, {"id": 487, "type": "game", "source": "go", "section": "Solution status", "text": "Go — Solution status\n\n19×19 Go is **unsolved** and is the standard example of a game whose sheer size\ndefeats solving. Its ~2 × 10^170 legal positions and ~10^360 game-tree complexity\ndwarf every other classical game.\n\n- **Small boards are solved.** [Van der Werf, van den Herik & Uiterwijk (2003)](../references.md#vanderwerf-go2003)\n  weakly solved **5×5 Go**: the first player (Black) wins, capturing the whole\n  board for a 25-point win. Boards up to roughly 5×6 / 6×6 have also been solved\n  in subsequent work; **7×7** has been very extensively analysed and its value\n  is widely agreed (a small Black win under common komi) though \"solved\" status\n  there is more nuanced.\n- **Superhuman play is not a solution.** [AlphaGo](../references.md#silver-alphago2016)\n  (2016) and its successors decisively surpassed top human players, and\n  [AlphaZero](../references.md#silver-alphazero2018) reached that level from\n  self-play alone — but this is [strong play, not solving](../lexicon/README.md#solving-vs-strong-play):\n  it establishes no proven game-theoretic value for 19×19."}, {"id": 488, "type": "game", "source": "go", "section": "Consensus on optimal play", "text": "Go — Consensus on optimal play\n\n- **Influence over territory early, territory over influence late** — early in the game, strong players build frameworks (moyo) that threaten large territories; converting influence into solid territory before the opponent invades is the central tension.\n- **Two eyes or die** — any group without two distinct eye spaces is eventually captured; building eyes (or the potential for them) is the unconditional requirement for group survival.\n- **Do not attach to weak stones** — attaching a stone to an opponent's already-weak group strengthens that group while thickening their position; instead, attack from a distance (the knight's move or two-space extension) to maintain flexibility.\n- **Komi calibrates the first-move advantage** — professional consensus has settled on 6.5 or 7.5 points komi as roughly fair; playing for a narrow margin win as Black (or neutralising it as White) shapes endgame priorities.\n- **Sente (initiative) is a resource** — a move that demands a response grants the player the next \"free\" move elsewhere; counting sente/gote sequences is essential in the middle and late game.\n- **Reducing while maintaining your own thickness** — invasions succeed when the invader can run or live; ensure your invasion point is not adjacent to a strong opponent wall that would make escape impossible."}, {"id": 489, "type": "game", "source": "go", "section": "Engines & current best play", "text": "Go — Engines & current best play\n\n- **Strongest known program(s):** KataGo ([https://github.com/lightvector/KataGo](https://github.com/lightvector/KataGo) ([archive](http://web.archive.org/web/20260510144301/https://github.com/lightvector/KataGo))) and Leela Zero ([https://github.com/leela-zero/leela-zero](https://github.com/leela-zero/leela-zero)) — both deep-learning Monte Carlo tree search engines in the AlphaGo/AlphaZero lineage.\n- **Strength:** Super-human on 19×19; all top engines vastly exceed professional human level.\n- **Where the proof / tablebase lives (if solved):** 5×5 solved (van der Werf et al., 2003); 19×19 not solved — no tablebase.\n- **Notes:** Engine consensus on \"fair komi\" (~7 points) is the closest thing to a settled game-theoretic claim for 19×19; the exact value remains unproven."}, {"id": 490, "type": "game", "source": "go", "section": "Complexity", "text": "Go — Complexity\n\nState-space ~2 × 10^170; game-tree ~10^360\n([van den Herik et al., 2002](../references.md#vandenherik2002)). Generalised Go\nis EXPTIME-complete."}, {"id": 491, "type": "game", "source": "go", "section": "References", "text": "Go — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Go_(game)) ([archive](http://web.archive.org/web/20260508155030/https://en.wikipedia.org/wiki/Go_(game)))\n- [van der Werf, E. C. D., van den Herik, H. J. & Uiterwijk, J. W. H. M. (2003). *Solving Go on Small Boards*.](../references.md#vanderwerf-go2003)\n- [Silver et al. (2016). *Mastering the game of Go…* (AlphaGo).](../references.md#silver-alphago2016)\n- [Silver et al. (2018). *AlphaZero*.](../references.md#silver-alphazero2018)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 492, "type": "game", "source": "go", "section": "See also", "text": "Go — See also\n\n- [Atari Go / capture Go](arimaa.md) — see [Arimaa](arimaa.md) for another \"designed to be hard for computers\" game · [Chess](chess.md) · [Amazons](amazons.md)\n- Lexicon: [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play) · [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 493, "type": "game", "source": "gomoku", "section": "overview", "text": "Gomoku\nFree-style five-in-a-row — weakly solved in 1993 as a first-player win.\nSolution status: Weakly solved (free-style, 15×15). Game-theoretic value: First-player win. Players: 2. Type: Partisan positional (k-in-a-row) game."}, {"id": 494, "type": "game", "source": "gomoku", "section": "Description", "text": "Gomoku — Description\n\nPlayed on the intersections of a 15×15 board. Players alternately place a stone\nof their colour; the winner is the first to make **five (or, in free-style,\nfive or more) in a row** — horizontally, vertically, or diagonally. \"Free-style\"\nGomoku imposes no restrictions on either player."}, {"id": 495, "type": "game", "source": "gomoku", "section": "Solution status", "text": "Gomoku — Solution status\n\nFree-style Gomoku on the standard 15×15 board is **weakly solved**.\n[Allis, van den Herik & Huntjens (1996)](../references.md#allis-gomoku1996)\n(workshop version 1993) proved it a **first-player win**, using\n[proof-number search](../lexicon/README.md#proof-number-search) and\nthreat-space search — the latter a key innovation that exploits the way forcing\nthreats (open fours, double threats) chain together. The first player can force\na win against any defence.\n\nThe strong first-player advantage in free-style play is exactly why competitive\nvariants exist: [Renju](renju.md) adds restrictions on the first player, and\nother rule sets (swap, swap2) re-balance the game. Allis's threat-space methods\nremain foundational for solving k-in-a-row games."}, {"id": 496, "type": "game", "source": "gomoku", "section": "Consensus on optimal play", "text": "Gomoku — Consensus on optimal play\n\n- **Play the centre first** — the exact centre of the 15×15 board connects to the most five-in-a-row lines; deviating from the centre as first player surrenders the strongest winning basis.\n- **Build double threats** — a \"double open four\" (two directions each one stone from five) cannot both be blocked; creating such forks is the immediate goal of Black's winning strategy.\n- **Respond to open threes immediately** — an unblocked open three becomes an open four on the opponent's next move, which then forces a block; respond before the forcing chain escalates.\n- **Avoid clustering all stones on one diagonal** — spreading threats across horizontal, vertical, and both diagonals makes your position harder to address with a single response.\n- **White must complicate and avoid open board** — White has no path to a forced win; the best defence creates a blocked, tactical fight where Black's forcing advantage is hardest to convert.\n- **Threat-space search wins games** — strong players calculate sequences of \"urgent\" threat moves (open fours, forks) many steps ahead; a player who sees one more forcing move in the chain will prevail."}, {"id": 497, "type": "game", "source": "gomoku", "section": "Engines & current best play", "text": "Gomoku — Engines & current best play\n\n- **Strongest known program(s):** Renju/Gomoku engines such as Yixin and Rapfi — Monte Carlo / threat-space hybrid engines used in online competition.\n- **Strength:** Super-human; top engines find the forced winning lines that humans miss.\n- **Where the proof / tablebase lives (if solved):** [Allis, van den Herik & Huntjens (1996)](../references.md#allis-gomoku1996) — proof of first-player win via threat-space search; no exhaustive tablebase.\n- **Notes:** Competitive play uses swap or swap2 opening rules to neutralise the proven first-player advantage; free-style Gomoku is a formal first-player win."}, {"id": 498, "type": "game", "source": "gomoku", "section": "Complexity", "text": "Gomoku — Complexity\n\nState-space ~10^105, game-tree ~10^70\n([van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 499, "type": "game", "source": "gomoku", "section": "References", "text": "Gomoku — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Gomoku) ([archive](http://web.archive.org/web/20260506025335/https://en.wikipedia.org/wiki/Gomoku))\n- [Allis, van den Herik & Huntjens (1996). *Go-Moku Solved by New Search Techniques*.](../references.md#allis-gomoku1996)\n- [Allis, V. (1994). *Searching for Solutions in Games and Artificial Intelligence*.](../references.md#allis1994)"}, {"id": 500, "type": "game", "source": "gomoku", "section": "See also", "text": "Gomoku — See also\n\n- [Renju](renju.md) · [Pente](pente.md) · [Connect6](connect6.md) · [Connect Four](connect-four.md) · [Qubic](qubic.md)\n- Lexicon: [proof-number search](../lexicon/README.md#proof-number-search) · [weakly solved](../lexicon/README.md#weakly-solved)"}, {"id": 501, "type": "game", "source": "gonnect", "section": "overview", "text": "Gonnect\nA Go-based connection game on the Go board — ultra-weakly solved as a\nSolution status: Ultra-weakly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan connection game on a Go board."}, {"id": 502, "type": "game", "source": "gonnect", "section": "Description", "text": "Gonnect — Description\n\nGonnect (João Pedro Neto, 2000) plays on a Go board and uses Go's capture rules,\nbut with a **Hex-style connection win condition**: connect any two opposite\nsides of the board. Combining Go captures with a connection goal makes a game\nwith rich tactics in a small ruleset."}, {"id": 503, "type": "game", "source": "gonnect", "section": "Rules", "text": "Gonnect — Rules\n\n1. Go board (commonly 13×13 or 19×19).\n2. Players alternate placing one stone of their colour on an empty intersection,\n   subject to Go's no-suicide rule and an extra no-pass rule: a player must\n   move if any legal move exists.\n3. Standard Go captures: a group with no liberties is removed.\n4. The first player to form a connected group spanning their two opposite sides\n   (orthogonal connection) wins. If a player has **no legal move**, they win\n   (no-passing forced-no-move rule). The ko rule applies."}, {"id": 504, "type": "game", "source": "gonnect", "section": "Solution status", "text": "Gonnect — Solution status\n\nUltra-weakly solved: the **first player wins** by a strategy-stealing argument\n[(Neto, 2000)](../references.md#schensted-titus1975) **[verify]** — Gonnect\ncannot be drawn (the win condition is symmetric and an extra stone never hurts).\nThe proof is non-constructive."}, {"id": 505, "type": "game", "source": "gonnect", "section": "Consensus on optimal play", "text": "Gonnect — Consensus on optimal play\n\n- **Build groups with multiple connection paths** — a single-path chain across the board is easily cut by captures; maintain at least two separate pathways to each side so the opponent must deal with both simultaneously.\n- **Exploit the no-pass rule** — unlike Go, you cannot pass; creating positions where any opponent move either completes your connection or puts their own group in atari can be decisive.\n- **Prioritise liberties in contested areas** — because groups can be captured Go-style, a connection attempt through a low-liberty group is fragile; connect through living or unkillable groups whenever possible.\n- **Cutting the opponent's chain is often better than extending yours** — inserting a stone that divides the opponent's path forces them to rescue one branch, letting you extend the other leg of your connection uncontested.\n- **Central stones serve both connection directions** — a stone in the middle of the board contributes to horizontal and vertical connection alike; edge stones commit to only one side."}, {"id": 506, "type": "game", "source": "gonnect", "section": "Engines & current best play", "text": "Gonnect — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** Ultra-weak solution (first-player win) via strategy-stealing argument; no constructive proof or tablebase.\n- **Notes:** The strategy-stealing proof is non-constructive; the actual winning strategy for the first player is not known."}, {"id": 507, "type": "game", "source": "gonnect", "section": "Complexity", "text": "Gonnect — Complexity\n\nComparable to Go on the same board."}, {"id": 508, "type": "game", "source": "gonnect", "section": "References", "text": "Gonnect — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Gonnect) ([archive](http://web.archive.org/web/20260315150103/https://en.wikipedia.org/wiki/Gonnect))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework for connection-game strategy stealing)"}, {"id": 509, "type": "game", "source": "gonnect", "section": "See also", "text": "Gonnect — See also\n\n- [Hex](hex.md) · [Go](go.md) · [Crossway](crossway.md)\n- Lexicon: [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved) · [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 510, "type": "game", "source": "grundys-game", "section": "overview", "text": "Grundy's game\nAn impartial game whose nim-value sequence is computed for billions of values\nSolution status: Partially solved (algorithmically solved for all practical heap sizes; structure unproven). Game-theoretic value: Computable for any given position, but no closed form. Players: 2. Type: Impartial combinatorial game."}, {"id": 511, "type": "game", "source": "grundys-game", "section": "Description", "text": "Grundy's game — Description\n\nA heap of objects. On a turn a player splits one heap into two **non-empty,\nunequal** parts. A player who cannot move (all heaps are size 1 or 2) loses\nunder [normal play](../lexicon/README.md#normal-play-convention)."}, {"id": 512, "type": "game", "source": "grundys-game", "section": "Solution status", "text": "Grundy's game — Solution status\n\nGrundy's game is **partially solved**. Any *specific* position can be analysed:\nthe [nim-value](../lexicon/README.md#nim-value) of a heap is computed by the\n[mex](../lexicon/README.md#mex) recurrence, and positions decompose by\n[nim-sum](../lexicon/README.md#nim-sum). The single-heap nim-value sequence has\nbeen computed for billions of heap sizes.\n\nWhat is **open** — and a celebrated unsolved problem in combinatorial game\ntheory — is whether that sequence is *eventually periodic*. It is widely\nconjectured to be (most octal-style games whose sequences have been studied are\nperiodic), but no proof exists, and no period has been found despite enormous\ncomputation. So the game is solved in the operational sense for any heap a human\nor computer will encounter, but not solved in the theoretical sense."}, {"id": 513, "type": "game", "source": "grundys-game", "section": "Consensus on optimal play", "text": "Grundy's game — Consensus on optimal play\n\n- **Compute nim-values, then nim-sum to zero** — with multiple heaps, calculate the nim-value (Grundy value) of each heap using the mex recurrence, then make a move that sets the nim-sum of all heaps to 0; this is the exact winning condition.\n- **A heap of size 2 is a dead end** — heaps of size 1 and 2 cannot be split (size-1 is indivisible; size-2 cannot be split into two unequal non-empty parts); track them as terminal heaps with nim-value 0.\n- **Heap size 3 has nim-value 1** — split into {1, 2}; this is the smallest non-trivial move and a useful anchor for hand calculation.\n- **Use precomputed tables for larger heaps** — beyond small heaps, the nim-value sequence is irregular enough that memorisation or table lookup is the only practical approach for over-the-board play.\n- **With a single large heap, nim-value 0 is a loss for the player to move** — if your only heap has nim-value 0, every split you make will give the opponent a nim-value-0 position to respond to; you lose with optimal opponent play."}, {"id": 514, "type": "game", "source": "grundys-game", "section": "Engines & current best play", "text": "Grundy's game — Engines & current best play\n\n- **Strongest known program(s):** No dedicated game engine; any CGT toolkit implementing mex/nim-sum (e.g., Aaron Siegel's CGSuite) solves any given position exactly.\n- **Strength:** Perfectly solvable for any position that fits in memory; the nim-value sequence has been computed for billions of heap sizes.\n- **Where the proof / tablebase lives (if solved):** Precomputed nim-value tables exist up to large heap sizes; see [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** The celebrated open question — whether the nim-value sequence is eventually periodic — does not affect practical play; any specific position can be evaluated exactly."}, {"id": 515, "type": "game", "source": "grundys-game", "section": "Complexity", "text": "Grundy's game — Complexity\n\nComputing the nim-value of a heap of size *n* by the naive recurrence is\npolynomial in *n*."}, {"id": 516, "type": "game", "source": "grundys-game", "section": "References", "text": "Grundy's game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Grundy%27s_game) ([archive](http://web.archive.org/web/20251004213435/https://en.wikipedia.org/wiki/Grundy%27s_game))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)"}, {"id": 517, "type": "game", "source": "grundys-game", "section": "See also", "text": "Grundy's game — See also\n\n- [Kayles](kayles.md) · [Dawson's chess](dawsons-chess.md) · [Nim](nim.md) · [Subtract-a-square](subtract-a-square.md)\n- Lexicon: [nim-value](../lexicon/README.md#nim-value) · [octal game](../lexicon/README.md#octal-game)"}, {"id": 518, "type": "game", "source": "hackenbush", "section": "overview", "text": "Hackenbush\nThe game that taught combinatorial game theory how to do arithmetic — its\nSolution status: Strongly solved (as a theory). Game-theoretic value: Any position has a computable CGT value (a surreal number, or a value with infinitesimals/nimbers). Players: 2. Type: Partisan combinatorial game (Green Hackenbush is impartial)."}, {"id": 519, "type": "game", "source": "hackenbush", "section": "Description", "text": "Hackenbush — Description\n\nA drawing of coloured line segments (\"edges\") connected to the ground. In\n**Blue-Red Hackenbush** one player may remove blue edges, the other red edges;\nin **Green Hackenbush** all edges are green and either player may remove any.\nRemoving an edge also removes anything no longer connected to the ground. Under\n[normal play](../lexicon/README.md#normal-play-convention) the player unable to\nmove loses."}, {"id": 520, "type": "game", "source": "hackenbush", "section": "Solution status", "text": "Hackenbush — Solution status\n\nHackenbush is **strongly solved as a theory**: every position has an exactly\ncomputable combinatorial-game value, and from that value the winner (and how\nmuch \"in hand\" each side is) follows immediately.\n\n- **Blue-Red Hackenbush** values are exactly the [surreal numbers](../lexicon/README.md#surreal-number);\n  a string of edges evaluates by a simple sign-expansion rule, and arbitrary\n  pictures add. This is the historical origin of surreal numbers in\n  [Conway (1976)](../references.md#conway1976).\n- **Green Hackenbush** is [impartial](../lexicon/README.md#impartial-game); its\n  values are [nimbers](../lexicon/README.md#nim-value), computed by the\n  \"colon\" and \"fusion\" principles for trees and general graphs.\n- **Blue-Red-Green** combines both, yielding values with numbers,\n  infinitesimals, and nimbers.\n\nHackenbush is \"solved\" in the strongest theoretical sense — there is a complete\ncalculus — even though evaluating a specific large picture can still take work."}, {"id": 521, "type": "game", "source": "hackenbush", "section": "Consensus on optimal play", "text": "Hackenbush — Consensus on optimal play\n\n- **Blue-Red strings: read the value by sign expansion** — in a vertical string, each edge is a bit (Blue = positive step, Red = negative step); the value is the surreal number given by reading the string as a binary fraction after the first sign change; remove whichever edge collapses the most positive (or negative) value.\n- **Trees: evaluate bottom-up** — the colon principle lets you replace any branch with its nimber (Green) or game value (Blue-Red); work leaf-to-root rather than root-to-leaf.\n- **Cycles: apply the fusion principle** — in Green Hackenbush, any cycle contributes a nimber equal to its length modulo 2 (odd cycle → nimber 1, even cycle → nimber 0); merge vertices on the cycle to a single ground-connected node.\n- **Blue-Red: aim for positive total value as Blue, negative as Red** — if the sum of all component values is positive, Blue wins under optimal play regardless of who moves first; the magnitude is the \"number of free moves\" in hand.\n- **Match losing components** — if your position has a negative component (bad for you), try to make a move in an equal-and-opposite good component to cancel it; hedging is the arithmetic of Hackenbush strategy."}, {"id": 522, "type": "game", "source": "hackenbush", "section": "Engines & current best play", "text": "Hackenbush — Engines & current best play\n\n- **Strongest known program(s):** CGSuite (Aaron Siegel) — a general CGT toolkit that evaluates Hackenbush positions exactly.\n- **Strength:** Exact solution for any position that can be represented; no uncertainty remains.\n- **Where the proof / tablebase lives (if solved):** [Conway (1976)](../references.md#conway1976) and [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001) contain the complete theory.\n- **Notes:** Hackenbush is \"solved\" in the deepest theoretical sense; every position has an exact surreal/nimber value and optimal play follows mechanically."}, {"id": 523, "type": "game", "source": "hackenbush", "section": "Complexity", "text": "Hackenbush — Complexity\n\nEvaluation cost depends on the picture; trees are easy, general graphs require\nthe fusion principle."}, {"id": 524, "type": "game", "source": "hackenbush", "section": "References", "text": "Hackenbush — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hackenbush)\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 525, "type": "game", "source": "hackenbush", "section": "See also", "text": "Hackenbush — See also\n\n- [Nim](nim.md) · [Domineering](domineering.md) · [Toads and Frogs](toads-and-frogs.md)\n- Lexicon: [surreal number](../lexicon/README.md#surreal-number) · [combinatorial game theory](../lexicon/README.md#combinatorial-game-theory)"}, {"id": 526, "type": "game", "source": "halatafl", "section": "overview", "text": "Halatafl\nScandinavian fox-and-geese hunt game — partially analysed.\nSolution status: Partially analysed **[verify]** (small variants). Game-theoretic value: Geese win with optimal play on standard board **[verify]**. Players: 2 (asymmetric). Type: Partisan asymmetric hunt game."}, {"id": 527, "type": "game", "source": "halatafl", "section": "Description", "text": "Halatafl — Description\n\nHalatafl is a Norse variant of fox-and-geese — described in the 13th-century\n*Grettis saga* — played on a cross-shaped board where one player controls a\nsingle fox and the other controls a flock of geese trying to corner it."}, {"id": 528, "type": "game", "source": "halatafl", "section": "Rules", "text": "Halatafl — Rules\n\n1. Board: cross-shaped grid of 33 cells (the standard fox-and-geese\n   diagram).\n2. One player has 13 geese starting on the lower arm and centre; the other\n   has 1 fox starting in the upper centre.\n3. Geese move one step in any orthogonal direction (no backward in some\n   variants — **[verify]**). Geese never capture.\n4. The fox moves one step orthogonally or **jumps** an adjacent goose along\n   an orthogonal line to an empty cell beyond, removing the goose.\n5. The fox wins by reducing the geese to a number too small to trap it (e.g.,\n   fewer than 6). The geese win by surrounding the fox so it cannot move."}, {"id": 529, "type": "game", "source": "halatafl", "section": "Solution status", "text": "Halatafl — Solution status\n\nHalatafl is closely related to classical Fox-and-Geese, for which several\nsmall variants are solved. The standard variant is broadly believed to be a\n**geese win** with optimal play; **[verify]** the specific Halatafl ruleset\nfor an authoritative solution."}, {"id": 530, "type": "game", "source": "halatafl", "section": "Consensus on optimal play", "text": "Halatafl — Consensus on optimal play\n\n- **Geese: maintain a gapless advancing line** — as in classical Fox and Geese, the key is never leaving a hole in the formation that the fox can jump through; advance the line uniformly.\n- **Geese: use the flanks to contain, not just chase** — wrapping geese around the fox's sides prevents diagonal escapes and compresses its space without relying solely on a head-on push.\n- **Fox: head for the corners or flanks immediately** — the fox's best escape route is along the board edge or toward a corner where the geese's wider formation cannot follow efficiently.\n- **Fox: create forced goose moves** — a jump that captures one goose while threatening another forces the geese to react inefficiently, potentially opening a gap in their line.\n- **Geese: never leave a single isolated goose ahead of the line** — an isolated advanced goose is a free capture for the fox, reducing the flock below the trapping threshold."}, {"id": 531, "type": "game", "source": "halatafl", "section": "Engines & current best play", "text": "Halatafl — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** Partially analysed; no published complete solution specific to the Halatafl ruleset.\n- **Notes:** Closely related to classical Fox and Geese (see [fox-and-geese.md](fox-and-geese.md)); strategic principles carry over."}, {"id": 532, "type": "game", "source": "halatafl", "section": "Complexity", "text": "Halatafl — Complexity\n\nSmall."}, {"id": 533, "type": "game", "source": "halatafl", "section": "References", "text": "Halatafl — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hnefatafl) ([archive](http://web.archive.org/web/20251004212953/https://en.wikipedia.org/wiki/Hnefatafl))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 534, "type": "game", "source": "halatafl", "section": "See also", "text": "Halatafl — See also\n\n- [Fox and Geese](fox-and-geese.md) · [Catch the Hare](catch-the-hare.md) · [Tablut](tablut.md)\n- Lexicon: [hunt game](../lexicon/README.md#hunt-game)"}, {"id": 535, "type": "game", "source": "hanabi", "section": "overview", "text": "Hanabi\nCooperative card game of hidden information — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown (cooperative — expected max score). Players: 2–5 (fully cooperative). Type: Cooperative imperfect-information card game."}, {"id": 536, "type": "game", "source": "hanabi", "section": "Description", "text": "Hanabi — Description\n\nHanabi (Antoine Bauza, 2010) is a fully cooperative card game in which each\nplayer holds their hand **facing away from themselves** — they see everyone\nelse's cards but not their own. The team works together to play fireworks\n(suit colours in ascending order from 1 to 5) using limited information\ntokens."}, {"id": 537, "type": "game", "source": "hanabi", "section": "Rules", "text": "Hanabi — Rules\n\n1. Deck: 50 cards in 5 colours; each colour has three 1s, two each of 2/3/4,\n   and one 5.\n2. Each player is dealt a hand (4 or 5 cards depending on player count) held\n   facing outward.\n3. The team starts with 8 **information** tokens and 3 **fuse** tokens.\n4. On a turn the active player does **one** of:\n   - **Give a clue** (spends 1 information token): pick another player and\n     indicate either a colour or a rank; identify all cards in that player's\n     hand matching the clue.\n   - **Discard a card** (returns 1 information token, up to the cap of 8).\n   - **Play a card**: place it on the table starting (or continuing) a\n     firework of its colour 1, 2, 3, 4, 5. A wrong play discards the card\n     and burns 1 fuse token.\n5. After a card is played or discarded, the player draws a new card if any\n   remain.\n6. Game ends when the deck is exhausted (one final round is played), all 5\n   fireworks reach 5, or all 3 fuse tokens are spent. The score is the sum\n   of the top card of each firework, max 25."}, {"id": 538, "type": "game", "source": "hanabi", "section": "Solution status", "text": "Hanabi — Solution status\n\nHanabi is **not solved**. The cooperative-information aspect attracted\nmachine-learning research (DeepMind's Hanabi Learning Environment, 2019)\nshowing AI struggles with conventional human play."}, {"id": 539, "type": "game", "source": "hanabi", "section": "Consensus on optimal play", "text": "Hanabi — Consensus on optimal play\n\n- **Clues should convey the most information possible** — in the H-group convention system, a clue carries meaning beyond its literal content (\"this colour clue to the newest card means play it immediately\"); teams pre-agree on conventions to pack maximum information into each token.\n- **Protect 5s and unique cards** — cards that exist only once in the deck (each colour's 5, and any card whose duplicates have been discarded) are irreplaceable; clue or protect them before discarding becomes necessary.\n- **The discard pile is a shared board state** — track discards carefully; a discarded card that makes a later play impossible changes the team's scoring ceiling and should inform clue priorities.\n- **Save information tokens by discarding lowest-value unneeded cards** — when the information token pool is full (8) you must play or clue; maintain a 3–5 token reserve to avoid forced bad plays.\n- **Finesse and prompt clues** — advanced convention: a clue pointing at a card in position N can implicitly tell an intermediate player that their newest unclued card is the \"bridge\" card to be played first; this allows complex sequences with a single token.\n- **Never waste the last fuse token on a speculative play** — with one fuse token remaining, only play a card when the team has established its identity beyond doubt; the game ends on the third misplay."}, {"id": 540, "type": "game", "source": "hanabi", "section": "Engines & current best play", "text": "Hanabi — Engines & current best play\n\n- **Strongest known program(s):** WTFWhere (competitive self-play agent) and various DeepMind/academic bots (Sad, SmartBot) from the Hanabi Learning Environment benchmark.\n- **Strength:** Top self-play agents score near-perfect (24–25/25) against other bots of the same convention but underperform when paired with humans using different conventions.\n- **Where the proof / tablebase lives (if solved):** Not solved; DeepMind's Hanabi Learning Environment (2019) is the standard benchmark.\n- **Notes:** Human-AI cooperative play is an active research challenge; the game's hidden-information structure makes cross-agent generalisation hard."}, {"id": 541, "type": "game", "source": "hanabi", "section": "Complexity", "text": "Hanabi — Complexity\n\nLarge."}, {"id": 542, "type": "game", "source": "hanabi", "section": "References", "text": "Hanabi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hanabi_(card_game)) ([archive](http://web.archive.org/web/20260306094445/https://en.wikipedia.org/wiki/Hanabi_(card_game)))\n- [Yoshizoe et al. *Hanabi study (2019).*](../references.md#yoshizoe-hanabi) **[verify]**"}, {"id": 543, "type": "game", "source": "hanabi", "section": "See also", "text": "Hanabi — See also\n\n- [Bridge](bridge.md) · [Skat](skat.md)\n- Lexicon: [imperfect information](../lexicon/README.md#imperfect-information)"}, {"id": 544, "type": "game", "source": "hare-and-hounds", "section": "overview", "text": "Hare and Hounds\nA small asymmetric pursuit game — fully solved by exhaustive search; with\nSolution status: Weakly solved. Game-theoretic value: Hounds win with correct play; hare escapes if hounds err **[verify]**. Players: 2 (asymmetric: one hare vs. three hounds). Type: Partisan pursuit game."}, {"id": 545, "type": "game", "source": "hare-and-hounds", "section": "Description", "text": "Hare and Hounds — Description\n\nPlayed on a small board of 11 points connected by lines. Three **hounds** start\nat one end, a single **hare** in the middle. Hounds may move along lines\nvertically or forward but never backward (toward their own side); the hare moves\nalong any line in any direction. The hounds win by trapping the hare so it\ncannot move; the hare wins by slipping past all three hounds, or — in the\nstandard rule set — if the hounds make a number of non-advancing \"side\" moves in\na row (a stalling rule that prevents the hounds from playing for a draw)."}, {"id": 546, "type": "game", "source": "hare-and-hounds", "section": "Solution status", "text": "Hare and Hounds — Solution status\n\nHare and Hounds is **weakly solved** by trivial exhaustive search: the position\ngraph has only a few thousand states. With correct play the **hounds confine the\nhare** — they advance in a connected line that the hare cannot pierce. The hare\nwins only if the hounds break formation or violate the no-stalling rule. The\nexact value statement is sensitive to which traditional stalling/repetition rule\nis used. **[verify]** the precise value against a primary combinatorial analysis."}, {"id": 547, "type": "game", "source": "hare-and-hounds", "section": "Consensus on optimal play", "text": "Hare and Hounds — Consensus on optimal play\n\n- **Hounds: advance as an unbroken wall** — the three hounds must maintain contact (no gaps between adjacent pieces) as they advance; a single gap in the line lets the hare slip through and escape.\n- **Hounds: don't stall on side moves** — stalling rules prevent the hounds from playing indefinitely without advancing; every move should bring the formation one step closer to the end of the board, otherwise the hare wins on the stalling count.\n- **Hounds: coordinate the outer pair** — the two edge hounds must keep pace with the centre hound; letting an edge hound fall behind opens a flank lane for the hare.\n- **Hare: immediately probe the flanks** — the hare cannot wait in the middle while the hounds advance; sprint toward an edge and force the outer hound to make a choice between closing and leaving a gap.\n- **Hare: provoke asymmetry** — move in ways that require two different hounds to respond, making it impossible for both to act without one falling out of formation."}, {"id": 548, "type": "game", "source": "hare-and-hounds", "section": "Engines & current best play", "text": "Hare and Hounds — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. The game is small enough that any tree-search program solves it instantly.\n- **Strength:** Perfectly solvable by exhaustive search; state space is a few thousand positions.\n- **Where the proof / tablebase lives (if solved):** Folklore / multiple analyses; see [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** A classic teaching example of how a coordinated group confines a faster lone piece; the full game graph can be enumerated in milliseconds."}, {"id": 549, "type": "game", "source": "hare-and-hounds", "section": "Complexity", "text": "Hare and Hounds — Complexity\n\nTiny: the board and piece count make the full game graph small enough to\nenumerate by hand-assisted search."}, {"id": 550, "type": "game", "source": "hare-and-hounds", "section": "References", "text": "Hare and Hounds — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hare_and_Hounds) ([archive](http://web.archive.org/web/20251007041441/https://en.wikipedia.org/wiki/Hare_and_Hounds))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 551, "type": "game", "source": "hare-and-hounds", "section": "See also", "text": "Hare and Hounds — See also\n\n- [Fox and Geese](fox-and-geese.md) · [Tigers and Goats (Bagh-Chal)](tigers-and-goats.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 552, "type": "game", "source": "hashiwokakero", "section": "overview", "text": "Hashiwokakero\nBridge-building logic puzzle — NP-complete.\nSolution status: NP-complete. Game-theoretic value: Per-puzzle (unique solution by convention). Players: 1. Type: Solo logic puzzle."}, {"id": 553, "type": "game", "source": "hashiwokakero", "section": "Description", "text": "Hashiwokakero — Description\n\nHashiwokakero (橋をかけろ — \"build bridges\"; Nikoli, 1990) is a logic puzzle\non a grid of \"islands\" each labelled with a number. The solver draws straight\nhorizontal or vertical bridges connecting islands; each island's degree must\nequal its number, the bridge graph must be connected, and bridges may not\ncross."}, {"id": 554, "type": "game", "source": "hashiwokakero", "section": "Rules", "text": "Hashiwokakero — Rules\n\n1. Board: rectangular grid with some cells marked as islands; each island\n   carries a label 1–8.\n2. The solver draws bridges between islands subject to:\n   - Bridges run only **horizontally or vertically** between two distinct\n     islands.\n   - **At most two** bridges may connect the same pair of islands.\n   - Bridges may not cross other bridges and may not pass through islands.\n3. The number of bridge endpoints at each island must equal its label.\n4. The graph formed by all bridges must be **connected** (single component)."}, {"id": 555, "type": "game", "source": "hashiwokakero", "section": "Solution status", "text": "Hashiwokakero — Solution status\n\nHashiwokakero is **NP-complete** (Andersson 2009 and others)."}, {"id": 556, "type": "game", "source": "hashiwokakero", "section": "Consensus on optimal play", "text": "Hashiwokakero — Consensus on optimal play\n\n- **Max-capacity islands first** — an island labelled 8 in the interior must have exactly two bridges in all four directions; resolve these immediately with no deduction required.\n- **Force-fill constrained islands** — an island labelled N that has exactly N/2 neighbours (where each can bear at most 2 bridges) must use both bridges to every neighbour; identify and fill these early.\n- **Avoid premature isolation** — never draw bridges that would create a connected component cut off from the rest of the grid (no further bridge can reach in or out); connectivity is the hardest global constraint to undo.\n- **Use \"must connect\" logic near the boundary** — corner and edge islands have fewer neighbour directions; a label of 3 in a corner with only two neighbours forces at least one double-bridge.\n- **Propagate through chains** — once one bridge is placed, update all islands in both the row and column, rechecking forced moves from high-label islands; many puzzles cascade-solve with pure propagation.\n- **Branch only as a last resort** — well-designed Hashiwokakero puzzles are solvable without backtracking; if forced to guess, pick the choice that most constrains subsequent islands."}, {"id": 557, "type": "game", "source": "hashiwokakero", "section": "Engines & current best play", "text": "Hashiwokakero — Engines & current best play\n\n- **Strongest known program(s):** Simon Tatham's Bridges solver (part of his Portable Puzzle Collection) — constraint propagation with backtracking.\n- **Strength:** Solves all standard Nikoli-published puzzles (which are designed to have unique solutions deducible without backtracking).\n- **Where the proof / tablebase lives (if solved):** NP-completeness proof: Andersson (2009); individual puzzles have unique solutions by construction.\n- **Notes:** As a single-player puzzle, \"solving\" means finding the unique answer; the NP-hardness applies to arbitrary instances, not well-formed published puzzles."}, {"id": 558, "type": "game", "source": "hashiwokakero", "section": "Complexity", "text": "Hashiwokakero — Complexity\n\nNP-complete."}, {"id": 559, "type": "game", "source": "hashiwokakero", "section": "References", "text": "Hashiwokakero — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hashiwokakero) ([archive](http://web.archive.org/web/20260425160159/https://en.wikipedia.org/wiki/Hashiwokakero))\n- [Andersson. *Hashiwokakero is NP-complete*.](../references.md#slitherlink-np)\n- [Yato & Seta (2003). *Complexity and Completeness of Finding Another Solution and its Application to Puzzles*.](../references.md#selman-sudoku2003)"}, {"id": 560, "type": "game", "source": "hashiwokakero", "section": "See also", "text": "Hashiwokakero — See also\n\n- [Sudoku](sudoku.md) · [Slitherlink](slitherlink.md) · [Nonograms](nonograms.md)\n- Lexicon: [NP-completeness](../lexicon/README.md#np-completeness)"}, {"id": 561, "type": "game", "source": "havannah", "section": "overview", "text": "Havannah\nA connection game with three different winning shapes — solved only on small\nSolution status: Partially solved (small boards). Game-theoretic value: Unknown for tournament-size boards. Players: 2. Type: Partisan connection game."}, {"id": 562, "type": "game", "source": "havannah", "section": "Description", "text": "Havannah — Description\n\nPlayed on a hexagonal board of hexagons (tournament size: 10 cells per side).\nPlayers alternately place stones. A player wins by completing any one of three\nshapes: a **ring** (a loop around at least one cell), a **bridge** (connecting\ntwo of the six corners), or a **fork** (connecting three of the six edges)."}, {"id": 563, "type": "game", "source": "havannah", "section": "Solution status", "text": "Havannah — Solution status\n\nHavannah is **partially solved**: exhaustive search has settled small boards\n(side length up to roughly 4–5 cells, depending on the study), and these tend\nto be first-player wins. The tournament-size board is far too large for current\nexhaustive methods and is **unsolved**. Havannah was for some years a noted\nchallenge for game AI — its large branching factor and the difficulty of\nrecognising the ring/bridge/fork goals made it resistant to classical search —\nbefore Monte-Carlo tree search and later neural methods produced strong\nplayers."}, {"id": 564, "type": "game", "source": "havannah", "section": "Consensus on optimal play", "text": "Havannah — Consensus on optimal play\n\n- **Threaten multiple win shapes simultaneously** — a stone that advances toward both a fork and a bridge is much harder to counter than one aimed at a single goal; forcing the opponent to block two threats at once is the central attacking principle.\n- **Corners are double-edged** — a corner counts as one edge point for a fork AND as a corner piece for a bridge; occupying or contesting corners early gives threats in both categories.\n- **Respond to ring attempts aggressively** — rings require encircling at least one cell; if the opponent is building a loose loop, inserting a stone inside the potential ring breaks it; don't let rings grow uncontested.\n- **Use the swap rule wisely** — if swap is in effect, the first move should occupy a modestly strong but not obviously dominant cell; too-powerful first moves will be swapped.\n- **Keep groups connected** — disconnected stones give the opponent opportunities to cut and isolate; a network of stones with short bridge-connections (moving to adjacent hexes through two cells) maintains both fork and ring potential.\n- **Deny the opponent's key junction cells** — cells where several of the opponent's groups would connect (completing a fork or bridge) are worth contesting even at material cost."}, {"id": 565, "type": "game", "source": "havannah", "section": "Engines & current best play", "text": "Havannah — Engines & current best play\n\n- **Strongest known program(s):** No single dominant public engine; Wanderer and various MCTS-based programs competed in computer games tournaments (e.g., ICGA).\n- **Strength:** Strong amateur to competitive; modern MCTS engines play well above casual human level.\n- **Where the proof / tablebase lives (if solved):** Small boards (up to roughly side-5) solved by exhaustive search; tournament board unsolved.\n- **Notes:** The swap rule is standard in competitive play; first player is generally considered advantaged on full boards."}, {"id": 566, "type": "game", "source": "havannah", "section": "Complexity", "text": "Havannah — Complexity\n\nTournament board (271 cells): state and game-tree complexity comparable to\nlarge connection games; precise standardised figures are not well established."}, {"id": 567, "type": "game", "source": "havannah", "section": "References", "text": "Havannah — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Havannah) ([archive](http://web.archive.org/web/20250906172227/https://en.wikipedia.org/wiki/Havannah))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 568, "type": "game", "source": "havannah", "section": "See also", "text": "Havannah — See also\n\n- [Hex](hex.md) · [Y](y.md) · [TwixT](twixt.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity) · [partially solved](../lexicon/README.md#solved-game)"}, {"id": 569, "type": "game", "source": "heads-up-limit-holdem", "section": "overview", "text": "Heads-up limit hold'em\nThe first competitively-played imperfect-information game to be essentially\nSolution status: Essentially weakly solved (ε-Nash equilibrium, ε below lifetime-detectable). Game-theoretic value: Near-draw — a small first-player (dealer) advantage. Players: 2. Type: Stochastic imperfect-information game (poker)."}, {"id": 570, "type": "game", "source": "heads-up-limit-holdem", "section": "Description", "text": "Heads-up limit hold'em — Description\n\nTwo-player Texas hold'em with **fixed bet sizes** (limit betting): each player is\ndealt two private cards, five community cards are revealed across four betting\nrounds, and bets/raises are capped per round. Because hole cards are hidden and\nthe deck is shuffled, this is a game of both **imperfect information** and\n**chance** — \"solving\" it means computing a Nash-equilibrium strategy, not a\nsingle win/lose/draw value."}, {"id": 571, "type": "game", "source": "heads-up-limit-holdem", "section": "Solution status", "text": "Heads-up limit hold'em — Solution status\n\nHeads-up limit hold'em is **essentially weakly solved**.\n[Bowling, Burch, Johanson & Tammelin (2015)](../references.md#bowling2015), with\nthe program **Cepheus**, used a variant of counterfactual regret minimisation\n(CFR+) running on a large cluster to compute a strategy whose exploitability is\n**ε-small** — small enough that it could not be beaten with statistical\nsignificance even over a human lifetime of play. That is not a *perfect*\nequilibrium (the qualifier \"essentially\"), but it is as close as is practically\nmeaningful. The result: the game is a **near-draw**, with a measured **small\nadvantage to the dealer** (first player). It was the first nontrivial\nimperfect-information game played competitively by humans to be solved to this\nstandard."}, {"id": 572, "type": "game", "source": "heads-up-limit-holdem", "section": "Consensus on optimal play", "text": "Heads-up limit hold'em — Consensus on optimal play\n\n- **Mix your actions to stay unexploitable** — Nash-equilibrium play requires randomising bet/call/fold frequencies so that the opponent cannot profitably deviate; pure strategies (always bet with X, always fold with Y) are exploitable.\n- **The dealer (BTN) has a persistent edge** — acting last on every post-flop street is a structural advantage; the button should play a slightly wider range and apply more pressure in position.\n- **Defend big blind wide in limit** — because pot odds in limit are fixed and generous when facing a raise, the big blind must call with a wide range to prevent the button from profitably raising with any two cards.\n- **Thin value bets are correct** — in limit hold'em, the fixed bet-to-pot ratio is small; betting one pair for thin value on the river is correct far more often than in no-limit, where sizing risk is larger.\n- **Cepheus's equilibrium mixes heavily on the river** — even with strong hands the correct equilibrium strategy sometimes checks back to protect checking ranges; don't polarise bet ranges completely.\n- **Statistical exploitability over a human lifetime is near zero** — the Cepheus solution proved that even a slightly sub-optimal strategy (within ε of equilibrium) cannot be beaten with statistical significance in any realistic number of hands."}, {"id": 573, "type": "game", "source": "heads-up-limit-holdem", "section": "Engines & current best play", "text": "Heads-up limit hold'em — Engines & current best play\n\n- **Strongest known program(s):** Cepheus (Bowling, Burch, Johanson & Tammelin, 2015) — CFR+ equilibrium solver.\n- **Strength:** Essentially unexploitable; below the statistical detection threshold over a human lifetime.\n- **Where the proof / tablebase lives (if solved):** [Bowling et al. (2015)](../references.md#bowling2015); Cepheus strategy table was publicly released by the University of Alberta.\n- **Notes:** First nontrivial competitively-played imperfect-information game to be essentially solved; the result is a near-draw with a small dealer advantage."}, {"id": 574, "type": "game", "source": "heads-up-limit-holdem", "section": "Complexity", "text": "Heads-up limit hold'em — Complexity\n\nAbout 3.2 × 10^14 decision points were tracked during the solve; the full game\nhas roughly 10^17 states. The achievement was managing this with CFR+ and disk\nstorage, not enumerating the game outright."}, {"id": 575, "type": "game", "source": "heads-up-limit-holdem", "section": "References", "text": "Heads-up limit hold'em — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Texas_hold_%27em) ([archive](http://web.archive.org/web/20260506223844/https://en.wikipedia.org/wiki/Texas_hold_%27em))\n- [Bowling, Burch, Johanson & Tammelin (2015). *Heads-up Limit Hold'em Poker is Solved*.](../references.md#bowling2015)"}, {"id": 576, "type": "game", "source": "heads-up-limit-holdem", "section": "See also", "text": "Heads-up limit hold'em — See also\n\n- [Heads-up no-limit hold'em](heads-up-nolimit-holdem.md) · [Liar's dice](liars-dice.md)\n- Lexicon: [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved) · [Nash equilibrium](../lexicon/README.md#nash-equilibrium) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 577, "type": "game", "source": "heads-up-nolimit-holdem", "section": "overview", "text": "Heads-up no-limit hold'em\nSuperhuman AI exists and beat top professionals — but the game itself, with\nSolution status: Unsolved (superhuman play exists; no equilibrium computed for the full game). Game-theoretic value: Unknown — small first-player edge expected. Players: 2. Type: Stochastic imperfect-information game (poker)."}, {"id": 578, "type": "game", "source": "heads-up-nolimit-holdem", "section": "Description", "text": "Heads-up no-limit hold'em — Description\n\nTwo-player Texas hold'em with **no-limit betting**: at any point a player may bet\nany amount up to their entire stack. Like the limit version it has hidden hole\ncards and a shuffled deck, but the **unrestricted bet sizing** makes the action\nspace effectively continuous, exploding the game tree far beyond the limit\nvariant."}, {"id": 579, "type": "game", "source": "heads-up-nolimit-holdem", "section": "Solution status", "text": "Heads-up no-limit hold'em — Solution status\n\nHeads-up no-limit hold'em is **not solved**. Its game tree is astronomically\nlarger than limit hold'em — roughly 10^160 states, with a continuum of legal bet\nsizes — so no ε-Nash equilibrium for the full game has been computed. What was\nachieved in 2017 is **superhuman play**: [Libratus](../references.md#brown-libratus2018)\n(Brown & Sandholm) and [DeepStack](../references.md#moravcik-deepstack2017)\n(Moravčík et al.) both decisively beat top human professionals, using\nabstraction, continual re-solving and CFR-based techniques. But beating\nprofessionals is [strong play, not solving](../lexicon/README.md#solving-vs-strong-play):\nthese programs use bet-size abstractions and depth-limited re-solving, and their\nresidual exploitability for the full continuous game is not bounded to the\nlifetime-undetectable standard that [limit hold'em](heads-up-limit-holdem.md)\nreached."}, {"id": 580, "type": "game", "source": "heads-up-nolimit-holdem", "section": "Consensus on optimal play", "text": "Heads-up no-limit hold'em — Consensus on optimal play\n\n- **Use solvers to build range-balanced strategies** — modern GTO solvers (PioSOLVER, GTO+, etc.) solve abstracted bet-tree versions of specific spots; professional players study solver outputs and internalise frequency-based strategies rather than playing purely by feel.\n- **Bet-size selection is a strategic weapon** — unlike limit hold'em, choosing between small, medium, and pot-sized bets lets you polarise or protect your range; solvers show that different board textures call for different sizing menus.\n- **Balance your bluffs with your value bets** — a player who bets the river only with strong hands is exploited by folding; the equilibrium mixes bluffs into every bet size so that the opponent cannot profitably deviate.\n- **Positional advantage is amplified by stack depth** — acting last post-flop allows the IP (in-position) player to control pot size and choose when to bluff; deep stacks magnify this advantage because the threat of large future bets is more credible.\n- **3-bet / 4-bet ranges must include bluffs** — preflop re-raising ranges that contain only strong hands are easily countered by folding everything below the threshold; mix in suited connectors and suited aces as bluff candidates.\n- **Exploit population leaks, not GTO** — against recreational players, pure GTO play leaves money on the table; identify systematic over-folds or over-calls and deviate from equilibrium to maximise expected value against that specific opponent."}, {"id": 581, "type": "game", "source": "heads-up-nolimit-holdem", "section": "Engines & current best play", "text": "Heads-up no-limit hold'em — Engines & current best play\n\n- **Strongest known program(s):** Libratus ([Brown & Sandholm, 2018](../references.md#brown-libratus2018)) and DeepStack ([Moravčík et al., 2017](../references.md#moravcik-deepstack2017)) — CFR-based re-solving with abstraction; GTO study tools include PioSOLVER and GTO+.\n- **Strength:** Super-human at defeating top professionals in tournament-format matches.\n- **Where the proof / tablebase lives (if solved):** Not formally solved; no full-game ε-Nash equilibrium computed — see [heads-up-limit-holdem.md](heads-up-limit-holdem.md) for the solved limit variant.\n- **Notes:** Standard professional training now uses solver-generated strategies; the \"equilibrium\" studied is always an approximation over a discretised bet-size tree."}, {"id": 582, "type": "game", "source": "heads-up-nolimit-holdem", "section": "Complexity", "text": "Heads-up no-limit hold'em — Complexity\n\nOn the order of 10^160 game states, with an effectively continuous action space.\nThis is the central reason the full game is unsolved while the limit variant is\nessentially solved."}, {"id": 583, "type": "game", "source": "heads-up-nolimit-holdem", "section": "References", "text": "Heads-up no-limit hold'em — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Texas_hold_%27em) ([archive](http://web.archive.org/web/20260506223844/https://en.wikipedia.org/wiki/Texas_hold_%27em))\n- [Brown & Sandholm (2018). *Superhuman AI for heads-up no-limit poker: Libratus beats top professionals*.](../references.md#brown-libratus2018)\n- [Moravčík et al. (2017). *DeepStack: Expert-level artificial intelligence in heads-up no-limit poker*.](../references.md#moravcik-deepstack2017)"}, {"id": 584, "type": "game", "source": "heads-up-nolimit-holdem", "section": "See also", "text": "Heads-up no-limit hold'em — See also\n\n- [Heads-up limit hold'em](heads-up-limit-holdem.md) · [Liar's dice](liars-dice.md)\n- Lexicon: [Nash equilibrium](../lexicon/README.md#nash-equilibrium) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play) · [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 585, "type": "game", "source": "hex", "section": "overview", "text": "Hex\nThe connection game that *cannot* be drawn — proven a first-player win, but\nSolution status: Ultra-weakly solved (standard 11×11); weakly solved for small boards (through ~10×10). Game-theoretic value: First-player win (on any symmetric n×n board). Players: 2. Type: Partisan connection game."}, {"id": 586, "type": "game", "source": "hex", "section": "Description", "text": "Hex — Description\n\nPlayed on a rhombic board of hexagons (commonly 11×11). One player connects the\ntop and bottom edges with a chain of their stones, the other connects left and\nright. Players alternately place one stone; stones are never moved or removed."}, {"id": 587, "type": "game", "source": "hex", "section": "Solution status", "text": "Hex — Solution status\n\nHex has a striking split between *value* and *strategy*.\n\n- **Hex cannot end in a draw** — a filled board always contains exactly one\n  winning chain ([Gale, 1979](../references.md#gale1979) ties this to the\n  Brouwer fixed-point theorem).\n- Because there are no draws, the [strategy-stealing argument](../lexicon/README.md#strategy-stealing)\n  ([Nash, c. 1949](../references.md#nash-hex)) proves the **first player has a\n  winning strategy** on any symmetric board. This makes Hex\n  **[ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved)** — the\n  value is known — but the proof is non-constructive and exhibits *no*\n  strategy.\n- **Explicit (weak) solutions** have been found for small boards: Jing Yang\n  gave winning strategies for 7×7, 8×8, and 9×9 in the early 2000s, and Ryan\n  Hayward's group weakly solved 8×8 ([Henderson, Arneson & Hayward, 2009](../references.md#hayward-hex2009))\n  and later 9×9 and 10×10.\n\nThe standard **11×11** board is still only ultra-weakly solved: we know the\nfirst player wins, but no full winning strategy is known. (The opening-move\n\"swap\" rule is used in play precisely to neutralise this proven first-player\nadvantage.)"}, {"id": 588, "type": "game", "source": "hex", "section": "Consensus on optimal play", "text": "Hex — Consensus on optimal play\n\n- **Virtual connections are the currency of Hex** — two groups of the same colour are \"virtually connected\" if they can be joined regardless of the opponent's next move; maintaining virtual connections across the board is the core calculation.\n- **The acute corners belong to no one, and to both** — the corner cells are weak entry points for both sides; the critical real estate is the cells adjacent to the corner that control the corner approaches.\n- **Ladders and ladder escapes decide games** — a ladder (a forced sequence pushing a chain along an edge) is unavoidable unless a pre-placed \"escape\" stone breaks it; recognising potential ladders and placing escape stones early is essential.\n- **Take the short-path cells** — cells that lie on most shortest winning paths between your two sides have the highest value; prioritise them and contest the opponent's equivalent cells.\n- **The swap rule changes first-move selection** — with swap in effect, the first move should be on a moderately strong cell; too-central or too-corner openings will be swapped; the classic swappable cell is the exact centre.\n- **Block by building, not by responding** — placing a stone that advances your own connection while also threatening the opponent's chain is more efficient than pure defence; pure response play cedes tempo."}, {"id": 589, "type": "game", "source": "hex", "section": "Engines & current best play", "text": "Hex — Engines & current best play\n\n- **Strongest known program(s):** Mohex (Ryan Hayward's group) — Monte Carlo tree search with hex-specific knowledge; MoHex-3HNN adds neural networks.\n- **Strength:** Super-human on 11×11; top engines vastly exceed expert human play.\n- **Where the proof / tablebase lives (if solved):** 8×8–10×10 weakly solved by Hayward et al. ([Henderson, Arneson & Hayward, 2009](../references.md#hayward-hex2009)); 11×11 ultra-weakly solved (first-player win proven, strategy unknown).\n- **Notes:** The swap rule is standard in competitive 11×11 play; without it the first player wins with best play."}, {"id": 590, "type": "game", "source": "hex", "section": "Complexity", "text": "Hex — Complexity\n\nState-space ~10^56, game-tree ~10^98 for 11×11\n([van den Herik et al., 2002](../references.md#vandenherik2002)). Generalised\nHex is PSPACE-complete."}, {"id": 591, "type": "game", "source": "hex", "section": "References", "text": "Hex — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hex_(board_game)) ([archive](http://web.archive.org/web/20260511121115/https://en.wikipedia.org/wiki/Hex_(board_game)))\n- [Nash, J. — strategy-stealing argument for Hex.](../references.md#nash-hex)\n- [Gale, D. (1979). *The Game of Hex and the Brouwer Fixed-Point Theorem*.](../references.md#gale1979)\n- [Henderson, Arneson & Hayward (2009). *Solving 8×8 Hex*.](../references.md#hayward-hex2009)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 592, "type": "game", "source": "hex", "section": "See also", "text": "Hex — See also\n\n- [Y](y.md) · [Havannah](havannah.md) · [TwixT](twixt.md) · [Bridg-it](bridg-it.md) · [Chomp](chomp.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing) · [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved)"}, {"id": 593, "type": "game", "source": "hexapawn", "section": "overview", "text": "Hexapawn\nA 3×3 pawns-only chess miniature, famous as the game a matchbox computer can\nSolution status: Strongly solved. Game-theoretic value: Second-player win (standard 3×3 board). Players: 2. Type: Partisan combinatorial game (chess-derived)."}, {"id": 594, "type": "game", "source": "hexapawn", "section": "Description", "text": "Hexapawn — Description\n\nPlayed on a 3×3 board, each side starting with three pawns on its back rank.\nPawns move and capture exactly as chess pawns (one step forward, capture one\nstep diagonally). A player wins by advancing a pawn to the far rank, by\ncapturing all enemy pawns, or by leaving the opponent with no legal move.\nIntroduced by [Gardner (1962)](../references.md#gardner-hexapawn1962)."}, {"id": 595, "type": "game", "source": "hexapawn", "section": "Solution status", "text": "Hexapawn — Solution status\n\nHexapawn is **strongly solved**. The position space is tiny — only a few dozen\nreachable positions — so the entire game tree is trivial to evaluate exhaustively.\nWith perfect play the **second player wins** on the standard 3×3 board.\n\nHexapawn is historically important less for the result than as a teaching\ndevice: Gardner used it to describe a *learning* machine, and Donald Michie's\ncontemporaneous MENACE ([Michie, 1963](../references.md#michie-menace1963))\nshowed a matchbox-and-bead \"computer\" converging on perfect play of such small\ngames — an early, tangible demonstration of reinforcement learning."}, {"id": 596, "type": "game", "source": "hexapawn", "section": "Consensus on optimal play", "text": "Hexapawn — Consensus on optimal play\n\n- **Second player's core defence: mirror or block** — in the standard 3×3 start, White (first player) has only three opening moves; for each one the optimal Black response is known and can be memorised; Black's goal is to either advance a pawn to promotion or leave White with no legal move.\n- **Avoid diagonal captures that open lanes for promotion** — capturing an opponent's pawn can clear a path for their adjacent pawn to advance; count promotable pawn lines before capturing.\n- **Block all three advance lanes** — with three files, controlling the path of each opposing pawn is the whole game; a pawn that reaches the far rank wins immediately, so no lane can be left open.\n- **The second player wins by steering into the unique drawn/winning lines** — the full game tree is tiny; memorise the three or four key branching points and the correct response at each; there is nothing more."}, {"id": 597, "type": "game", "source": "hexapawn", "section": "Engines & current best play", "text": "Hexapawn — Engines & current best play\n\n- **Strongest known program(s):** Any minimax search, including MENACE (Michie's matchbox computer, 1963) — the entire game tree is trivially small.\n- **Strength:** Perfect play is achievable by exhaustive search and learnable by humans with minimal study.\n- **Where the proof / tablebase lives (if solved):** [Gardner (1962)](../references.md#gardner-hexapawn1962); complete game tree solvable by hand.\n- **Notes:** Famous as a reinforcement-learning teaching example (MENACE); the result (second-player win) is a memorisable fact, not a computational challenge."}, {"id": 598, "type": "game", "source": "hexapawn", "section": "Complexity", "text": "Hexapawn — Complexity\n\nNegligible; solvable by hand."}, {"id": 599, "type": "game", "source": "hexapawn", "section": "References", "text": "Hexapawn — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hexapawn) ([archive](http://web.archive.org/web/20260405012433/https://en.wikipedia.org/wiki/Hexapawn))\n- [Gardner, M. (1962). *Mathematical Games* (Hexapawn).](../references.md#gardner-hexapawn1962)\n- [Michie, D. (1963). *Experiments on the mechanization of game-learning* (MENACE).](../references.md#michie-menace1963)"}, {"id": 600, "type": "game", "source": "hexapawn", "section": "See also", "text": "Hexapawn — See also\n\n- [Minichess](minichess.md) · [Tic-tac-toe](tic-tac-toe.md) · [Breakthrough](breakthrough.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 601, "type": "game", "source": "hive", "section": "overview", "text": "Hive\nBoardless insect-piece game — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan boardless placement-and-movement game."}, {"id": 602, "type": "game", "source": "hive", "section": "Description", "text": "Hive — Description\n\nHive (John Yianni, 2001) is a 2-player abstract played without a board:\nhexagonal insect tiles are placed and moved to surround the opposing\n**queen**. The lack of a board means the playing area expands with the\n\"hive\" of placed pieces."}, {"id": 603, "type": "game", "source": "hive", "section": "Rules", "text": "Hive — Rules\n\n1. No board: pieces are hexagonal tiles placed in a connected cluster.\n2. Each player has 11 tiles: 1 Queen Bee, 2 Spiders, 2 Beetles, 3 Grasshoppers,\n   3 Soldier Ants (plus expansion pieces in extensions, not used here).\n3. On each turn a player either **places** a new tile from their reserve next\n   to a friendly tile (and not next to any opposing tile after the first\n   move), or **moves** one tile already in play.\n4. The Queen Bee must be placed by each player's fourth move.\n5. Piece movement:\n   - **Queen**: 1 step around the hive edge.\n   - **Spider**: exactly 3 steps around the hive edge.\n   - **Beetle**: 1 step including the option of climbing on top of any\n     stack, blocking the covered piece.\n   - **Grasshopper**: leaps in a straight line over a contiguous run of\n     pieces, landing on the empty cell beyond.\n   - **Soldier Ant**: any number of steps around the hive edge.\n6. The **one-hive** rule: no move may disconnect the hive, and at no point in\n   a slide may the moving tile lose contact with the hive.\n7. A player wins when the **opposing Queen Bee** is completely surrounded by\n   six tiles of either colour."}, {"id": 604, "type": "game", "source": "hive", "section": "Solution status", "text": "Hive — Solution status\n\nHive is **not solved**. Strong engines exist (Mzinga, Nokamute) but no\ngame-theoretic value is published."}, {"id": 605, "type": "game", "source": "hive", "section": "Consensus on optimal play", "text": "Hive — Consensus on optimal play\n\n- **Place your queen early but not first** — delaying queen placement to the fourth move is the latest allowed; placing it on move 2 or 3 is generally stronger than waiting, as it unlocks movement options; but never place it as the very first tile or it is immediately targetable.\n- **Ants are the most mobile attackers** — soldier ants can reach any position on the hive perimeter in one move; getting ants into attacking positions around the opponent's queen while keeping your own queen shielded is the dominant mid-game goal.\n- **Beetles pin queens** — a beetle climbing onto the queen immobilises it and begins surrounding it; threatening a beetle pin forces the opponent to keep escape hexes open.\n- **Grasshoppers threaten gaps** — a grasshopper can jump over a contiguous line and land in a gap adjacent to the queen; keep at least one grasshopper primed for a queen-surrounding jump.\n- **One-hive rule creates tactical constraints** — identify \"pillars\" (pieces whose removal would disconnect the hive); these pieces cannot move, which limits your options; force the opponent into positions where their key pieces become pillars.\n- **Surround your queen with your own pieces** — a queen surrounded by two of your own soldiers on her flanks is harder to complete-surround; keeping friendly blocking tiles adjacent to your queen buys time while you attack."}, {"id": 606, "type": "game", "source": "hive", "section": "Engines & current best play", "text": "Hive — Engines & current best play\n\n- **Strongest known program(s):** Mzinga and Nokamute — dedicated Hive engines using MCTS or alpha-beta search.\n- **Strength:** Strong amateur; engines exceed casual human play but the gap is smaller than in solved games.\n- **Where the proof / tablebase lives (if solved):** Not solved; no tablebase or published game-theoretic value.\n- **Notes:** A small first-player advantage is suspected from engine self-play but not proven; opening theory for the base game is well developed in the competitive community."}, {"id": 607, "type": "game", "source": "hive", "section": "Complexity", "text": "Hive — Complexity\n\nLarge."}, {"id": 608, "type": "game", "source": "hive", "section": "References", "text": "Hive — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hive_(game)) ([archive](http://web.archive.org/web/20260421231217/https://en.wikipedia.org/wiki/Hive_(game)))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 609, "type": "game", "source": "hive", "section": "See also", "text": "Hive — See also\n\n- [Hex](hex.md) · [Onyx](onyx.md) · [Cathedral](cathedral.md)\n- Lexicon: [perfect information](../lexicon/README.md#perfect-information)"}, {"id": 610, "type": "game", "source": "horde-chess", "section": "overview", "text": "Horde chess\nAsymmetric chess variant: a single army versus a horde of pawns — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan asymmetric chess variant."}, {"id": 611, "type": "game", "source": "horde-chess", "section": "Description", "text": "Horde chess — Description\n\nIn Horde chess, Black plays a standard chess army and White plays 36 pawns\n(four ranks of pawns). White wins by checkmating the Black king; Black wins\nby capturing every White pawn."}, {"id": 612, "type": "game", "source": "horde-chess", "section": "Rules", "text": "Horde chess — Rules\n\n1. Setup: Black has the orthodox starting army; White has 36 pawns occupying\n   ranks 1–4 (with the second rank shifted to fill out the 36-pawn pattern in\n   the common online variant).\n2. White moves first. White pawns move and capture as ordinary pawns; pawns on\n   their starting rank may move one or two squares.\n3. White has **no king** — White is not checked, and is mated only when no\n   pawns remain.\n4. Black plays orthodox chess; Black wins by removing all White pawns.\n5. White wins by delivering checkmate to the Black king.\n6. Draws by stalemate are draws as usual."}, {"id": 613, "type": "game", "source": "horde-chess", "section": "Solution status", "text": "Horde chess — Solution status\n\nHorde chess is **not solved**. Engines play it well but its asymmetry and\nmaterial imbalance defeat standard endgame theory."}, {"id": 614, "type": "game", "source": "horde-chess", "section": "Consensus on optimal play", "text": "Horde chess — Consensus on optimal play\n\nHeuristics from strong play (Black, the chess side, is widely judged to win at top level despite material deficit):\n\n- **For Black: never trade pieces for pawns one-for-one** — Black is outnumbered 36-to-16; even-material trades favour the horde. Black must use pieces as long-range mowing machines, often with knights and bishops attacking *backward* into the horde from outside its reach.\n- **For Black: target the back ranks** — the horde must keep its front line intact to threaten promotions; piling up rooks behind enemy lines forces it to concede tempo.\n- **For Black: blockade promotion squares** — a knight on the seventh rank stops a file of pawns indefinitely.\n- **For White (horde): march in waves, not lines** — keep the front rank advancing only when the second can immediately fill gaps; isolated advanced pawns get plucked.\n- **For White: prefer captures that gain tempo over promotions** — promoting a pawn is only valuable when the new queen survives the next move."}, {"id": 615, "type": "game", "source": "horde-chess", "section": "Engines & current best play", "text": "Horde chess — Engines & current best play\n\n- **Strongest known programs:** [Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish)) (open source); engine analysis is available on [Lichess](https://lichess.org/variant/horde).\n- **Strength:** Super-human at both sides.\n- **Notes:** Engine self-play consistently shows Black (the army side) winning a clear majority — counter to naive material intuition."}, {"id": 616, "type": "game", "source": "horde-chess", "section": "Complexity", "text": "Horde chess — Complexity\n\nSimilar to chess."}, {"id": 617, "type": "game", "source": "horde-chess", "section": "References", "text": "Horde chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/List_of_chess_variants) ([archive](http://web.archive.org/web/20260508095621/https://en.wikipedia.org/wiki/List_of_chess_variants))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 618, "type": "game", "source": "horde-chess", "section": "See also", "text": "Horde chess — See also\n\n- [Chess](chess.md) · [King of the Hill](king-of-the-hill.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 619, "type": "game", "source": "international-draughts", "section": "overview", "text": "International draughts\nDraughts on a 10×10 board with flying kings — much bigger than English\nSolution status: Unsolved (endgame tables computed). Game-theoretic value: Unknown (widely believed to be a draw). Players: 2. Type: Partisan board game."}, {"id": 620, "type": "game", "source": "international-draughts", "section": "Description", "text": "International draughts — Description\n\nPlayed on the 50 dark squares of a 10×10 board, 20 pieces per side. Men capture\nboth forward and backward; captures are compulsory and the **maximum capture**\nmust be taken; kings are \"flying\" — they move and capture any distance along a\ndiagonal. These rules make captures long and forcing."}, {"id": 621, "type": "game", "source": "international-draughts", "section": "Solution status", "text": "International draughts — Solution status\n\nInternational draughts is **unsolved**. It is roughly **ten orders of magnitude\nlarger** than [English checkers](checkers.md) (~10^30 vs ~5 × 10^20 positions),\nwhich puts a checkers-style weak solution well out of current reach. Substantial\n**endgame tablebases** have been computed (covering positions up to several\npieces), giving exact solutions to those sub-games and supporting very strong\nengine play; competitive games between top humans and engines draw at a very\nhigh rate. But the game-theoretic value of the standard opening has not been\nproven."}, {"id": 622, "type": "game", "source": "international-draughts", "section": "Consensus on optimal play", "text": "International draughts — Consensus on optimal play\n\n- **Maximum-capture obligation is a tactical fulcrum** — you must take the largest number of pieces in a capture sequence; skilled play involves setting up \"shots\" that force the opponent into a long capture that leaves their pieces badly positioned after it completes.\n- **Flying kings control the long diagonals** — a king on an open diagonal threatens pieces at any range and restricts the opponent's movement; centralise kings to long diagonals that cross the board.\n- **Guard against backwards captures on your men** — unlike English checkers, men must capture backwards; leaving a man that can be captured backwards while extending your own chain weakens your structure.\n- **Endgame: king vs. two or three men is tablebase-decided** — many such endings are well-studied; knowing the theoretical outcome from your endgame tables prevents wasting moves in drawn or lost positions.\n- **Avoid isolated men on the wings** — wing pieces are harder to retreat to safety and easier to surround; maintain a connected front that can shift laterally.\n- **Tempo in the opening determines midgame piece activity** — losing tempo by retreating or making obligatory bad captures early lets the opponent seize the long diagonals; opening systems focus on compact, tempo-preserving development."}, {"id": 623, "type": "game", "source": "international-draughts", "section": "Engines & current best play", "text": "International draughts — Engines & current best play\n\n- **Strongest known program(s):** Kingsrow International (Ed Trice variant), Damage, and Tornado — all use alpha-beta search with extensive endgame tablebases.\n- **Strength:** Super-human; engines vastly exceed top human play at elite time controls.\n- **Where the proof / tablebase lives (if solved):** Endgame tablebases computed for positions up to roughly 6–8 pieces; game-theoretic value of starting position unproven.\n- **Notes:** Elite human-vs-engine matches draw at very high rates, reinforcing the consensus that the starting position is drawn, though this is not proven."}, {"id": 624, "type": "game", "source": "international-draughts", "section": "Complexity", "text": "International draughts — Complexity\n\nState-space ~10^30; game-tree ~10^54\n([van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 625, "type": "game", "source": "international-draughts", "section": "References", "text": "International draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/International_draughts) ([archive](http://web.archive.org/web/20260511121115/https://en.wikipedia.org/wiki/International_draughts))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [Schaeffer, J. et al. (2007). *Checkers Is Solved*.](../references.md#schaeffer2007) (the smaller, solved cousin)"}, {"id": 626, "type": "game", "source": "international-draughts", "section": "See also", "text": "International draughts — See also\n\n- [Checkers (English draughts)](checkers.md) · [Fanorona](fanorona.md) · [Lasca](lasca.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 627, "type": "game", "source": "italian-draughts", "section": "overview", "text": "Italian draughts\n8×8 draughts variant with men that cannot capture kings — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan draughts."}, {"id": 628, "type": "game", "source": "italian-draughts", "section": "Description", "text": "Italian draughts — Description\n\nItalian draughts is played on an 8×8 board with the dark squares oriented to\nthe right of each player (a 90° rotation of English placement). Men cannot\ncapture kings, and the rules require players to capture the maximum number of\npieces possible."}, {"id": 629, "type": "game", "source": "italian-draughts", "section": "Rules", "text": "Italian draughts — Rules\n\n1. Board: 8×8, with dark squares to each player's right. Each side has 12 men.\n2. Men move and capture diagonally forward only; capture is mandatory.\n3. **A man may never capture a king.**\n4. When multiple captures are available, the player must choose the line that\n   captures the **most pieces**; ties are broken by preferring sequences with\n   more kings captured (the \"majority rule\").\n5. Men reaching the back rank promote to king; kings move one square in any\n   diagonal direction (short kings — not flying).\n6. A player who cannot move loses."}, {"id": 630, "type": "game", "source": "italian-draughts", "section": "Solution status", "text": "Italian draughts — Solution status\n\nItalian draughts is **not solved**. The smaller mobility of kings and the\nmajority rule make engine analysis distinct from English draughts."}, {"id": 631, "type": "game", "source": "italian-draughts", "section": "Consensus on optimal play", "text": "Italian draughts — Consensus on optimal play\n\n- **Protect your kings from men — but not the reverse** — since men cannot capture kings, a king is safe from any man's attack; conversely, your men are vulnerable to the opponent's kings; concentrate on promoting men and shielding them until they crown.\n- **Majority-rule triggers force long capture sequences** — when multiple captures are available you must take the most pieces, and when tied, prefer sequences capturing more kings; plan sequences with this in mind so the obligation benefits you, not your opponent.\n- **Short kings require close-range tactics** — Italian kings move only one square at a time (no flying); king-vs-king endings are slower and more positional than in international draughts; avoid exchanging men for kings unless you gain a structural advantage.\n- **Control the centre with men, not kings** — centralised men advance to promotion faster and cannot be captured by kings; push central men forward early while keeping flank men as backup.\n- **Force the opponent's men to capture your kings** — the opponent's men cannot capture your kings; use this asymmetry by placing kings in paths of advancing enemy men, which the opponent cannot remove with those men, creating road-blocks."}, {"id": 632, "type": "game", "source": "italian-draughts", "section": "Engines & current best play", "text": "Italian draughts — Engines & current best play\n\n- **Strongest known program(s):** No widely-known public engine specific to Italian draughts; general draughts programs adapted to its rules are used in the Italian competitive scene.\n- **Strength:** Not benchmarked publicly; engine play is stronger than amateur human level.\n- **Where the proof / tablebase lives (if solved):** Not solved; no published tablebase.\n- **Notes:** Small competitive scene in Italy; the \"men cannot capture kings\" rule is the defining asymmetry that separates Italian from English and international draughts."}, {"id": 633, "type": "game", "source": "italian-draughts", "section": "Complexity", "text": "Italian draughts — Complexity\n\nSimilar to English draughts."}, {"id": 634, "type": "game", "source": "italian-draughts", "section": "References", "text": "Italian draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Italian_draughts) ([archive](http://web.archive.org/web/20260111103728/https://en.wikipedia.org/wiki/Italian_draughts))\n- [Schaeffer et al. (2007). *Checkers is Solved*.](../references.md#schaeffer2007) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 635, "type": "game", "source": "italian-draughts", "section": "See also", "text": "Italian draughts — See also\n\n- [English draughts](checkers.md) · [Russian draughts](russian-draughts.md) · [Turkish draughts](turkish-draughts.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 636, "type": "game", "source": "janggi", "section": "overview", "text": "Janggi\nKorean chess — closely related to xiangqi, and likewise unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 637, "type": "game", "source": "janggi", "section": "Description", "text": "Janggi — Description\n\nPlayed on a 9×10 board derived from [xiangqi](xiangqi.md), but with several rule\ndifferences: there is **no river**, pieces sit and move on intersections, the\nElephant has a longer move, players may **pass**, and a distinctive opening\n\"setup\" choice lets each side swap the starting squares of their horse and\nelephant on either wing. A bare-general \"draw by counting\" (material score)\nrule resolves long endgames."}, {"id": 638, "type": "game", "source": "janggi", "section": "Solution status", "text": "Janggi — Solution status\n\nJanggi is **unsolved**. Its complexity is broadly comparable to its relative\n[xiangqi](xiangqi.md) — on the order of 10^40 positions — placing it far beyond\nexhaustive solving. The ability to pass and the opening setup choices add\nfurther branching. Janggi has competitive engines but no proof of the standard\ngame's value, and it has received noticeably less computational-solving\nattention than chess, shogi, or xiangqi."}, {"id": 639, "type": "game", "source": "janggi", "section": "Consensus on optimal play", "text": "Janggi — Consensus on optimal play\n\n- **Opening setup choice sets the character of the game** — choosing \"horse-elephant\" vs. \"elephant-horse\" order on each wing changes attack patterns for the whole game; standard competitive practice favours specific setup pairings based on what the opponent selects.\n- **The pass rule is a tempo weapon** — unlike xiangqi or chess, a legal pass is allowed; passing to force the opponent into zugzwang (a position where any move worsens their position) is a key endgame and certain middlegame technique.\n- **Elephants are stronger in Janggi than in xiangqi** — the Janggi elephant has a slightly different leap and is more active; treat it as a major piece and don't trade it casually.\n- **Palace diagonals are critical attack lines** — the general (king) moves freely within the nine-cell palace, including diagonally; threatening the general along a palace diagonal forces defensive responses and can enable back-rank tactics.\n- **Material count resolves long endgames** — the draw-by-counting rule means a player with more material can claim a draw after 100 moves; know the piece values and when to invoke or avoid this rule.\n- **Cannons weaken as pieces are traded** — cannons must jump over exactly one piece to capture; in open positions with few pieces, cannons become passive; plan exchanges with cannon activity in mind."}, {"id": 640, "type": "game", "source": "janggi", "section": "Engines & current best play", "text": "Janggi — Engines & current best play\n\n- **Strongest known program(s):** No dominant public engine known to the cataloguer; Korean-language competitive programs are used in the online Janggi community.\n- **Strength:** Strong amateur; engines exceed casual human play but are less developed than chess or shogi engines.\n- **Where the proof / tablebase lives (if solved):** Not solved; no published complete endgame tablebase.\n- **Notes:** Less computational-solving attention than chess/shogi/xiangqi; competitive theory is primarily human-derived and passed through Korean competitive channels."}, {"id": 641, "type": "game", "source": "janggi", "section": "Complexity", "text": "Janggi — Complexity\n\nComparable to xiangqi: state-space on the order of 10^40, game-tree far beyond\nsearch."}, {"id": 642, "type": "game", "source": "janggi", "section": "References", "text": "Janggi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Janggi) ([archive](http://web.archive.org/web/20260513200804/https://en.wikipedia.org/wiki/Janggi))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 643, "type": "game", "source": "janggi", "section": "See also", "text": "Janggi — See also\n\n- [Xiangqi](xiangqi.md) · [Chess](chess.md) · [Shogi](shogi.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 644, "type": "game", "source": "kalah", "section": "overview", "text": "Kalah\nThe Western commercial mancala — weakly solved for many board sizes, and\nSolution status: Weakly solved (many (holes, seeds) configurations). Game-theoretic value: First-player win for most configurations (often by a large margin). Players: 2. Type: Partisan sowing (mancala) game."}, {"id": 645, "type": "game", "source": "kalah", "section": "Description", "text": "Kalah — Description\n\nPlayed on two rows of pits plus a scoring \"store\" (kalah) for each player.\nPlayers sow seeds counter-clockwise; landing the last seed in your own store\ngrants an **extra turn**, and landing in an empty pit on your side **captures**\nthe opposite pit. The game is parameterised by *h* holes per side and *s* seeds\nper hole — \"Kalah(6,4)\" is the common commercial version."}, {"id": 646, "type": "game", "source": "kalah", "section": "Solution status", "text": "Kalah — Solution status\n\nKalah is **weakly solved** for a wide range of configurations.\n[Irving, Donkers & Uiterwijk (2000)](../references.md#irving-kalah2000) solved\nKalah(*h*,*s*) for many small-to-moderate (*h*,*s*) by full-game search with\nendgame [databases](../lexicon/README.md#endgame-tablebase), and later work\n(notably by Anders Carstensen) extended this to the standard **Kalah(6,4)** and\nbeyond. The recurring verdict: **the first player wins**, frequently by a large\nscore margin — the \"extra turn\" rule rewards the first mover heavily. (Some very\nsmall configurations are draws or first-player wins by a single seed; the value\ngenuinely depends on (*h*,*s*).)"}, {"id": 647, "type": "game", "source": "kalah", "section": "Consensus on optimal play", "text": "Kalah — Consensus on optimal play\n\n- **The first extra-turn chain is decisive** — chains of extra turns (last seed lands in your store) can empty your side before the opponent moves much; counting seeds to maximise your opening chain length is the most important skill.\n- **Capture the opponent's largest pit whenever possible** — the capture rule (land last seed in an empty pit on your side, take opposite pit) is a high-value action; preferentially set up captures from pits facing full opponent pits.\n- **Deplete pits that enable opponent chains** — if the opponent has a pit whose count lands in their store, emptying or disrupting that pit before they can use it breaks their chain.\n- **Keep your store count ahead early** — the game ends when one side's pits are all empty; if your store is well ahead when this happens you win regardless of the opponent sweeping their remaining seeds.\n- **Avoid leaving a full strip for your opponent** — if your side has many seeds scattered evenly, the opponent can set up a series of captures; unevenness on your side (some full, some empty) is harder to exploit.\n- **For Kalah(6,4) the first-player winning line starts with pit 3 or 4** — computer analysis shows specific first moves that initiate winning chains; knowing even one winning opening line is sufficient for a human to win against non-computer opponents."}, {"id": 648, "type": "game", "source": "kalah", "section": "Engines & current best play", "text": "Kalah — Engines & current best play\n\n- **Strongest known program(s):** Solver programs by Irving/Donkers/Uiterwijk and Anders Carstensen; no well-known named public engine, but Kalah(6,4) is fully solved.\n- **Strength:** Perfect play computable; first player wins with optimal play in Kalah(6,4).\n- **Where the proof / tablebase lives (if solved):** [Irving, Donkers & Uiterwijk (2000)](../references.md#irving-kalah2000); Carstensen's extended work.\n- **Notes:** The strong first-player advantage makes commercial Kalah a poor competitive game; the interest is mathematical."}, {"id": 649, "type": "game", "source": "kalah", "section": "Complexity", "text": "Kalah — Complexity\n\nConfiguration-dependent; standard Kalah(6,4) was within reach of search plus\nendgame databases."}, {"id": 650, "type": "game", "source": "kalah", "section": "References", "text": "Kalah — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Kalah) ([archive](http://web.archive.org/web/20260422060748/https://en.wikipedia.org/wiki/Kalah))\n- [Irving, G., Donkers, J. & Uiterwijk, J. (2000). *Solving Kalah*.](../references.md#irving-kalah2000)"}, {"id": 651, "type": "game", "source": "kalah", "section": "See also", "text": "Kalah — See also\n\n- [Awari (Oware)](awari.md) · [Bao](bao.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 652, "type": "game", "source": "kayles", "section": "overview", "text": "Kayles\nThe archetypal octal game: knock down one pin, or two adjacent pins, from a\nSolution status: Strongly solved. Game-theoretic value: Depends on the row(s); determined by nim-values. Players: 2. Type: Impartial combinatorial game (octal game 0.77)."}, {"id": 653, "type": "game", "source": "kayles", "section": "Description", "text": "Kayles — Description\n\nA row of pins. On a turn a player removes either a single pin or two *adjacent*\npins, which may split the row into two independent shorter rows. Under\n[normal play](../lexicon/README.md#normal-play-convention) the player removing\nthe last pin wins."}, {"id": 654, "type": "game", "source": "kayles", "section": "Solution status", "text": "Kayles — Solution status\n\nKayles is **strongly solved**. [Guy & Smith (1956)](../references.md#guy-smith1956)\nshowed that the [nim-value](../lexicon/README.md#nim-value) sequence of a single\nrow of *n* pins is **eventually periodic** — periodic with period 12 for all\n*n* ≥ 71, with only finitely many exceptional values before that. Because a\nposition with several rows decomposes into a [nim-sum](../lexicon/README.md#nim-sum)\nof independent rows (by the [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)),\nthe entire game is solved: any position's value, and an optimal move, is\ncomputable instantly from the periodic table.\n\nKayles is the historically important first example showing octal-game nim-value\nsequences can be proven periodic — a property that is *known* for many octal\ngames and *conjectured but unproven* for others (see\n[Grundy's game](grundys-game.md))."}, {"id": 655, "type": "game", "source": "kayles", "section": "Consensus on optimal play", "text": "Kayles — Consensus on optimal play\n\n- **Look up the nim-value from the period-12 table** — for any single row of n pins, the nim-value is given by the precomputed periodic table (period 12 for n ≥ 71); this is a one-step table lookup.\n- **Nim-sum all row values to get the position value** — with multiple rows, XOR their nim-values; if the nim-sum is non-zero you are in a winning position and can find the correct move.\n- **Winning move: reduce nim-sum to zero** — find a move in one of the rows that changes its nim-value so the new XOR of all rows equals 0; this is always possible from a non-zero (winning) position.\n- **Removing two adjacent pins splits the row** — removing two adjacent pins from the interior of a row of n creates two independent rows of sizes k and (n−k−2); calculate both halves' nim-values and choose the split that zeroes the nim-sum.\n- **Rows of size 0 (empty) are zero** — rows that have been completely removed contribute 0 to the nim-sum and can be ignored; focus on rows with non-zero nim-values."}, {"id": 656, "type": "game", "source": "kayles", "section": "Engines & current best play", "text": "Kayles — Engines & current best play\n\n- **Strongest known program(s):** Any CGT toolkit implementing the period-12 table and nim-sum (e.g., CGSuite by Aaron Siegel) solves any Kayles position instantly.\n- **Strength:** Perfectly and efficiently solvable; O(number of rows) per-move computation.\n- **Where the proof / tablebase lives (if solved):** [Guy & Smith (1956)](../references.md#guy-smith1956); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Historically the first octal game proven to have an eventually-periodic nim-value sequence; contrasts with Grundy's game whose periodicity remains unproven."}, {"id": 657, "type": "game", "source": "kayles", "section": "Complexity", "text": "Kayles — Complexity\n\nThe period-12 table makes per-position analysis O(number of rows)."}, {"id": 658, "type": "game", "source": "kayles", "section": "References", "text": "Kayles — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Kayles) ([archive](http://web.archive.org/web/20251201221417/https://en.wikipedia.org/wiki/Kayles))\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 659, "type": "game", "source": "kayles", "section": "See also", "text": "Kayles — See also\n\n- [Dawson's chess](dawsons-chess.md) · [Treblecross](treblecross.md) · [Nim](nim.md) · [Grundy's game](grundys-game.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 660, "type": "game", "source": "king-of-the-hill", "section": "overview", "text": "King of the Hill\nChess variant where the king reaching the centre wins — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 661, "type": "game", "source": "king-of-the-hill", "section": "Description", "text": "King of the Hill — Description\n\nKing of the Hill keeps orthodox chess unchanged except that a player wins\nimmediately by safely moving the king to one of the four central squares\n(d4, e4, d5, e5). The new objective makes king activity matter from move one."}, {"id": 662, "type": "game", "source": "king-of-the-hill", "section": "Rules", "text": "King of the Hill — Rules\n\n1. Standard chess setup, movement, and rules.\n2. Additional winning condition: a player wins immediately if they move (or\n   leave) their own king onto **d4, e4, d5, or e5** such that the king is not\n   in check on that square.\n3. Standard checkmate, stalemate, and draw rules also apply.\n4. A move that would place the king on a central square while in check is\n   illegal as usual."}, {"id": 663, "type": "game", "source": "king-of-the-hill", "section": "Solution status", "text": "King of the Hill — Solution status\n\nKing of the Hill is **not solved**. Engine play is strong and many opening\nlines are deeply analysed but no formal value is proven."}, {"id": 664, "type": "game", "source": "king-of-the-hill", "section": "Consensus on optimal play", "text": "King of the Hill — Consensus on optimal play\n\nHeuristics from strong online play:\n\n- **Restrain the centre, then race** — the same four central squares that decide chess strategically now win the game outright. Standard openings that fight for d4/e4/d5/e5 (1.e4, 1.d4) carry over, but the priority shifts toward controlling those squares with pieces (not just pawns) so the opposing king cannot safely walk there.\n- **Don't castle into a wall** — long castling moves the king *away* from the hill; short castling keeps it within striking distance later in the endgame.\n- **Trade the queens early** — without queens, sending the king to the centre becomes safe. Strong players often happily trade queens once the centre is locked.\n- **King marches in the endgame** — the classical chess maxim that the king is a strong endgame piece becomes a primary winning plan."}, {"id": 665, "type": "game", "source": "king-of-the-hill", "section": "Engines & current best play", "text": "King of the Hill — Engines & current best play\n\n- **Strongest known programs:** [Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish)) (open source) — the win-by-king-on-d4 condition is encoded as a terminal condition.\n- **Strength:** Super-human; [Lichess King of the Hill leaderboard](https://lichess.org/variant/kingOfTheHill) ([archive](http://web.archive.org/web/20260511091235/https://lichess.org/variant/kingOfTheHill)) shows engines dominating top humans.\n- **Notes:** No specific tablebases; orthodox-chess tablebases give wrong answers because they ignore the centre-square win."}, {"id": 666, "type": "game", "source": "king-of-the-hill", "section": "Complexity", "text": "King of the Hill — Complexity\n\nSimilar to chess."}, {"id": 667, "type": "game", "source": "king-of-the-hill", "section": "References", "text": "King of the Hill — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Chess_variant) ([archive](http://web.archive.org/web/20260508053202/https://en.wikipedia.org/wiki/Chess_variant))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 668, "type": "game", "source": "king-of-the-hill", "section": "See also", "text": "King of the Hill — See also\n\n- [Chess](chess.md) · [Three-check chess](three-check-chess.md) · [Horde chess](horde-chess.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 669, "type": "game", "source": "klondike-solitaire", "section": "overview", "text": "Klondike solitaire\nThe default Windows card game — its exact win rate under perfect play is still\nSolution status: Partially solved (per-deal solvers exist; overall win rate estimated, not exact). Game-theoretic value: N/A (solitaire) — fraction of solvable deals not known exactly. Players: 1 (puzzle / solitaire). Type: Single-player stochastic card game."}, {"id": 670, "type": "game", "source": "klondike-solitaire", "section": "Description", "text": "Klondike solitaire — Description\n\nA standard 52-card deck is dealt into seven tableau piles (with face-down cards),\na stock, and four empty foundations. The player builds the foundations up by suit\nfrom Ace to King, moving cards among the tableau in alternating colours. Standard\nKlondike has **hidden information** (face-down tableau cards) and **chance** (the\nshuffle), so it sits apart from the perfect-information games in this archive."}, {"id": 671, "type": "game", "source": "klondike-solitaire", "section": "Solution status", "text": "Klondike solitaire — Solution status\n\nKlondike is **partially solved**. Two distinct questions matter:\n\n- **\"Thoughtful\" Klondike** assumes the player knows the location of every card\n  (perfect information). [Bjarnason, Fern & Tadepalli (2009)](../references.md#bjarnason-klondike2009)\n  used Monte-Carlo planning to estimate that roughly **82%** of deals are\n  winnable under thoughtful play — and individual deals can be settled exactly by\n  a solver. This is an estimate with confidence bounds, not an exact enumeration.\n- **Standard Klondike** (cards genuinely hidden) has no known optimal policy and\n  a lower, not-exactly-known win rate; it is a game of imperfect information.\n\nSo: any *given* deal can be solved by computer, but the **overall** value — the\nexact fraction of deals winnable — remains unknown, a situation sometimes called\n\"one of the embarrassments of applied mathematics.\""}, {"id": 672, "type": "game", "source": "klondike-solitaire", "section": "Consensus on optimal play", "text": "Klondike solitaire — Consensus on optimal play\n\n- **Expose face-down cards as the first priority** — uncovering buried face-down cards gives information and new options; always prefer a move that flips a new card over a move of equal apparent value that does not.\n- **Delay sending cards to the foundation if they may be needed** — moving a card to the foundation is often irreversible in practice; a black 6 sent to the foundation cannot be used to unblock a red 5 later; only move foundations when it does not restrict future tableau moves.\n- **Build tableau columns down in alternating colours** — this is mandatory, but strategically prefer keeping columns orderly so that longer ordered sequences can be moved as blocks.\n- **Empty a short column to create a free space** — an empty tableau column is a temporary holding spot for a card or sequence; clearing the shortest pile first is usually faster.\n- **When in doubt, play the move that gives the most options next turn** — in the hidden-information game, uncertainty about face-down cards means \"maximise future options\" is the best guide; avoid moves that commit the tableau to a dead-end arrangement."}, {"id": 673, "type": "game", "source": "klondike-solitaire", "section": "Engines & current best play", "text": "Klondike solitaire — Engines & current best play\n\n- **Strongest known program(s):** Various open-source Klondike solvers — exhaustive search or IDA* for \"thoughtful\" (fully-revealed) deals.\n- **Strength:** Perfect for thoughtful Klondike (fully-revealed); strong heuristic play for standard hidden-information Klondike.\n- **Where the proof / tablebase lives (if solved):** [Bjarnason, Fern & Tadepalli (2009)](../references.md#bjarnason-klondike2009) — ~82% win rate estimate for thoughtful Klondike.\n- **Notes:** The exact fraction of winnable deals remains unknown; individual deals can be definitively solved or proved unsolvable by computer search."}, {"id": 674, "type": "game", "source": "klondike-solitaire", "section": "Complexity", "text": "Klondike solitaire — Complexity\n\nThe deal space is 52! ≈ 8 × 10^67; per-deal search trees are large but tractable\nfor modern solvers. The exact solvable fraction is the open quantity."}, {"id": 675, "type": "game", "source": "klondike-solitaire", "section": "References", "text": "Klondike solitaire — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Klondike_(solitaire)) ([archive](http://web.archive.org/web/20260513195301/https://en.wikipedia.org/wiki/Klondike_(solitaire)))\n- [Bjarnason, Fern & Tadepalli (2009). *Lower Bounding Klondike Solitaire with Monte-Carlo Planning*.](../references.md#bjarnason-klondike2009)"}, {"id": 676, "type": "game", "source": "klondike-solitaire", "section": "See also", "text": "Klondike solitaire — See also\n\n- [Yahtzee](yahtzee.md) · [Peg solitaire](pegs-solitaire.md)\n- Lexicon: [perfect information](../lexicon/README.md#perfect-information) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 677, "type": "game", "source": "klotski", "section": "overview", "text": "Klotski\nSliding-block puzzle whose generalisation is PSPACE-complete.\nSolution status: Generalised n×n sliding-block puzzle PSPACE-complete (Hearn & Demaine, 2005). Game-theoretic value: Per-puzzle. Players: 1. Type: Solo sliding-block puzzle."}, {"id": 678, "type": "game", "source": "klotski", "section": "Description", "text": "Klotski — Description\n\nKlotski (commonly the *L'Âne Rouge* / Huarong Pass puzzle) is a sliding-block\npuzzle in a 4×5 frame: one 2×2 block plus rectangular and square pieces must\nbe rearranged to slide the 2×2 block out the bottom opening. The generalised\nsliding-block puzzle is **PSPACE-complete** (Hearn & Demaine 2005)."}, {"id": 679, "type": "game", "source": "klotski", "section": "Rules", "text": "Klotski — Rules\n\n1. Board: 4×5 frame (in the classical Huarong Pass version) containing one\n   2×2 piece, four 1×2 pieces (vertical or horizontal), and four 1×1 pieces.\n2. The 4×5 frame has a 2-cell opening on the bottom edge.\n3. On a move the solver slides one piece one cell orthogonally to an empty\n   space (no rotation, no jumping).\n4. The puzzle is solved when the 2×2 piece reaches the bottom-centre and can\n   slide out the opening."}, {"id": 680, "type": "game", "source": "klotski", "section": "Solution status", "text": "Klotski — Solution status\n\nPer-instance Klotski puzzles are solved by BFS in seconds. The **generalised\nsliding-block problem** (with arbitrary block shapes on an n×n board) is\n**PSPACE-complete** (Hearn & Demaine 2005)."}, {"id": 681, "type": "game", "source": "klotski", "section": "Consensus on optimal play", "text": "Klotski — Consensus on optimal play\n\n- **Minimise moves by planning the 2×2 block's route first** — identify the path the 2×2 block must travel from its start to the exit; then work out which other pieces must move out of the way for each step, in reverse order.\n- **Create space at the top before pushing down** — in *L'Âne Rouge* / Huarong Pass the 2×2 block starts near the top; the 1×2 and 1×1 pieces must be rotated out of the block's path by first consolidating them in corners.\n- **Cycle small pieces through corners** — the four 1×1 squares are the most flexible pieces; route them into corners to open lanes for the larger pieces.\n- **BFS gives the provably shortest solution** — for the classical 4×5 Huarong Pass layout, BFS finds the minimum-move solution (81 moves) exactly; hand-solving is only a puzzle challenge, not strategically interesting beyond the minimum.\n- **Generalised instances: plan \"corridors\" for big pieces** — in arbitrary sliding-block puzzles, long pieces need unobstructed corridors; identifying corridor-blocking pieces and clearing them is the key sub-problem."}, {"id": 682, "type": "game", "source": "klotski", "section": "Engines & current best play", "text": "Klotski — Engines & current best play\n\n- **Strongest known program(s):** Standard BFS/IDA* sliding-block solvers (e.g., implementations in Simon Tatham's Puzzle Collection) — exact minimum-move solution in seconds.\n- **Strength:** Perfect; BFS is complete and optimal for any fixed-size Klotski instance.\n- **Where the proof / tablebase lives (if solved):** Classical *L'Âne Rouge* minimum: 81 moves (well-established); PSPACE-completeness for generalised problem: Hearn & Demaine (2005).\n- **Notes:** A single-player puzzle; \"solving\" means finding the minimum-move path to the exit — BFS accomplishes this trivially for classical board sizes."}, {"id": 683, "type": "game", "source": "klotski", "section": "Complexity", "text": "Klotski — Complexity\n\nPSPACE-complete in general."}, {"id": 684, "type": "game", "source": "klotski", "section": "References", "text": "Klotski — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Klotski) ([archive](http://web.archive.org/web/20260410144029/https://en.wikipedia.org/wiki/Klotski))\n- [Hearn & Demaine (2005). *Nondeterministic Constraint Logic and PSPACE-Completeness*.](../references.md#demaine-rushhour2002)"}, {"id": 685, "type": "game", "source": "klotski", "section": "See also", "text": "Klotski — See also\n\n- [Rush Hour](rush-hour.md) · [Sokoban](sokoban.md) · [Tower of Hanoi](tower-of-hanoi.md)\n- Lexicon: [PSPACE](../lexicon/README.md#pspace)"}, {"id": 686, "type": "game", "source": "konane", "section": "overview", "text": "Kōnane\nHawaiian checkers — a capture-by-jump game that is a prime testbed for\nSolution status: Partially solved (CGT theory of positions; small boards and many endgame shapes). Game-theoretic value: Known for small boards and many decomposed positions; standard large boards unsolved. Players: 2. Type: Partisan combinatorial game."}, {"id": 687, "type": "game", "source": "konane", "section": "Description", "text": "Kōnane — Description\n\nPlayed on a rectangular board initially filled with black and white stones in a\ncheckerboard pattern. After two opening removals, a player moves by **jumping\none of their stones over an orthogonally adjacent enemy stone** into an empty\ncell, capturing the jumped stone; multi-jumps in a straight line are allowed.\nA player with no legal move loses ([normal play](../lexicon/README.md#normal-play-convention))."}, {"id": 688, "type": "game", "source": "konane", "section": "Solution status", "text": "Kōnane — Solution status\n\nKōnane is **partially solved**. It is unusually friendly to\n[combinatorial game theory](../lexicon/README.md#combinatorial-game-theory)\nbecause late positions break into independent regions whose exact CGT values can\nbe computed and summed — Kōnane is a standard example used to demonstrate the\nCGT value calculus on a \"real\" cultural game. Exact values are known for many\nsmall boards and for large catalogues of endgame fragments, and small full\nboards have been solved by search. The standard large playing boards, however,\nare not solved as a whole. The generalised game has also been studied from a\ncomputational-complexity standpoint."}, {"id": 689, "type": "game", "source": "konane", "section": "Consensus on optimal play", "text": "Kōnane — Consensus on optimal play\n\n- **Multi-jump chains are decisive** — a stone that can jump multiple enemy stones in a straight line removes several opponents in one move; seek positions that set up long chains and deny the opponent similar opportunities.\n- **Preserve jumping ability for your key stones** — once a stone has no orthogonally adjacent enemy stone it can no longer move; avoid allowing your active stones to become isolated islands with no targets.\n- **Opening removal choice shapes the game** — the two opening removals (one from each player) determine which lanes become active; removing an edge stone opens a long jump lane along that side; central removals create more branching paths.\n- **In endgame, apply CGT value analysis** — late Kōnane positions decompose into independent rectangular regions; compute the CGT value of each region (often a small integer or fraction) and nim-sum them to find the winning move.\n- **Temperature of components guides move selection** — in decomposed endgames, play in the highest-temperature (hottest) component first; deferring hot moves is a losing strategy in CGT-valued games."}, {"id": 690, "type": "game", "source": "konane", "section": "Engines & current best play", "text": "Kōnane — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. CGT software (e.g., CGSuite) can evaluate decomposed endgame fragments exactly.\n- **Strength:** Not benchmarked for full-board play; endgame positions solvable exactly via CGT.\n- **Where the proof / tablebase lives (if solved):** M. Ernst (1995) and subsequent CGT analyses; no full-board tablebase for large boards.\n- **Notes:** Kōnane is a standard CGT teaching example because its late-game positions cleanly decompose into independent sums; the opening game is a harder search problem."}, {"id": 691, "type": "game", "source": "konane", "section": "Complexity", "text": "Kōnane — Complexity\n\nDepends on board dimensions; CGT decomposition makes endgames tractable, openings\nremain hard."}, {"id": 692, "type": "game", "source": "konane", "section": "References", "text": "Kōnane — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Konane) ([archive](http://web.archive.org/web/20251004213934/https://en.wikipedia.org/wiki/Konane))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- M. Ernst (1995). *Playing Konane mathematically: A combinatorial game-theoretic analysis*. **[verify]**"}, {"id": 693, "type": "game", "source": "konane", "section": "See also", "text": "Kōnane — See also\n\n- [Checkers (English draughts)](checkers.md) · [Clobber](clobber.md) · [Amazons](amazons.md)\n- Lexicon: [combinatorial game theory](../lexicon/README.md#combinatorial-game-theory) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 694, "type": "game", "source": "l-game", "section": "overview", "text": "L game\nEdward de Bono's minimalist game on a 4×4 board — tiny, elegant, and a\nSolution status: Strongly solved. Game-theoretic value: Draw (neither player can force a win). Players: 2. Type: Partisan combinatorial game."}, {"id": 695, "type": "game", "source": "l-game", "section": "Description", "text": "L game — Description\n\nPlayed on a 4×4 board. Each player has one L-shaped piece (covering 4 squares);\nthere are also two 1×1 neutral pieces. On a turn a player **must** pick up their\nL-piece and place it back in a different position (any orientation, including\nflipped), and **may** then move one neutral piece to any empty square. A player\nwho cannot move their L-piece to a new position loses."}, {"id": 696, "type": "game", "source": "l-game", "section": "Solution status", "text": "L game — Solution status\n\nThe L game is **strongly solved**. The whole game has only **2,296** distinct\npositions, so exhaustive analysis — done both by computer and, famously, by hand\n— covers every position. The result: with perfect play **neither player can\nforce a win; the game is a draw**. A player loses only by making a mistake; from\nany position, the side to move can always avoid loss.\n\nDe Bono designed the game deliberately as the simplest possible \"real\" strategy\ngame, and its complete solvability is part of the point."}, {"id": 697, "type": "game", "source": "l-game", "section": "Consensus on optimal play", "text": "L game — Consensus on optimal play\n\n- **You can always draw with correct play** — the full position graph (2,296 positions) confirms that from any reachable position, the player to move can find at least one drawing response; losing requires an actual error.\n- **Move your L-piece before considering the neutrals** — evaluate all legal L-piece placements first, identify which ones are safe (no immediate losing response), then use neutral placement to maximise your flexibility or restrict the opponent's next L-placement.\n- **Neutral pieces are powerful blockers** — placing a neutral in a cell that an opponent's L-piece would need can cut off many of the opponent's legal moves; use neutrals proactively to reduce the opponent's options, not just to \"waste\" the option.\n- **Avoid leaving only one legal L-placement** — if your next position has only one legal L-placement, the opponent can potentially block it next turn with a neutral; maintain at least two valid placements from any position you enter.\n- **Symmetry traps are the main winning motif** — most wins in the L game occur when one player reduces the other to a single legal L-placement and then blocks it with a neutral; recognising when you are one neutral-move away from this is the core tactical pattern."}, {"id": 698, "type": "game", "source": "l-game", "section": "Engines & current best play", "text": "L game — Engines & current best play\n\n- **Strongest known program(s):** Any exhaustive solver over the 2,296-position graph; the full game tree is trivially small and perfect play is lookup-based.\n- **Strength:** Perfect; the complete position graph has been enumerated.\n- **Where the proof / tablebase lives (if solved):** Complete exhaustive analysis; see [Wikipedia](https://en.wikipedia.org/wiki/L_game) for a summary.\n- **Notes:** Designed by Edward de Bono as the simplest possible \"real\" strategy game; the draw result with perfect play is a deliberate feature of the design."}, {"id": 699, "type": "game", "source": "l-game", "section": "Complexity", "text": "L game — Complexity\n\n2,296 positions — trivially exhaustible."}, {"id": 700, "type": "game", "source": "l-game", "section": "References", "text": "L game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/L_game)\n- E. de Bono (1969). *The Five-Day Course in Thinking* (introduces the L-Game). **[verify]**"}, {"id": 701, "type": "game", "source": "l-game", "section": "See also", "text": "L game — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Hexapawn](hexapawn.md) · [Mū tōrere](mu-torere.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 702, "type": "game", "source": "lasca", "section": "overview", "text": "Lasca\nEmanuel Lasker's draughts variant where captured pieces are stacked, not\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 703, "type": "game", "source": "lasca", "section": "Description", "text": "Lasca — Description\n\nInvented by chess world champion Emanuel Lasker (1911). Played on a 7×7 board\n(25 playing squares), 11 pieces per side. The defining rule: when you capture by\njumping, the captured piece is **not removed** — it is placed *underneath* the\njumping piece, forming a **column (tower)**. Only the top piece of a column\ndetermines its owner and movement; capturing the top piece can \"liberate\" an\nenemy piece beneath it. A player wins when the opponent cannot move."}, {"id": 704, "type": "game", "source": "lasca", "section": "Solution status", "text": "Lasca — Solution status\n\nLasca is **unsolved**. The stacking rule is what makes it hard: a single square\ncan hold a tower of many pieces in many colour-orderings, so the state space is\nvastly larger than a same-sized flat draughts board — pieces are never\npermanently removed, only re-buried and re-exposed. This blocks the\n[retrograde-analysis](../lexicon/README.md#retrograde-analysis) \"few pieces\nleft\" simplification that made [checkers](checkers.md) solvable. Lasca has only\na small competitive and programming community, and no game-theoretic value has\nbeen established."}, {"id": 705, "type": "game", "source": "lasca", "section": "Consensus on optimal play", "text": "Lasca — Consensus on optimal play\n\n- **Control the top of your towers** — only the top piece determines who controls a tower; capturing the top liberates the piece beneath and hands it to the opponent; prioritise maintaining friendly tops on your valuable towers.\n- **Build tall towers carefully** — a tall tower with enemy pieces buried beneath it is a liability: losing the top piece in a jump turns those enemy pieces into mobile attackers; keep your deepest towers guarded.\n- **Forcing captures that liberate strong enemy pieces is losing** — before making a jump, check what colour the piece beneath your target is; liberating a strong enemy piece from under a tower can instantly swing the position.\n- **The player who can access buried pieces first has an advantage** — jumping a tower to claim the top can unearth a piece of the correct colour that becomes a new attacker; plan sequences with the tower composition in mind.\n- **Mobility wins in the endgame** — a player with more moveable towers wins by attrition; avoid positions where all your towers are locked behind opponent towers."}, {"id": 706, "type": "game", "source": "lasca", "section": "Engines & current best play", "text": "Lasca — Engines & current best play\n\n- **Strongest known program(s):** No widely-known public engine; a handful of amateur programs exist.\n- **Strength:** Not benchmarked; engine community is small.\n- **Where the proof / tablebase lives (if solved):** Not solved; the tower mechanic prevents straightforward retrograde analysis.\n- **Notes:** Emanuel Lasker's 1911 invention has a tiny competitive community; the stacking rule makes the state space much larger than standard 7×7 draughts."}, {"id": 707, "type": "game", "source": "lasca", "section": "Complexity", "text": "Lasca — Complexity\n\nLarge and not well quantified in the literature; the tower mechanic is the key\nsource of blow-up."}, {"id": 708, "type": "game", "source": "lasca", "section": "References", "text": "Lasca — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Lasca) ([archive](http://web.archive.org/web/20251219095839/https://en.wikipedia.org/wiki/Lasca))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 709, "type": "game", "source": "lasca", "section": "See also", "text": "Lasca — See also\n\n- [Checkers (English draughts)](checkers.md) · [International draughts](international-draughts.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [state-space complexity](../lexicon/README.md#state-space-complexity)"}, {"id": 710, "type": "game", "source": "lasker-morris", "section": "overview", "text": "Lasker Morris\nEmanuel Lasker's variant of Nine Men's Morris that fixes its \"stalling\"\nSolution status: Weakly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan placement+movement game."}, {"id": 711, "type": "game", "source": "lasker-morris", "section": "Description", "text": "Lasker Morris — Description\n\nA modification of [Nine Men's Morris](nine-mens-morris.md) proposed by world\nchess champion Emanuel Lasker (1931). The key change: a player may **either\nplace a new stone or move an existing one on every turn**, instead of completing\nthe entire placement phase first. This eliminates the rigid two-phase structure\nthat, in standard morris, gives a strong advantage to the placement-phase\ndefender."}, {"id": 712, "type": "game", "source": "lasker-morris", "section": "Rules", "text": "Lasker Morris — Rules\n\n1. Same 24-point board as [Nine Men's Morris](nine-mens-morris.md), with nine\n   stones per player.\n2. On each turn a player may **either** place a new (reserve) stone on an empty\n   point **or** move one of their on-board stones to an adjacent empty point.\n3. Forming a row of three (\"mill\") allows the player to remove one opposing\n   stone, as in standard morris.\n4. The \"flying\" endgame rule (move anywhere when down to 3 stones) usually\n   applies.\n5. A player reduced to 2 stones, or unable to move, loses."}, {"id": 713, "type": "game", "source": "lasker-morris", "section": "Solution status", "text": "Lasker Morris — Solution status\n\nWeakly solved by [Gasser (1996)](../references.md#gasser-laskermorris1996) at\nthe same time as standard Nine Men's Morris and using the same retrograde\nanalysis: the game is a **draw** with perfect play. Lasker Morris fixes the\n\"who-places-first\" tempo issue without changing the headline result."}, {"id": 714, "type": "game", "source": "lasker-morris", "section": "Consensus on optimal play", "text": "Lasker Morris — Consensus on optimal play\n\n- **Mix placements and moves from the start** — unlike standard Morris, you can move an existing stone instead of placing a new one; use this to set up mills while simultaneously developing stone positions, gaining the \"chess-like\" tempo advantage Lasker intended.\n- **Prevent the opponent's mills before they form** — placing or moving a stone to break a near-complete opponent mill takes priority over building your own; a mill gives the opponent a removal and is very difficult to recover from.\n- **Keep stones mobile** — stones on board points with no adjacent empty points are stuck; maintain each stone near at least one empty adjacent point so it can contribute to mills later.\n- **Removal targets: take the opponent's most mobile stone** — when you form a mill and must remove a stone, take the one that would be hardest to replace — typically the stone closest to the opponent's two other mills.\n- **Flying (3-stone) phase is a drawing resource** — being reduced to 3 stones triggers the flying rule (move anywhere); if you have set up a pattern the opponent cannot break and you still have 3 stones, you may hold a draw; knowing this prevents premature resignation."}, {"id": 715, "type": "game", "source": "lasker-morris", "section": "Engines & current best play", "text": "Lasker Morris — Engines & current best play\n\n- **Strongest known program(s):** Gasser's solver (retrograde analysis, 1996) — provides perfect play from any position in the ~10^10-position database.\n- **Strength:** Perfect; the complete game has been solved.\n- **Where the proof / tablebase lives (if solved):** [Gasser (1996)](../references.md#gasser-laskermorris1996) — same retrograde analysis that solved Nine Men's Morris.\n- **Notes:** The game is a draw with perfect play; Lasker's rule change produces a more dynamic game than standard Morris without changing the game-theoretic outcome."}, {"id": 716, "type": "game", "source": "lasker-morris", "section": "Complexity", "text": "Lasker Morris — Complexity\n\nState-space and tablebase size on the order of 10^10 — within the same range as\nNine Men's Morris and handled by Gasser's solver."}, {"id": 717, "type": "game", "source": "lasker-morris", "section": "References", "text": "Lasker Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nine_men%27s_morris) ([archive](http://web.archive.org/web/20260328233608/https://en.wikipedia.org/wiki/Nine_Men%27s_Morris))\n- [Gasser (1996). *Solving Nine Men's Morris* — Lasker variant.](../references.md#gasser-laskermorris1996)"}, {"id": 718, "type": "game", "source": "lasker-morris", "section": "See also", "text": "Lasker Morris — See also\n\n- [Nine Men's Morris](nine-mens-morris.md) · [Six Men's Morris](six-mens-morris.md) · [Twelve Men's Morris](twelve-mens-morris.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 719, "type": "game", "source": "liars-dice", "section": "overview", "text": "Liar's dice\nA bluffing dice game — small variants yield to game-theoretic equilibrium\nSolution status: Unsolved in general; small two-player variants solved (Nash equilibria computed). Game-theoretic value: Variant-dependent. Players: 2+. Type: Stochastic imperfect-information bluffing game."}, {"id": 720, "type": "game", "source": "liars-dice", "section": "Description", "text": "Liar's dice — Description\n\nEach player rolls a set of dice and keeps them hidden under a cup. Players take\nturns making escalating **bids** about the dice on the table as a whole (e.g.\n\"four 5s\"), or **challenging** the previous bid. When a bid is challenged the\ndice are revealed: whoever was wrong loses a die. A player eliminated of all dice\nis out; the last player standing wins. The whole game turns on bluffing and\nprobabilistic inference from hidden information."}, {"id": 721, "type": "game", "source": "liars-dice", "section": "Solution status", "text": "Liar's dice — Solution status\n\nLiar's dice is **not solved** in general. As an imperfect-information game,\n\"solving\" means computing a Nash-equilibrium strategy; this is tractable for\n**small two-player variants** (few dice per player), where the game can be\nexpressed as a manageable extensive-form game and solved by linear programming or\ncounterfactual regret minimisation — and such small variants are a common\nteaching and benchmark example in the equilibrium-computation literature. But the\n**standard multi-die, multi-player game** has a state space that grows too fast\nfor a published full-game equilibrium, and multi-player equilibria are not even\nuniquely defined. No comprehensive solution exists."}, {"id": 722, "type": "game", "source": "liars-dice", "section": "Consensus on optimal play", "text": "Liar's dice — Consensus on optimal play\n\n- **Compute the expected number of each face before bidding** — with N total dice on the table, each face appears roughly N/6 times; wild-1 variants count 1s as wild, raising expected counts; bid slightly above the expected count and call bluffs when bids go well above it.\n- **Bluff proportionally to the uncertainty** — when you have none of the bid face yourself, the bid relies entirely on opponents' dice; with many opponents still in, high bids are plausible; with few opponents left, high bids become statistically unlikely and should be challenged.\n- **Randomise your honest/bluff ratio to stay unexploitable** — a player who only bids truthfully or only bluffs is quickly read; mix strategies so opponents cannot reliably deduce your dice from your bidding pattern.\n- **Challenge conservatively with many dice remaining** — early in the game, losing a die is a small setback; challenge only when a bid significantly exceeds statistical plausibility.\n- **Challenge aggressively with few dice remaining** — with 1 or 2 dice left, losing a die is catastrophic; call anything that requires opponents to have more of a face than is likely given remaining dice.\n- **Track what faces opponents have bid confidently** — repeated high bids of a specific face by one player likely reflect their actual dice; use this information to update your probability estimates for challenges."}, {"id": 723, "type": "game", "source": "liars-dice", "section": "Engines & current best play", "text": "Liar's dice — Engines & current best play\n\n- **Strongest known program(s):** CFR-based bots for fixed configurations; no publicly named dominant engine for the standard multi-player game.\n- **Strength:** Strong for small two-player variants (Nash equilibria computable); heuristic-level for standard multi-player game.\n- **Where the proof / tablebase lives (if solved):** Small two-player cases solved via linear programming / CFR; no published full solution for the standard game.\n- **Notes:** Multi-player equilibria are not uniquely defined even in theory; practical play relies on probabilistic heuristics and opponent modelling."}, {"id": 724, "type": "game", "source": "liars-dice", "section": "Complexity", "text": "Liar's dice — Complexity\n\nScales steeply with the number of dice and players; the bidding tree plus the\nhidden-information structure put the standard game beyond current full-solution\nmethods."}, {"id": 725, "type": "game", "source": "liars-dice", "section": "References", "text": "Liar's dice — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Liar%27s_dice) ([archive](http://web.archive.org/web/20260417235754/https://en.wikipedia.org/wiki/Liar%27s_dice))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 726, "type": "game", "source": "liars-dice", "section": "See also", "text": "Liar's dice — See also\n\n- [Heads-up limit hold'em](heads-up-limit-holdem.md) · [Battleship](battleship.md) · [Mastermind](mastermind.md)\n- Lexicon: [Nash equilibrium](../lexicon/README.md#nash-equilibrium) · [perfect information](../lexicon/README.md#perfect-information)"}, {"id": 727, "type": "game", "source": "lights-out", "section": "overview", "text": "Lights Out\nLinear-algebra puzzle: toggle lights to turn them all off.\nSolution status: Strongly solved. Game-theoretic value: Solvable iff initial state is in the image of the toggle matrix over GF(2). Players: 1. Type: Solo puzzle."}, {"id": 728, "type": "game", "source": "lights-out", "section": "Description", "text": "Lights Out — Description\n\nLights Out (Tiger Electronics, 1995) is a 5×5 grid of buttons; pressing a\nbutton toggles itself and its orthogonal neighbours. The goal is to turn off\nevery light from a given initial pattern. The puzzle reduces to solving a\nlinear system over GF(2)."}, {"id": 729, "type": "game", "source": "lights-out", "section": "Rules", "text": "Lights Out — Rules\n\n1. Board: 5×5 grid; each cell is either lit or dark.\n2. On a move the player presses one cell; that cell and its orthogonal\n   neighbours (up, down, left, right, if any) toggle on/off.\n3. The objective is to reach the all-dark configuration.\n4. The order in which cells are pressed does not matter (each cell needs only\n   to be pressed 0 or 1 times in total)."}, {"id": 730, "type": "game", "source": "lights-out", "section": "Solution status", "text": "Lights Out — Solution status\n\n**Strongly solved**: Anderson & Feil (1998) showed that the set of\nsolvable initial states is the image of the toggle matrix *M* over GF(2);\nexactly 2^23 of the 2^25 initial states on the 5×5 board are solvable. For\n*n*×*n* boards the analysis generalises straightforwardly."}, {"id": 731, "type": "game", "source": "lights-out", "section": "Consensus on optimal play", "text": "Lights Out — Consensus on optimal play\n\n- **\"Chase the lights\" row by row** — working from row 1 to row 4, press the cell in the current row that toggles the lit cell in the row above; this systematically eliminates lights one row at a time.\n- **Order of presses does not matter** — because toggling is an XOR operation, each button needs to be pressed 0 or 1 times total; rearranging the order gives the same result; simplify by planning a press-set, not a sequence.\n- **Check the bottom row after chasing** — after chasing through rows 1–4, the bottom row will have a specific lit pattern; there are only 5 cells (32 possible patterns); a lookup table maps each pattern to the required top-row presses that correct it (a second sweep solves).\n- **First check solvability** — approximately 1/4 of random 5×5 initial states are unsolvable (outside the image of the toggle matrix); if the puzzle does not resolve after the standard procedure, it may be genuinely unsolvable.\n- **For the minimum-press solution, use Gaussian elimination** — the linear-algebra formulation over GF(2) finds not just a solution but the minimum-button-press solution directly."}, {"id": 732, "type": "game", "source": "lights-out", "section": "Engines & current best play", "text": "Lights Out — Engines & current best play\n\n- **Strongest known program(s):** Any GF(2) Gaussian-elimination solver — exact minimum-press solution in polynomial time.\n- **Strength:** Perfect; every solvable instance has an exact minimum-press solution computable instantly.\n- **Where the proof / tablebase lives (if solved):** [Anderson & Feil (1998)](../references.md#anderson-feil1998) — complete mathematical analysis; 2^23 of 2^25 5×5 states are solvable.\n- **Notes:** A purely mathematical puzzle; \"strategy\" reduces to linear algebra over GF(2) — there is no opponent and no game-theoretic content beyond the solvability question."}, {"id": 733, "type": "game", "source": "lights-out", "section": "Complexity", "text": "Lights Out — Complexity\n\nPolynomial via Gaussian elimination."}, {"id": 734, "type": "game", "source": "lights-out", "section": "References", "text": "Lights Out — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Lights_Out_(game)) ([archive](http://web.archive.org/web/20260504175703/https://en.wikipedia.org/wiki/Lights_Out_%28game%29))\n- [Anderson & Feil (1998). *Turning Lights Out with Linear Algebra*.](../references.md#anderson-feil1998)"}, {"id": 735, "type": "game", "source": "lights-out", "section": "See also", "text": "Lights Out — See also\n\n- [Sudoku](sudoku.md) · [Slitherlink](slitherlink.md) · [Nonograms](nonograms.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 736, "type": "game", "source": "lines-of-action", "section": "overview", "text": "Lines of Action\nA connection game on a chessboard; played strongly by engines but not solved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 737, "type": "game", "source": "lines-of-action", "section": "Description", "text": "Lines of Action — Description\n\nPlayed on an 8×8 board. Each player has pieces (Black on the top and bottom\nedges, White on the left and right). A piece moves in a straight line exactly as\nmany squares as there are pieces (of either colour) on that line; it may jump\nits own pieces but not enemy pieces, and may capture by landing on an enemy\npiece. **The goal is to connect all of one's own pieces into a single\ngroup** (orthogonally or diagonally)."}, {"id": 738, "type": "game", "source": "lines-of-action", "section": "Solution status", "text": "Lines of Action — Solution status\n\nLines of Action is **unsolved**. Its complexity is modest by board-game\nstandards — comparable to Othello — and it has been a strong-AI success story:\nLOA programs reached a level widely regarded as superhuman, and the game was a\nregular Computer Olympiad event. But no proof of the game-theoretic value of the\nstandard opening position exists, so it remains unsolved.\n\nThe mismatch — a fairly small game that is nonetheless unsolved — reflects that\nno group has invested the focused effort a weak solution would require, rather\nthan any fundamental obstacle."}, {"id": 739, "type": "game", "source": "lines-of-action", "section": "Consensus on optimal play", "text": "Lines of Action — Consensus on optimal play\n\n- **Keep your pieces in a compact cluster** — the goal is a single connected group; pieces that wander to the edges become hard to reconnect; keep the cluster tight and avoid isolated outliers.\n- **Moves that both connect and disrupt** — the best moves advance your own connectivity (reduce the number of components your pieces form) while simultaneously splitting the opponent's group; evaluate moves by counting components before and after.\n- **The move-distance rule rewards centrality** — a piece on a full row or column moves far; a piece on a sparse row or column moves only a little; use dense lines to make long reaching moves and sparse lines for fine positioning.\n- **Restrict opponent mobility by occupying shared lines** — placing your pieces on lines the opponent needs to traverse forces their pieces to move longer distances (more of your pieces on that line means longer moves for both), which can overshoot their intended landing squares.\n- **Sacrifice pieces on the periphery if they join the core** — capturing an opponent piece that brings your outlier piece into your cluster is often worth the trade; count connectivity gain per move.\n- **Force asymmetry early** — symmetric positions reward the second player; break symmetry in a direction that compresses your pieces faster than the opponent's."}, {"id": 740, "type": "game", "source": "lines-of-action", "section": "Engines & current best play", "text": "Lines of Action — Engines & current best play\n\n- **Strongest known program(s):** MIA (Mark Winands, Maastricht University) and YL — alpha-beta search with Lines-of-Action-specific evaluation functions; MIA was the dominant Computer Olympiad program.\n- **Strength:** Super-human; top LOA engines are widely regarded as playing above the strongest humans.\n- **Where the proof / tablebase lives (if solved):** Not solved; no published game-theoretic value for the standard opening position.\n- **Notes:** The game is a regular Computer Olympiad event; empirical engine consensus on strong play is high despite the absence of a formal proof."}, {"id": 741, "type": "game", "source": "lines-of-action", "section": "Complexity", "text": "Lines of Action — Complexity\n\nState-space ~10^23–10^24, game-tree ~10^56 (order-of-magnitude figures from the\ngames-solved literature)."}, {"id": 742, "type": "game", "source": "lines-of-action", "section": "References", "text": "Lines of Action — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Lines_of_Action) ([archive](http://web.archive.org/web/20260307215220/https://en.wikipedia.org/wiki/Lines_of_Action))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 743, "type": "game", "source": "lines-of-action", "section": "See also", "text": "Lines of Action — See also\n\n- [Amazons](amazons.md) · [Hex](hex.md) · [Breakthrough](breakthrough.md)\n- Lexicon: [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 744, "type": "game", "source": "losing-chess", "section": "overview", "text": "Losing chess\nChess with the goal inverted — lose all your pieces to win — and, remarkably,\nSolution status: Weakly solved. Game-theoretic value: First-player (White) win. Players: 2. Type: Partisan board game (misère-flavoured chess variant)."}, {"id": 745, "type": "game", "source": "losing-chess", "section": "Description", "text": "Losing chess — Description\n\nThe pieces and board are as in [chess](chess.md), but the **objective is\ninverted**: you win by losing all your pieces or being stalemated. Crucially,\n**captures are compulsory** — if you can capture, you must (choosing among\ncaptures if several exist). The forced-capture rule makes play extremely\nforcing and is what brings the game within reach of solving."}, {"id": 746, "type": "game", "source": "losing-chess", "section": "Solution status", "text": "Losing chess — Solution status\n\nLosing chess is **weakly solved**. [Mark Watkins (2016)](../references.md#watkins-losingchess2016)\ncompleted a long, distributed proof-tree computation establishing that **White\nwins with perfect play**, and that the winning first move is **1. e3**. The\ncompulsory-capture rule prunes the game tree enormously — most positions have\nvery few legal moves — which is why a full-blown chess variant could be solved\nwhen chess itself cannot be.\n\nThis makes losing chess one of the most complex chess-family games to be\nweakly solved, and a striking case where *changing the win condition* turns an\nintractable game into a tractable one.\n\n> **[verify]** — Widely reported and accepted; the primary write-up should be\n> confirmed and pinned in [references.md](../references.md#watkins-losingchess2016)."}, {"id": 747, "type": "game", "source": "losing-chess", "section": "Consensus on optimal play", "text": "Losing chess — Consensus on optimal play\n\n- **Play 1. e3 as White** — this is the proven winning first move; any other first move has not been proven to win and may not be a win; the solution is specific to 1. e3.\n- **Force captures relentlessly** — losing chess is won by losing pieces fastest; moves that force the opponent to capture (by offering pieces they must take) are almost always best; every forced capture the opponent makes is a piece you no longer need to lose.\n- **Sacrifice major pieces early** — getting rid of the queen and rooks quickly by forcing the opponent to capture them reduces your material burden dramatically; don't hoard strong pieces.\n- **Use the compulsory-capture rule as a tactical weapon** — if you can offer multiple captures at once, the opponent can only take one; the others become free pieces for the opponent to leave on the board while you wait; offer sacrifices that force sequential captures.\n- **Stalemate is also a win** — being unable to move is a win condition; in complex endings, steering toward a position where all your pieces are blocked can win outright without losing all pieces.\n- **Against sub-optimal opponent play, engines are the guide** — the full winning tree from 1. e3 is large; Fairy-Stockfish in losing-chess mode plays near-optimally and is the practical reference for over-the-board training."}, {"id": 748, "type": "game", "source": "losing-chess", "section": "Engines & current best play", "text": "Losing chess — Engines & current best play\n\n- **Strongest known program(s):** Fairy-Stockfish ([https://github.com/ianfab/Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish))) supports antichess/losing-chess mode; Lichess hosts antichess with Fairy-Stockfish.\n- **Strength:** Super-human; consistent with the solved weak solution.\n- **Where the proof / tablebase lives (if solved):** [Watkins (2016)](../references.md#watkins-losingchess2016) — the distributed proof that 1. e3 wins; also playable on [Lichess antichess variant](https://lichess.org/variant/antichess).\n- **Notes:** One of the most complex chess-family games to be weakly solved; the compulsory-capture rule is what makes the game tree small enough to solve."}, {"id": 749, "type": "game", "source": "losing-chess", "section": "Complexity", "text": "Losing chess — Complexity\n\nNominally chess-scale, but compulsory captures make real game trees far smaller\nand far more forcing than ordinary chess."}, {"id": 750, "type": "game", "source": "losing-chess", "section": "References", "text": "Losing chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Losing_chess) ([archive](http://web.archive.org/web/20260420060404/https://en.wikipedia.org/wiki/Losing_chess))\n- [Watkins, M. (2016). *Losing Chess: 1. e3 Wins*.](../references.md#watkins-losingchess2016)"}, {"id": 751, "type": "game", "source": "losing-chess", "section": "See also", "text": "Losing chess — See also\n\n- [Chess](chess.md) · [Minichess](minichess.md) · [Dawson's chess](dawsons-chess.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [misère play](../lexicon/README.md#misère-play)"}, {"id": 752, "type": "game", "source": "lyngk", "section": "overview", "text": "LYNGK\nThe cooperative-style GIPF-project closer — stacking discs to capture colour\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2–4. Type: Partisan stacking game."}, {"id": 753, "type": "game", "source": "lyngk", "section": "Description", "text": "LYNGK — Description\n\nLYNGK (Kris Burm, 2017) is the seventh GIPF-project release, designed to close\nthe series. Each player secretly chooses **two colours** out of six; the\ngoal is to capture stacks containing all six colours by claiming and stacking\ndiscs."}, {"id": 754, "type": "game", "source": "lyngk", "section": "Rules", "text": "LYNGK — Rules\n\n1. Board: hexagonal grid of 49 cells, each cell with one or two discs of one\n   of six colours.\n2. Each player secretly picks **two colours** to \"own.\"\n3. On a turn, the player moves a stack containing their colour, leaping over an\n   adjacent stack containing a colour they do **not** own, landing on a stack\n   in their direction of choice and **stacking on top**.\n4. A stack of height 5 containing all distinct colours (or specific\n   colour-counts depending on the variant) is **captured** by the player who\n   completes it.\n5. The first player to capture a target number of \"stacks-of-five\" wins."}, {"id": 755, "type": "game", "source": "lyngk", "section": "Solution status", "text": "LYNGK — Solution status\n\nLYNGK is **not solved**. The hidden-colour assignment makes the game effectively\nimperfect-information for some analyses; even the perfect-information version\nhas no published solution."}, {"id": 756, "type": "game", "source": "lyngk", "section": "Consensus on optimal play", "text": "LYNGK — Consensus on optimal play\n\n- **Claim your two colours early** — declaring ownership locks your strategy for the rest of the game; pick colours that are densely distributed on the board and offer many stacking opportunities.\n- **Track the opponent's likely colour claim** — their early moves reveal which colours they own; once you have identified both of their colours, you know which stacks they can legally move, allowing you to block or redirect their path to five-colour completion.\n- **Build partial stacks that only you can complete** — a four-colour stack missing one colour you own is one move from a capture for you but potentially unmoveable for the opponent; build these controlled near-complete stacks.\n- **Block stacks containing all the opponent's colours** — a stack that includes both of the opponent's colours is a potential five-colour completion threat; interfere with it by stacking a disc on top with a neutral or your own colour, adding height but changing the required remaining piece.\n- **Tempo matters on a 49-cell board** — with limited stacks and finite colours, each move either advances a five-colour completion or delays the opponent; don't make neutral moves unless they also constrain the opponent."}, {"id": 757, "type": "game", "source": "lyngk", "section": "Engines & current best play", "text": "LYNGK — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** Not solved; no published analysis.\n- **Notes:** The newest GIPF-project title (2017); competitive scene is small and published theory is minimal."}, {"id": 758, "type": "game", "source": "lyngk", "section": "Complexity", "text": "LYNGK — Complexity\n\nLarge."}, {"id": 759, "type": "game", "source": "lyngk", "section": "References", "text": "LYNGK — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/LYNGK) ([archive](http://web.archive.org/web/20251004213009/https://en.wikipedia.org/wiki/LYNGK))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 760, "type": "game", "source": "lyngk", "section": "See also", "text": "LYNGK — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [YINSH](yinsh.md) · [TZAAR](tzaar.md)\n- Lexicon: [perfect information](../lexicon/README.md#perfect-information)"}, {"id": 761, "type": "game", "source": "maharajah-and-the-sepoys", "section": "overview", "text": "Maharajah and the Sepoys\nAn asymmetric chess variant: one super-piece against a full army. The army\nSolution status: Weakly solved. Game-theoretic value: Second player (the Sepoys) wins with correct play. Players: 2 (asymmetric). Type: Partisan board game (asymmetric chess variant)."}, {"id": 762, "type": "game", "source": "maharajah-and-the-sepoys", "section": "Description", "text": "Maharajah and the Sepoys — Description\n\nAn asymmetric chess variant. One player has a **full standard chess army** (the\n\"Sepoys\"); the other has a **single piece**, the **Maharajah**, which moves as a\ncombined Queen + Knight (an \"amazon\"). The Maharajah wins by capturing the enemy\nking; the Sepoys win by capturing the Maharajah. The Sepoys move first."}, {"id": 763, "type": "game", "source": "maharajah-and-the-sepoys", "section": "Solution status", "text": "Maharajah and the Sepoys — Solution status\n\nMaharajah and the Sepoys is **weakly solved**, and the result is a classical\npiece of chess-variant folklore: **the Sepoys win with correct play.** The\nMaharajah, however powerful as a single piece, cannot survive against a\ncoordinated full army that simply advances its pawns in a connected phalanx and\nnever leaves the Maharajah a profitable capture. The standard winning method is\nwell known and humanly executable: keep pawns mutually defended, advance the\nwall, and the Maharajah is eventually trapped.\n\nThe Sepoys must avoid careless captures — a single undefended piece can let the\nMaharajah equalise material — which is why the result is \"weakly solved with\ncorrect play\" rather than \"trivially won.\""}, {"id": 764, "type": "game", "source": "maharajah-and-the-sepoys", "section": "Consensus on optimal play", "text": "Maharajah and the Sepoys — Consensus on optimal play\n\n- **Sepoys: advance pawns in a mutually defended wall** — never leave a pawn undefended; the Maharajah can afford one capture that equalises material; a defended pawn-phalanx denies all such opportunities and slowly compresses the Maharajah's space.\n- **Sepoys: do not rush pieces forward singly** — pieces advanced alone become targets for the Maharajah's Amazon (queen+knight) move; bring the whole army forward together, subordinating individual activity to collective safety.\n- **Maharajah: fork whenever possible** — the Amazon's combined queen and knight reach makes two-attack forks against undefended Sepoy pieces the only realistic path to material gain; look for cells the Amazon can reach that attack two pieces simultaneously.\n- **Maharajah: avoid the edge** — the Amazon is weakest near the board edge where its mobility is cut in half; stay central to maximise threat range and escape paths.\n- **Sepoys: trade material freely except for pawns** — giving up a piece to keep the pawn wall intact is usually correct; pawns form the impenetrable front that drives the Maharajah into a corner.\n- **Maharajah: stalemate is a draw** — if the Sepoys leave no legal move for the Maharajah (without capturing it), the game is drawn; as the Maharajah player, steer toward positions with minimal legal squares and hope for a stalemate error."}, {"id": 765, "type": "game", "source": "maharajah-and-the-sepoys", "section": "Engines & current best play", "text": "Maharajah and the Sepoys — Engines & current best play\n\n- **Strongest known program(s):** No dedicated public engine; any chess-variant program (e.g., Fairy-Stockfish with Amazon piece) can play this correctly.\n- **Strength:** Sepoys win with correct play; optimal Sepoy strategy is humanly teachable.\n- **Where the proof / tablebase lives (if solved):** Classical folklore; see [Wikipedia](https://en.wikipedia.org/wiki/Maharajah_and_the_Sepoys) for a summary of the winning method.\n- **Notes:** A demonstration game rather than a competitive one; the Sepoy win is well-known in chess-variant circles as a teaching example in piece coordination."}, {"id": 766, "type": "game", "source": "maharajah-and-the-sepoys", "section": "Complexity", "text": "Maharajah and the Sepoys — Complexity\n\nSmall by chess standards — one side has a single piece, sharply limiting\nbranching."}, {"id": 767, "type": "game", "source": "maharajah-and-the-sepoys", "section": "References", "text": "Maharajah and the Sepoys — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Maharajah_and_the_Sepoys) ([archive](http://web.archive.org/web/20251207100534/https://en.wikipedia.org/wiki/Maharajah_and_the_Sepoys))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 768, "type": "game", "source": "maharajah-and-the-sepoys", "section": "See also", "text": "Maharajah and the Sepoys — See also\n\n- [Chess](chess.md) · [Fox and Geese](fox-and-geese.md) · [Hare and Hounds](hare-and-hounds.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 769, "type": "game", "source": "makruk", "section": "overview", "text": "Makruk\nThai chess — a chess relative with weak long-range pieces and a distinctive\nSolution status: Unsolved (some endgame tables computed). Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 770, "type": "game", "source": "makruk", "section": "Description", "text": "Makruk — Description\n\nPlayed on an 8×8 board. Makruk descends, like Western chess, from the older\n[shatranj](shatranj.md) tradition: its Queen-equivalent (Met) moves only one\nsquare diagonally, its Bishop-equivalent (Khon) is also short-range, and pawns\npromote on the *sixth* rank. Pieces start one rank advanced compared to chess. A\ndetailed **counting rule** forces a draw if the stronger side cannot mate within\na bounded number of moves."}, {"id": 771, "type": "game", "source": "makruk", "section": "Solution status", "text": "Makruk — Solution status\n\nMakruk is **unsolved**. With weaker long-range pieces than chess, games tend to\nbe slower and more manoeuvring, but the state-space and game-tree complexity\nremain on the order of chess's — far beyond exhaustive search. Endgame\ntablebases have been generated for small material counts (and matter a great\ndeal in practice because of the counting rule), giving exact solutions to those\nsub-games; the full game's value is unknown."}, {"id": 772, "type": "game", "source": "makruk", "section": "Consensus on optimal play", "text": "Makruk — Consensus on optimal play\n\n- **The counting rule shapes the entire endgame** — once the last pawn is promoted or captured, the defending side starts counting; the attacker must mate within the allotted moves (determined by the count rule) or the game is drawn; knowing when the count begins and how many moves remain is essential.\n- **Pawns promote on the sixth rank, not the eighth** — promoted pawns (Met/queen equivalent) come into play faster; use pawn advances early to threaten early promotion and force defensive commitments.\n- **The Met is a short-range piece — treat it as a bishop/knight hybrid** — the Met (queen) moves only one square diagonally; centralise it to maximise its impact rather than leaving it on the flank where it can only cover one or two adjacent squares.\n- **The Khon (bishop-equivalent) covers only one colour** — like chess bishops, Khon pieces are colour-bound (one square diagonal); if both Khon are on the same colour, coordinate them as a pair; if on different colours, one will always cover gaps the other cannot.\n- **Endgame: consult the tablebase for small material counts** — makruk tablebases for positions with a few pieces are publicly used by Thai competitive players; a position that looks winning may be drawn due to the counting rule; check before committing to a piece exchange.\n- **Opening: advance both flanks to create Met activity** — because pieces start one rank closer than in chess, early activation is easier; develop both wings simultaneously to avoid giving the opponent a free centralisation advantage."}, {"id": 773, "type": "game", "source": "makruk", "section": "Engines & current best play", "text": "Makruk — Engines & current best play\n\n- **Strongest known program(s):** No dominant publicly-named engine; Thai-language programs and Fairy-Stockfish (with Makruk variant) are used in the Thai competitive community.\n- **Strength:** Strong amateur; stronger than casual human play.\n- **Where the proof / tablebase lives (if solved):** Endgame tablebases computed for small material counts; the counting rule makes these tablebases particularly important in practical play.\n- **Notes:** Makruk is the most widely played traditional chess variant in mainland Southeast Asia; the counting rule distinguishes its endgames sharply from chess."}, {"id": 774, "type": "game", "source": "makruk", "section": "Complexity", "text": "Makruk — Complexity\n\nComparable to chess: state-space on the order of 10^40, game-tree far beyond\nsearch."}, {"id": 775, "type": "game", "source": "makruk", "section": "References", "text": "Makruk — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Makruk) ([archive](http://web.archive.org/web/20260302012918/https://en.wikipedia.org/wiki/Makruk))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 776, "type": "game", "source": "makruk", "section": "See also", "text": "Makruk — See also\n\n- [Shatranj](shatranj.md) · [Chess](chess.md) · [Xiangqi](xiangqi.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase)"}, {"id": 777, "type": "game", "source": "mastermind", "section": "overview", "text": "Mastermind\nThe code-breaking pegboard game — solved: the codebreaker can always win, and\nSolution status: Weakly solved (standard 4-peg, 6-colour game). Game-theoretic value: Codebreaker wins; ≤5 guesses suffice (worst case). Players: 2 (asymmetric: codemaker vs. codebreaker). Type: Deductive code-breaking game (one-sided hidden information)."}, {"id": 778, "type": "game", "source": "mastermind", "section": "Description", "text": "Mastermind — Description\n\nThe **codemaker** secretly chooses a code of 4 pegs, each one of 6 colours\n(repeats allowed). The **codebreaker** makes guesses; after each, the codemaker\nreports how many pegs are the right colour in the right position (\"black\" pegs)\nand how many are the right colour in the wrong position (\"white\" pegs). The\ncodebreaker tries to identify the code in as few guesses as possible."}, {"id": 779, "type": "game", "source": "mastermind", "section": "Solution status", "text": "Mastermind — Solution status\n\nMastermind is **weakly solved** for the standard 4-peg, 6-colour game.\n[Knuth (1977)](../references.md#knuth1977) gave a strategy — the \"minimax\"\nguessing rule, which always picks a guess minimising the worst-case number of\nremaining possibilities — that **guarantees a solve within 5 guesses**, and\nshowed 5 is necessary in the worst case. Later authors computed the strategy\nminimising the *expected* number of guesses (about 4.34 guesses), and exhaustive\nsearch over all 1,296 codes confirms these optima. Because the codebreaker can\nalways force a win within the bound, the game is solved in the codebreaker's\nfavour."}, {"id": 780, "type": "game", "source": "mastermind", "section": "Consensus on optimal play", "text": "Mastermind — Consensus on optimal play\n\n- **Start with 1122** — Knuth's optimal first guess is 1122 (two distinct colours, each appearing twice); this guess maximises the worst-case information gain and is the standard opening for the 5-guess strategy.\n- **After each response, eliminate all codes inconsistent with the feedback** — maintain (mentally or on paper) the set of codes still possible; your next guess should be chosen to minimise the size of the largest remaining group after the codemaker's response.\n- **Minimax: pick the guess that minimises the worst-case remaining codes** — at each step, for every candidate guess, compute the worst-case number of possibilities that remain; choose the guess with the smallest worst-case; this guarantees ≤5 guesses.\n- **Your guess need not itself be a possible code** — a \"non-code\" guess (a pattern you already know is wrong) can still provide useful information; don't restrict guesses to remaining possibilities if a non-code guess splits the remaining pool better.\n- **Five guesses suffice; four do not always** — Knuth proved 5 is the worst-case minimum; no strategy can guarantee a win in ≤4 guesses for all codes in the standard 4-peg, 6-colour game.\n- **For minimum expected guesses (~4.34), use a different strategy** — the minimax (worst-case) and expected-case-optimal strategies differ; if you care about average performance rather than the worst case, use the expected-case lookup table instead."}, {"id": 781, "type": "game", "source": "mastermind", "section": "Engines & current best play", "text": "Mastermind — Engines & current best play\n\n- **Strongest known program(s):** Any minimax or expected-case solver over the 1,296-code space; Knuth's 1977 algorithm is directly implementable and runs instantly.\n- **Strength:** Perfect; the worst-case-optimal strategy (≤5 guesses) and expected-case-optimal strategy (~4.34 guesses) are both fully computed.\n- **Where the proof / tablebase lives (if solved):** [Knuth (1977)](../references.md#knuth1977); complete strategy tables in subsequent exhaustive analyses.\n- **Notes:** Only 1,296 possible codes; exhaustive search is trivial — the puzzle's interest is algorithmic and mathematical, not computational."}, {"id": 782, "type": "game", "source": "mastermind", "section": "Complexity", "text": "Mastermind — Complexity\n\nOnly 1,296 possible codes, so the entire game is exhaustively searchable; the\n\"difficulty\" is purely the elegance of the minimax argument, not computational\nscale."}, {"id": 783, "type": "game", "source": "mastermind", "section": "References", "text": "Mastermind — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Mastermind_(board_game)) ([archive](http://web.archive.org/web/20260511072108/https://en.wikipedia.org/wiki/Mastermind_(board_game)))\n- [Knuth (1977). *The Computer as Master Mind*.](../references.md#knuth1977)"}, {"id": 784, "type": "game", "source": "mastermind", "section": "See also", "text": "Mastermind — See also\n\n- [Battleship](battleship.md) · [Liar's dice](liars-dice.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [minimax](../lexicon/README.md#minimax)"}, {"id": 785, "type": "game", "source": "maze-conway", "section": "overview", "text": "Maze (Conway)\nConway's \"Maze\" — a partisan path-tracing game from *On Numbers and Games*.\nSolution status: Strongly solved as a theory. Game-theoretic value: Position-dependent. Players: 2. Type: Partisan combinatorial game."}, {"id": 786, "type": "game", "source": "maze-conway", "section": "Description", "text": "Maze (Conway) — Description\n\nA short, instructive partisan game from [Conway's *On Numbers and Games*](../references.md#conway1976):\nplayers take turns moving a token through a small directed maze with edges\ncoloured for Left, Right, or either."}, {"id": 787, "type": "game", "source": "maze-conway", "section": "Rules", "text": "Maze (Conway) — Rules\n\n1. A directed graph with edges coloured **blue (L)**, **red (R)**, or **green\n   (either)** is given, with a token on a designated start node.\n2. Left moves: traverse a blue or green out-edge from the current node.\n3. Right moves: traverse a red or green out-edge.\n4. The player unable to move loses (normal play)."}, {"id": 788, "type": "game", "source": "maze-conway", "section": "Solution status", "text": "Maze (Conway) — Solution status\n\nStrongly solved as a theory. Each maze position has a CGT value computed\nrecursively; the value algebra is the standard one of *On Numbers and Games*."}, {"id": 789, "type": "game", "source": "maze-conway", "section": "Consensus on optimal play", "text": "Maze (Conway) — Consensus on optimal play\n\n- **Compute the position value bottom-up from the terminal nodes** — nodes with no out-edges are losses for the player to move (value 0 for the player without moves, computed as a CGT value); work backwards from these to assign exact values to each node.\n- **Move to the node with the most negative value (for Left) or most positive value (for Right)** — Left wants to reach a position with the highest Left-advantage; always move to the successor node with the CGT value most favourable to you.\n- **Green edges are shared resources** — a green edge that both players can traverse is a flexible move option; capturing it (by traversing it yourself) denies the opponent a future move, which may be strategically important even if it leads to a less favourable node for you.\n- **Terminal paths of only one colour are decisive** — if one player's only remaining moves lead to a dead end while the other still has options, the game is effectively won by the one with remaining moves; identify such colour-exclusive dead ends early.\n- **The value is an exact CGT number** — unlike heuristic games, every Conway Maze position has an exact surreal-number or nimber value; two positions with the same value are interchangeable, which allows simplification of compound mazes."}, {"id": 790, "type": "game", "source": "maze-conway", "section": "Engines & current best play", "text": "Maze (Conway) — Engines & current best play\n\n- **Strongest known program(s):** CGSuite (Aaron Siegel) — general CGT toolkit that evaluates any Conway Maze position exactly.\n- **Strength:** Perfect; the full theory is a standard application of CGT value calculus.\n- **Where the proof / tablebase lives (if solved):** [Conway (1976)](../references.md#conway1976); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** Conway's Maze is primarily a pedagogical example in CGT; its positions are small by design and the theory is a direct application of the surreal-number value algebra."}, {"id": 791, "type": "game", "source": "maze-conway", "section": "Complexity", "text": "Maze (Conway) — Complexity\n\nSmall."}, {"id": 792, "type": "game", "source": "maze-conway", "section": "References", "text": "Maze (Conway) — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Maze_(solitaire)) ([archive](http://web.archive.org/web/20251230102226/https://en.wikipedia.org/wiki/Maze_(solitaire)))\n- [Conway (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 793, "type": "game", "source": "maze-conway", "section": "See also", "text": "Maze (Conway) — See also\n\n- [Generalized Geography](geography.md) · [Hackenbush](hackenbush.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [surreal number](../lexicon/README.md#surreal-number)"}, {"id": 794, "type": "game", "source": "megaminx", "section": "overview", "text": "Megaminx\nDodecahedral twist puzzle — God's number not yet established.\nSolution status: Not fully solved (lower/upper bounds known). Game-theoretic value: God's number: lower bound 45 face turns; upper bound ~ several dozen **[verify]**. Players: 1. Type: Solo permutation puzzle."}, {"id": 795, "type": "game", "source": "megaminx", "section": "Description", "text": "Megaminx — Description\n\nThe Megaminx is a dodecahedral analogue of Rubik's Cube: 12 pentagonal\nfaces, 30 edges, 20 corners. Its state space is roughly 10^68 positions —\nmany orders of magnitude beyond the Rubik's Cube — so the exact diameter\n(\"God's number\") of its move graph has not been pinned down."}, {"id": 796, "type": "game", "source": "megaminx", "section": "Rules", "text": "Megaminx — Rules\n\n1. Puzzle: dodecahedron with each of 12 faces rotatable by 72°.\n2. On a move the solver rotates one face by 72° clockwise or\n   counterclockwise.\n3. The puzzle is solved when every face shows a single colour."}, {"id": 797, "type": "game", "source": "megaminx", "section": "Solution status", "text": "Megaminx — Solution status\n\nMegaminx is **not fully solved**. Specific *lower bounds* on God's number\n(currently 45 face turns) are known; tight upper bounds remain\nelusive. **[verify]** the current authoritative best bound."}, {"id": 798, "type": "game", "source": "megaminx", "section": "Consensus on optimal play", "text": "Megaminx — Consensus on optimal play\n\n- **Use layer-by-layer methods adapted for 12 faces** — the standard speed-solving approach solves the top face and top layer first, then proceeds layer by layer to the bottom, using F2L (first two layers) and OLL/PLL analogues adapted for pentagonal faces.\n- **Learn commutators for edge and corner insertion** — the Megaminx has the same piece types as the Rubik's Cube (corners, edges, centres) and the same commutator/conjugate technique for inserting a piece without disturbing already-solved pieces; the pattern is the same, just more of it.\n- **Solve \"star\" on the first face first** — the first face plus its five adjacent edge pieces (the \"star\" pattern) is the natural starting sub-goal; getting the star right sets up the entire first layer.\n- **Last layer algorithms: carry over cube knowledge** — the OLL and PLL algorithms for the Rubik's Cube last layer translate to Megaminx last-layer cases; a speed-cuber with advanced cube knowledge can apply the same patterns with adapted move sequences.\n- **For speedsolving: learn fewer algorithms, use insertions** — because the Megaminx has more cases than the cube, beginners solve it with longer human-friendly methods; competitive solvers use full-algorithm sets, but efficient insertions reduce move count."}, {"id": 799, "type": "game", "source": "megaminx", "section": "Engines & current best play", "text": "Megaminx — Engines & current best play\n\n- **Strongest known program(s):** Optimal solvers exist but are computationally expensive given the ~10^68 state space; no standard named public optimal solver for Megaminx is widely known.\n- **Strength:** Near-optimal solving for practical scrambles; God's number not yet established.\n- **Where the proof / tablebase lives (if solved):** Not fully solved; lower bound 45 face turns is known; exact God's number unknown.\n- **Notes:** State space (~10^68) vastly exceeds the Rubik's Cube (~4.3 × 10^19); a complete BFS/tablebase approach is computationally infeasible with current resources."}, {"id": 800, "type": "game", "source": "megaminx", "section": "Complexity", "text": "Megaminx — Complexity\n\nAstronomically larger than the Rubik's Cube state space."}, {"id": 801, "type": "game", "source": "megaminx", "section": "References", "text": "Megaminx — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Megaminx) ([archive](http://web.archive.org/web/20260504110719/https://en.wikipedia.org/wiki/Megaminx))\n- [Rokicki *et al.* (2017). *The diameter of the Rubik's cube group is twenty*.](../references.md#rokicki2014) (related)"}, {"id": 802, "type": "game", "source": "megaminx", "section": "See also", "text": "Megaminx — See also\n\n- [Rubik's Cube](rubiks-cube.md) · [Pocket Cube](pocket-cube.md) · [Pyraminx](pyraminx.md) · [Skewb](skewb.md)\n- Lexicon: [God's number](../lexicon/README.md#gods-number)"}, {"id": 803, "type": "game", "source": "minesweeper", "section": "overview", "text": "Minesweeper\nLogic-deduction puzzle — inference problem is NP-complete.\nSolution status: Inference subproblem is NP-complete (Kaye, 2000). Game-theoretic value: Per-puzzle. Players: 1. Type: Solo logic puzzle."}, {"id": 804, "type": "game", "source": "minesweeper", "section": "Description", "text": "Minesweeper — Description\n\nMinesweeper is the well-known Windows-era logic puzzle: numeric clues on\nuncovered cells indicate the count of adjacent mines, and the solver must\nflag all mines without detonating one. Kaye (2000) proved that the\n**Minesweeper Consistency Problem** — given a partial board, can it be\ncompleted consistently? — is **NP-complete**."}, {"id": 805, "type": "game", "source": "minesweeper", "section": "Rules", "text": "Minesweeper — Rules\n\n1. Board: rectangular grid of hidden cells; a fixed number of mines are\n   placed randomly.\n2. On a turn the solver clicks one cell:\n   - If the cell contains a mine, the game ends in loss.\n   - Otherwise it reveals a number 0–8 indicating the count of adjacent\n     mines; if the number is 0 the cell auto-clears its neighbours.\n3. The solver may **flag** a cell as a suspected mine (no consequence except\n   marker).\n4. The puzzle is solved when every non-mine cell has been revealed."}, {"id": 806, "type": "game", "source": "minesweeper", "section": "Solution status", "text": "Minesweeper — Solution status\n\nThe pure inference subproblem (deciding whether a partial board has any\nconsistent mine arrangement) is **NP-complete**. Practical games often\nrequire guesses on configurations where inference cannot decide."}, {"id": 807, "type": "game", "source": "minesweeper", "section": "Consensus on optimal play", "text": "Minesweeper — Consensus on optimal play\n\n- **Exhaust constraint propagation before guessing** — assign mines and safe cells using basic constraint logic (if a \"3\" has exactly 3 unrevealed neighbours, all are mines; if a \"1\" has exactly 1 unrevealed neighbour, it is a mine); never guess when deduction is possible.\n- **Use set-difference deduction** — if the constraint of one cell is a subset of another's constraint region, the difference gives exact mine/safe information; e.g., if cells A and B each constrain a shared area plus unique cells, subtract to determine the unique cells.\n- **When forced to guess, choose the cell with the lowest mine probability** — compute approximate mine probabilities for ambiguous regions using the remaining mine count and configuration; open the cell with the smallest chance of being a mine.\n- **The corners and edges are riskier for opening guesses** — the first click is conventionally mine-free in most implementations; open in the centre area to maximise the auto-clear cascade and expose the most cells early.\n- **Mine-counting constraints span the whole board** — the global mine count minus flagged mines limits how many mines remain; when the remaining mine count equals the number of unrevealed cells, all remaining cells are mines and can be flagged without further deduction.\n- **Guessing is sometimes unavoidable** — in approximately 1–3% of standard Expert games, even perfect play requires a 50/50 guess to complete; accept this and choose the lower-probability cell systematically."}, {"id": 808, "type": "game", "source": "minesweeper", "section": "Engines & current best play", "text": "Minesweeper — Engines & current best play\n\n- **Strongest known program(s):** Minesweeper bots combining complete constraint propagation with global probability analysis (e.g., Minesweeper Arbiter solvers used in international tournaments).\n- **Strength:** Near-optimal; human-competitive on standard boards when guesses are avoided.\n- **Where the proof / tablebase lives (if solved):** NP-completeness of the consistency problem: [Kaye (2000)](../references.md#kaye-minesweeper2000); per-puzzle solutions via constraint propagation.\n- **Notes:** Tournament Minesweeper is timed; speed of constraint propagation matters more than perfect probability calculation; human world records exist for Expert, Intermediate, and Beginner boards."}, {"id": 809, "type": "game", "source": "minesweeper", "section": "Complexity", "text": "Minesweeper — Complexity\n\nPer-instance can be exponential; consistency is NP-complete."}, {"id": 810, "type": "game", "source": "minesweeper", "section": "References", "text": "Minesweeper — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Minesweeper_(video_game)) ([archive](http://web.archive.org/web/20260501224147/https://en.wikipedia.org/wiki/Minesweeper_(video_game)))\n- [Kaye (2000). *Minesweeper is NP-complete*.](../references.md#kaye-minesweeper2000)"}, {"id": 811, "type": "game", "source": "minesweeper", "section": "See also", "text": "Minesweeper — See also\n\n- [Sudoku](sudoku.md) · [Slitherlink](slitherlink.md) · [Hashiwokakero](hashiwokakero.md)\n- Lexicon: [NP-completeness](../lexicon/README.md#np-completeness)"}, {"id": 812, "type": "game", "source": "minichess", "section": "overview", "text": "Minichess\nChess on a smaller board — several variants exist, and at least one (Gardner's\nSolution status: Partially solved (Gardner 5×5 weakly solved; others unsolved). Game-theoretic value: Gardner 5×5: draw. Larger minichess variants: unknown. Players: 2. Type: Partisan board game (chess variant)."}, {"id": 813, "type": "game", "source": "minichess", "section": "Description", "text": "Minichess — Description\n\n\"Minichess\" covers a family of chess variants on reduced boards, designed to\nkeep chess's pieces and rules while shrinking the search space. The best-known\nsolving target is **Gardner Minichess**: a 5×5 board with a full set of chess\npiece types per side. Others include **Los Alamos chess** (6×6, no bishops),\n**MicroChess** (4×5), and various 5×6 and 4×8 layouts."}, {"id": 814, "type": "game", "source": "minichess", "section": "Solution status", "text": "Minichess — Solution status\n\nMinichess is **partially solved** — it depends on the variant.\n\n- **Gardner's 5×5 Minichess is weakly solved.** [Mhalla & Prost (2013)](../references.md#vandenherik2002)\n  proved, by full game-tree search, that with perfect play it is a **draw**.\n  *(Reference: M. Mhalla & F. Prost, \"Gardner's Minichess Variant is Solved,\"\n  ICGA Journal, 2013 — **[verify]** and add to the bibliography.)*\n- Larger variants (Los Alamos 6×6, 5×6 boards, etc.) are **not** solved,\n  although they are far smaller than full chess and are plausible future\n  targets.\n\nMinichess variants are valuable as scaled-down testbeds: small enough that\nexhaustive solving is feasible for the smallest, large enough to retain real\nchess tactics."}, {"id": 815, "type": "game", "source": "minichess", "section": "Consensus on optimal play", "text": "Minichess — Consensus on optimal play\n\n- **Gardner 5×5: perfect play is a draw** — both sides can avoid losing with correct play; humans who know the basic tactical patterns (forks, pins in a tiny space) should aim to draw against any opponent.\n- **The small board amplifies tactical immediacy** — in Gardner 5×5, there is almost no \"quiet\" development phase; pieces come into contact immediately and threats must be calculated from move one; tactical alertness matters more than strategic manoeuvring.\n- **Knight forks are especially powerful on a 5×5 board** — the knight's L-shaped jump covers a significant fraction of the entire board; a misplaced piece can be forked from many squares; keep pieces out of knight-fork range of the opponent's knight.\n- **Pawns promote very quickly** — with only a few ranks to traverse, pawn races to promotion are a constant danger; count pawn-race outcomes before making other moves.\n- **Los Alamos (6×6, no bishops): open files and rooks dominate** — without bishops, the open diagonal game disappears; rooks and queens control open files, and knights cover diagonal weaknesses; seize open files early.\n- **For larger unsolved variants, use standard chess heuristics** — development, king safety, and control of the centre apply; engines based on standard chess evaluation functions play these variants well despite the absence of a formal solution."}, {"id": 816, "type": "game", "source": "minichess", "section": "Engines & current best play", "text": "Minichess — Engines & current best play\n\n- **Strongest known program(s):** For Gardner 5×5: the solver by Mhalla & Prost (2013) provides exact play; for larger variants, any strong chess engine (Stockfish, etc.) adapted for the specific piece rules plays well.\n- **Strength:** Perfect for Gardner 5×5 (fully solved); strong for larger variants.\n- **Where the proof / tablebase lives (if solved):** Gardner 5×5: Mhalla & Prost (2013), ICGA Journal — draw proven by full game-tree search.\n- **Notes:** Minichess variants are valuable scaled-down research testbeds; Gardner 5×5 is the only chess-family game with all standard piece types that has been formally solved."}, {"id": 817, "type": "game", "source": "minichess", "section": "Complexity", "text": "Minichess — Complexity\n\nVariant-dependent; the 5×5 Gardner board is small enough for a complete search."}, {"id": 818, "type": "game", "source": "minichess", "section": "References", "text": "Minichess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Minichess) ([archive](http://web.archive.org/web/20260210010246/https://en.wikipedia.org/wiki/Minichess))\n- M. Mhalla & F. Prost (2013). *Gardner's Minichess Variant is Solved*. ICGA Journal. **[verify]**\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 819, "type": "game", "source": "minichess", "section": "See also", "text": "Minichess — See also\n\n- [Chess](chess.md) · [Losing chess](losing-chess.md) · [Hexapawn](hexapawn.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 820, "type": "game", "source": "misere-nim", "section": "overview", "text": "Misère Nim\nNim played so that taking the last object *loses*; solved by a small twist on\nSolution status: Strongly solved. Game-theoretic value: Depends on heap configuration (see below). Players: 2. Type: Impartial combinatorial game (misère play)."}, {"id": 821, "type": "game", "source": "misere-nim", "section": "Description", "text": "Misère Nim — Description\n\nIdentical to [Nim](nim.md) — heaps of objects, remove any number from one heap\nper turn — but under the [misère play convention](../lexicon/README.md#misère-play)\nthe player who takes the *last* object **loses**."}, {"id": 822, "type": "game", "source": "misere-nim", "section": "Solution status", "text": "Misère Nim — Solution status\n\nMisère Nim is **strongly solved**, and the solution is almost as simple as\nnormal-play Nim. [Bouton (1901)](../references.md#bouton1901) showed: play as in\nnormal Nim (move to [nim-sum](../lexicon/README.md#nim-sum) 0) *until* your move\nwould leave only heaps of size 1. At that point, move so as to leave an **odd**\nnumber of size-1 heaps. Equivalently: with all heaps of size ≤ 1, the player to\nmove wins iff the number of heaps is even; with some heap of size ≥ 2, the\nnormal-play nim-sum rule applies unchanged.\n\nMisère Nim is famously the *easy* case: misère play of impartial games is in\ngeneral vastly harder than normal play, and a full misère theory\n(Conway's *genus theory*, later misère quotients) was needed for other games.\nNim is the exception where the misère fix is a one-line special case."}, {"id": 823, "type": "game", "source": "misere-nim", "section": "Consensus on optimal play", "text": "Misère Nim — Consensus on optimal play\n\n- **If any heap has size ≥ 2, play exactly as in normal Nim** — compute the nim-sum (XOR) of all heap sizes; move to make it 0; this is optimal identical to normal play until only size-1 heaps remain.\n- **When only size-1 heaps remain, leave an odd number** — the single-exception rule: at the moment all remaining heaps are size 1 (or would be after your move), the correct play is to leave an **odd** number of such heaps; the player facing an odd number of single-object heaps must take one, leaving an even number for the opponent, who can mirror until the opponent takes the last one.\n- **The \"switch\" moment is the key calculation** — identify in advance the position where all heaps collapse to size 1; your last move with a heap of size ≥ 2 should also leave the correct (odd/even) parity of unit heaps.\n- **If all heaps are size 1 already, count and parity decides immediately** — even number of size-1 heaps: the player to move loses; odd number: wins; no further calculation needed.\n- **Misère Nim is the easy misère case** — for most other impartial games, misère theory is far harder; Nim's misère rule is a special one-line exception, not a general template."}, {"id": 824, "type": "game", "source": "misere-nim", "section": "Engines & current best play", "text": "Misère Nim — Engines & current best play\n\n- **Strongest known program(s):** Any nim-sum calculator solves Misère Nim in O(n) time; no dedicated software needed.\n- **Strength:** Perfect; the exact strategy is a closed-form rule.\n- **Where the proof / tablebase lives (if solved):** [Bouton (1901)](../references.md#bouton1901); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** The misère rule for Nim is exceptionally simple compared to misère play for other impartial games, which require the full genus theory or misère quotient machinery."}, {"id": 825, "type": "game", "source": "misere-nim", "section": "Complexity", "text": "Misère Nim — Complexity\n\nAs with Nim, a family of positions rather than one game; optimal play is\ncomputable in linear time per position."}, {"id": 826, "type": "game", "source": "misere-nim", "section": "References", "text": "Misère Nim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Bouton, C. L. (1901). *Nim, A Game with a Complete Mathematical Theory*.](../references.md#bouton1901)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)"}, {"id": 827, "type": "game", "source": "misere-nim", "section": "See also", "text": "Misère Nim — See also\n\n- [Nim](nim.md) · [Notakto](notakto.md) (misère X-only tic-tac-toe)\n- Lexicon: [misère play](../lexicon/README.md#misère-play) · [nim-sum](../lexicon/README.md#nim-sum)"}, {"id": 828, "type": "game", "source": "mnk-games", "section": "overview", "text": "m,n,k-games\nThe general family of \"tic-tac-toe on an m × n board with k-in-a-row to\nSolution status: Strongly classified (theory). Game-theoretic value: Draw or first-player win, depending on (m, n, k). Players: 2. Type: Partisan k-in-a-row games (family)."}, {"id": 829, "type": "game", "source": "mnk-games", "section": "Description", "text": "m,n,k-games — Description\n\nThe unified theory of \"tic-tac-toe family\" games. An m,n,k-game is played on an\nm × n grid; the goal is to be the first player to form an unbroken line of k\nof your marks in any straight direction."}, {"id": 830, "type": "game", "source": "mnk-games", "section": "Rules", "text": "m,n,k-games — Rules\n\n1. Board: m × n grid, empty initially.\n2. Players alternate placing one of their marks (X or O) on any empty cell.\n3. The first player to **align k of their marks** in a single horizontal,\n   vertical, or diagonal line wins.\n4. If the board fills with no winning line, the game is a draw."}, {"id": 831, "type": "game", "source": "mnk-games", "section": "Solution status", "text": "m,n,k-games — Solution status\n\nStrongly classified:\n\n- **Strategy-stealing** shows that the second player **cannot win** any\n  m,n,k-game — only first-player win or draw are possible.\n- **Tic-tac-toe (3,3,3)** is a draw — [classic](tic-tac-toe.md).\n- **Gomoku (15,15,5)** is a first-player win\n  ([Allis 1993/1996](../references.md#allis-gomoku1996)).\n- **Qubic (4,4,4,4)** (the 4-D version) and **(4,4,4)** are first-player wins;\n  see [Qubic](qubic.md).\n- For **k ≥ 8**, no m,n,k-game is a first-player win for any m,n —\n  ([Hales–Jewett type pairing argument, classical](../references.md#bcg2001)).\n- For **k = 6** and **k = 7**, full classification is known for many m,n.\n\nThe general \"for which (m, n, k) is the game a first-player win?\" question is\nnontrivial; see *Winning Ways* and Beck's *Combinatorial Games: Tic-Tac-Toe\nTheory* for the comprehensive treatment."}, {"id": 832, "type": "game", "source": "mnk-games", "section": "Consensus on optimal play", "text": "m,n,k-games — Consensus on optimal play\n\n- **The second player can never win** — strategy-stealing proves this universally; only first-player wins or draws occur; if you are second player, aim for a draw.\n- **For k ≥ 8, any board is a draw** — pairing/strategy arguments prove no m,n,k-game with k ≥ 8 is a first-player win regardless of board size; if the winning-line length is 8 or more, the first player cannot force a win.\n- **For small k (k = 3, 4, 5) on large boards, threat-tree attacks win for first player** — the first player builds double open-k−1 threats (two simultaneous nearly-complete lines); forcing sequences that create unblockable forks are the winning mechanism.\n- **Centre cells dominate for small boards** — cells that lie on the most k-length winning lines should be taken first; in (3,3,3) the centre is on 4 lines, corners on 2, edges on 2; in larger boards, central cells are similarly privileged.\n- **Pairing strategies give drawing algorithms** — for draw-valued games, a pairing argument assigns each cell a unique \"partner\"; whenever first player plays in a cell, second player responds in the partner; this ensures first player never completes a line.\n- **For specific cases, threat-space search gives exact results** — the Allis (1993/1996) technique for Gomoku applies to any m,n,k-game: chain compulsory threat sequences to prove a first-player win."}, {"id": 833, "type": "game", "source": "mnk-games", "section": "Engines & current best play", "text": "m,n,k-games — Engines & current best play\n\n- **Strongest known program(s):** Game-specific solvers for each (m,n,k) instance — threat-space search engines, retrograde analysis; no single universal engine for the whole family.\n- **Strength:** Exact for settled cases (3,3,3), (15,15,5), (4,4,4), etc.; heuristic for borderline unsettled cases.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001); [Allis, van den Herik & Huntjens (1996)](../references.md#allis-gomoku1996); case-by-case literature.\n- **Notes:** The complete classification of m,n,k-game values (for all m,n,k) remains an open problem in combinatorial game theory; individual instances are often solved but the general boundary between \"draw\" and \"first-player win\" is not fully characterised."}, {"id": 834, "type": "game", "source": "mnk-games", "section": "Complexity", "text": "m,n,k-games — Complexity\n\nm,n,k-games on growing boards are PSPACE-hard in general — the value problem\ngeneralises the standard hardness of generalised tic-tac-toe."}, {"id": 835, "type": "game", "source": "mnk-games", "section": "References", "text": "m,n,k-games — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/M,n,k-game) ([archive](http://web.archive.org/web/20260307135720/https://en.wikipedia.org/wiki/M,n,k-game))\n- [Allis, van den Herik & Huntjens (1996). *Go-Moku Solved by New Search Techniques*.](../references.md#allis-gomoku1996)\n- [Patashnik (1980). *Qubic*.](../references.md#patashnik1980)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways*.](../references.md#bcg2001)"}, {"id": 836, "type": "game", "source": "mnk-games", "section": "See also", "text": "m,n,k-games — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Gomoku](gomoku.md) · [Qubic](qubic.md) · [Connect Four](connect-four.md)\n- Lexicon: [pairing strategy](../lexicon/README.md#pairing-strategy) · [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 837, "type": "game", "source": "mock-turtles", "section": "overview", "text": "Mock Turtles\nA coin-turning game whose every position decomposes into a tidy nim-sum of\nSolution status: Strongly solved. Game-theoretic value: Depends on the coin configuration (nim-sum of component values). Players: 2. Type: Impartial combinatorial game (coin-turning game)."}, {"id": 838, "type": "game", "source": "mock-turtles", "section": "Description", "text": "Mock Turtles — Description\n\nA row of coins, each heads or tails. On a turn a player turns over up to **three**\ncoins, of which the **rightmost must go from heads to tails**. The game ends\nwhen all coins are tails; under [normal play](../lexicon/README.md#normal-play-convention)\nthe player making the last move wins."}, {"id": 839, "type": "game", "source": "mock-turtles", "section": "Solution status", "text": "Mock Turtles — Solution status\n\nMock Turtles is **strongly solved** as one of the classic *coin-turning games*.\nIn any coin-turning game, a position is the [nim-sum](../lexicon/README.md#nim-sum)\nof the single-coin games given by each heads coin, so the whole game reduces to\nknowing the [nim-value](../lexicon/README.md#nim-value) of \"a single head in\nposition *n*.\" For Mock Turtles those values are the **Mock Turtle numbers**:\nthe *n*-th value is 2n or 2n+1, chosen to have an odd number of binary 1s\n(an \"odious\" number). With this closed form every position's value and an\noptimal move are immediate."}, {"id": 840, "type": "game", "source": "mock-turtles", "section": "Consensus on optimal play", "text": "Mock Turtles — Consensus on optimal play\n\n- **XOR all Mock Turtle values for heads coins** — for each coin at position n (0-indexed from the right) that is heads, look up the Mock Turtle number for n; nim-sum (XOR) all those values; if the result is non-zero you are in a winning position.\n- **Winning move: choose coins to XOR the nim-sum to 0** — find a set of up to three coins (rightmost must be flipped from heads to tails) whose combined flip reduces the nim-sum to 0; this is always possible from a non-zero position.\n- **Mock Turtle value of position n is the odious number nearest to 2n** — \"odious\" means having an odd number of 1-bits in binary; the Mock Turtle value is 2n if 2n is odious, otherwise 2n+1; memorise or compute this quickly.\n- **Turning multiple coins can target multiple positions** — the flexibility to flip up to three coins (rightmost from heads to tails) means you can alter the nim-sum by changing up to three components; use this freedom to zeroise the nim-sum efficiently.\n- **All-tails is a losing position for the player to move** — the empty position (all tails) has nim-value 0 and is a P-position (previous player wins); steer toward leaving your opponent with all-tails."}, {"id": 841, "type": "game", "source": "mock-turtles", "section": "Engines & current best play", "text": "Mock Turtles — Engines & current best play\n\n- **Strongest known program(s):** Any nim-value calculator implementing the Mock Turtle closed form; no dedicated software needed.\n- **Strength:** Perfect; the strategy is a closed-form calculation requiring only XOR operations.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001); [Guy & Smith (1956)](../references.md#guy-smith1956).\n- **Notes:** Mock Turtles is a classic coin-turning game; the \"Mock Turtle numbers\" (odious numbers near 2n) provide a clean closed-form solution demonstrating the power of the nim-value calculus."}, {"id": 842, "type": "game", "source": "mock-turtles", "section": "Complexity", "text": "Mock Turtles — Complexity\n\nLinear in the row length to evaluate a position."}, {"id": 843, "type": "game", "source": "mock-turtles", "section": "References", "text": "Mock Turtles — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)"}, {"id": 844, "type": "game", "source": "mock-turtles", "section": "See also", "text": "Mock Turtles — See also\n\n- [Turning Turtles](turning-turtles.md) · [Nim](nim.md) · [Kayles](kayles.md)\n- Lexicon: [nim-sum](../lexicon/README.md#nim-sum) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 845, "type": "game", "source": "mock-wythoff", "section": "overview", "text": "Mock Wythoff\nA Wythoff-like two-pile game with a slightly modified diagonal move — its\nSolution status: Partial / largely characterised **[verify]**. Game-theoretic value: First-player win except on a sparse P-set. Players: 2. Type: Impartial two-pile game."}, {"id": 846, "type": "game", "source": "mock-wythoff", "section": "Description", "text": "Mock Wythoff — Description\n\nMock Wythoff modifies the diagonal move of [Wythoff's game](wythoffs-game.md):\nwhere Wythoff allows removal of the same number from both piles, Mock Wythoff\nallows the diagonal move only when the two amounts are nearly but not exactly\nequal (or restricts diagonal moves to a parity class). Different authors define\n\"mock Wythoff\" with small variations; specific rule statements differ\n**[verify]**."}, {"id": 847, "type": "game", "source": "mock-wythoff", "section": "Rules", "text": "Mock Wythoff — Rules\n\n1. Two piles of tokens, sizes (a, b).\n2. On your turn, pick any one of:\n   - Remove any positive number from a single pile (Nim move);\n   - Remove (k, k+1) tokens (or some other diagonal-with-offset move),\n     depending on the variant.\n3. The player who takes the last token wins (normal play)."}, {"id": 848, "type": "game", "source": "mock-wythoff", "section": "Solution status", "text": "Mock Wythoff — Solution status\n\nPartially solved. For specific variants of Mock Wythoff the **P-positions** can\nbe written using Beatty sequences with irrational moduli closely related to the\ngolden ratio, with proofs by induction analogous to Wythoff's classical\nanalysis. A clean general theory matching the elegance of plain Wythoff has not,\nto this archive's knowledge, been published; treat detailed value claims as\n**[verify]**."}, {"id": 849, "type": "game", "source": "mock-wythoff", "section": "Consensus on optimal play", "text": "Mock Wythoff — Consensus on optimal play\n\n- **Compute P-positions by induction** — for any specific rule set, enumerate P-positions from (0,0) upward: (a,b) is a P-position if no move from (a,b) lands on a P-position; the pattern becomes recognisable quickly for small piles.\n- **In Wythoff-like games, steer toward the P-set** — the P-positions in Mock Wythoff form a sparse set, likely following a Beatty-sequence pattern similar to Wythoff's (floor(nφ), floor(nφ²)); from a winning position, any move that reaches a P-position wins.\n- **The offset diagonal move is the key tactical tool** — the modified diagonal (e.g., remove (k, k+1)) is the move that differs from ordinary Nim; use it to reach specific pile-difference targets that land on P-positions.\n- **Pile differences matter** — in Wythoff-family games, the difference |a − b| between pile sizes plays a role analogous to a pile count in regular Nim; keep track of differences as you enumerate P-positions.\n- **When in doubt, enumerate small cases and look for periodicity** — the P-position table for reasonable pile sizes is quickly computed; once the pattern is recognised, it serves as a decision rule for the rest of the game."}, {"id": 850, "type": "game", "source": "mock-wythoff", "section": "Engines & current best play", "text": "Mock Wythoff — Engines & current best play\n\n- **Strongest known program(s):** Any two-pile nim-value enumerator; the P-positions are computable by straightforward retrograde analysis for any pile bound.\n- **Strength:** Perfect for specific rule sets once P-positions are enumerated.\n- **Where the proof / tablebase lives (if solved):** Partially solved for specific variants; [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001) covers Wythoff variants generally; specific Mock Wythoff publications vary by rule definition.\n- **Notes:** The exact rule set of \"Mock Wythoff\" varies across sources; verify the specific diagonal-move modification before applying any stated P-position formula."}, {"id": 851, "type": "game", "source": "mock-wythoff", "section": "Complexity", "text": "Mock Wythoff — Complexity\n\nSmall: the position graph for any reasonable bound on pile size is enumerable\nand the P-positions become recognisably structured very quickly."}, {"id": 852, "type": "game", "source": "mock-wythoff", "section": "References", "text": "Mock Wythoff — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Wythoff%27s_game) ([archive](http://web.archive.org/web/20251206112849/https://en.wikipedia.org/wiki/Wythoff%27s_game))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001) (general framework)\n- [Wythoff (1907). *A modification of the game of Nim*.](../references.md#wythoff1907)"}, {"id": 853, "type": "game", "source": "mock-wythoff", "section": "See also", "text": "Mock Wythoff — See also\n\n- [Wythoff's game](wythoffs-game.md) · [Nim](nim.md) · [Fibonacci Nim](fibonacci-nim.md)\n- Lexicon: [nim-value](../lexicon/README.md#nim-value) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)"}, {"id": 854, "type": "game", "source": "mogul", "section": "overview", "text": "Mogul\nAn octal-game cousin of Mock Turtles — its nim-values are tabulated.\nSolution status: Strongly solved. Game-theoretic value: Determined by nim-sum of single-coin nim-values. Players: 2. Type: Impartial turning game / octal game."}, {"id": 855, "type": "game", "source": "mogul", "section": "Description", "text": "Mogul — Description\n\nA \"turning-coins\" game in the *Winning Ways* tradition: a row of coins is given,\neach face-up or face-down, and players take turns flipping coins under\nconstraints chosen so the resulting nim-sequence is rich and irregular."}, {"id": 856, "type": "game", "source": "mogul", "section": "Rules", "text": "Mogul — Rules\n\n1. A row of n coins, each \"heads\" or \"tails,\" is laid out in positions\n   1, 2, ..., n.\n2. On your turn, choose one of the allowed flipping patterns from the octal\n   code defining Mogul (turning 2 or 3 coins at chosen positions, subject to\n   the rightmost flipped coin going from heads to tails). The exact octal code\n   is **.55... [verify]** — different sources differ on the precise digit.\n3. The player who cannot move loses (normal play).\n\nTurning-game positions decompose by **Mock-Turtles theorem**: the nim-value of a\nposition is the nim-sum of the nim-values of single heads-coins."}, {"id": 857, "type": "game", "source": "mogul", "section": "Solution status", "text": "Mogul — Solution status\n\nStrongly solved by [Guy & Smith (1956)](../references.md#guy-smith1956): the\nsingle-coin nim-values form a tabulated sequence (well-defined by the octal\ncode), and the game value of any position is the nim-sum of nim-values of\nheads-coins. Optimal play follows."}, {"id": 858, "type": "game", "source": "mogul", "section": "Consensus on optimal play", "text": "Mogul — Consensus on optimal play\n\n- **Precompute the single-coin nim-value table** — the nim-value of a single heads-coin in position n is determined by the game's octal code via the mex recurrence; tabulate these values for n = 0, 1, 2, ... up to the board size.\n- **XOR all single-coin values to get the position value** — once the table is available, XOR (nim-sum) the values for every heads-coin position; if the result is non-zero you are in a winning (N-) position.\n- **Winning move: find a flip that reduces the nim-sum to 0** — among all legal moves (flipping 2 or 3 coins, rightmost going from heads to tails), choose the one that makes the nim-sum of the resulting position equal to 0.\n- **The rightmost flipped coin constraint is the key tactical limitation** — every legal move must flip at least one coin from heads to tails (the rightmost in the chosen group); this is what makes the game finite and the nim-sequence well-defined.\n- **Use the Mock-Turtles decomposition principle** — positions decompose into independent single-coin games; updating only the affected single-coin values after each move (rather than recomputing from scratch) makes calculation efficient over the board."}, {"id": 859, "type": "game", "source": "mogul", "section": "Engines & current best play", "text": "Mogul — Engines & current best play\n\n- **Strongest known program(s):** Any nim-value calculator implementing the octal-code mex recurrence; no dedicated software needed beyond a lookup table.\n- **Strength:** Perfect; the game is strongly solved via Guy & Smith's nim-value tabulation.\n- **Where the proof / tablebase lives (if solved):** [Guy & Smith (1956)](../references.md#guy-smith1956); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001).\n- **Notes:** The exact octal code defining Mogul should be verified against a primary source before constructing the nim-value table; different authors use slightly different octal digits."}, {"id": 860, "type": "game", "source": "mogul", "section": "Complexity", "text": "Mogul — Complexity\n\nTrivial — O(n) to evaluate a position."}, {"id": 861, "type": "game", "source": "mogul", "section": "References", "text": "Mogul — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Combinatorial_game_theory) ([archive](http://web.archive.org/web/20260508023449/https://en.wikipedia.org/wiki/Combinatorial_game_theory))\n- [Guy & Smith (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 862, "type": "game", "source": "mogul", "section": "See also", "text": "Mogul — See also\n\n- [Mock Turtles](mock-turtles.md) · [Turning Turtles](turning-turtles.md) · [Ruler game](ruler-game.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [nim-sum](../lexicon/README.md#nim-sum)"}, {"id": 863, "type": "game", "source": "mu-torere", "section": "overview", "text": "Mū tōrere\nA Māori game on an eight-pointed star — small, elegant, and a draw with\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan sliding/blocking game."}, {"id": 864, "type": "game", "source": "mu-torere", "section": "Description", "text": "Mū tōrere — Description\n\nPlayed on a star with eight outer points (*kewai*) and one centre (*pūtahi*).\nEach player has four pieces, initially occupying four adjacent outer points. On\na turn a player slides a piece to an *adjacent empty* point — but a move from an\nouter point into the centre is allowed only if at least one of the moving\npiece's neighbours is an enemy piece. A player who cannot move loses (the\nopponent has blocked them)."}, {"id": 865, "type": "game", "source": "mu-torere", "section": "Solution status", "text": "Mū tōrere — Solution status\n\nMū tōrere is **strongly solved**. The state space is tiny — only a few dozen\npositions — so it is fully exhaustible, and [Ascher (1987)](../references.md#ascher-mutorere1987)\ngave a complete mathematical analysis. With perfect play the game is a **draw**:\nneither player can force the opponent into a no-move position against correct\ndefence. (The restriction on moving into the centre is exactly what prevents a\nquick forced win and makes the drawn structure work.)\n\nMū tōrere is a frequently cited example in ethnomathematics of a traditional\ngame whose full game-theoretic structure has been rigorously worked out."}, {"id": 866, "type": "game", "source": "mu-torere", "section": "Consensus on optimal play", "text": "Mū tōrere — Consensus on optimal play\n\n- **Keep your four pieces adjacent** — the initial configuration has each player's pieces occupying four consecutive outer points; separating your pieces by moving one too far from the group creates a piece that cannot easily return to contribute to a blocking formation.\n- **Guard the centre access condition** — a piece can only move to the centre if it is adjacent to at least one enemy piece; if you occupy the centre, your opponent may be denied access to it if their pieces are not adjacent to yours; use this to control centre traffic.\n- **Avoid zugzwang positions** — the game ends when a player cannot move; avoid moving into a position where all your pieces are blocked by friendly pieces and the centre is out of reach; every move should maintain at least two legal responses on your next turn.\n- **Mirror the opponent's tempo** — in the draw-structure of the game, matching the opponent's moves on the opposite arc of the star tends to preserve the balance; breaking symmetry carelessly can create the blocked position that loses.\n- **The restricted centre-entry rule is the whole game** — without the restriction on entering the centre, the game would be trivially won by rushing to the centre; understanding exactly when you can and cannot use the centre is the decisive tactical knowledge."}, {"id": 867, "type": "game", "source": "mu-torere", "section": "Engines & current best play", "text": "Mū tōrere — Engines & current best play\n\n- **Strongest known program(s):** Any exhaustive tree-search over the ~few-dozen-position graph; no dedicated software needed.\n- **Strength:** Perfect; the entire position graph has been enumerated.\n- **Where the proof / tablebase lives (if solved):** [Ascher (1987)](../references.md#ascher-mutorere1987) — complete mathematical analysis.\n- **Notes:** A Māori traditional game; cited in ethnomathematics as a clear example of a culturally significant game with a rigorously proven game-theoretic value (draw)."}, {"id": 868, "type": "game", "source": "mu-torere", "section": "Complexity", "text": "Mū tōrere — Complexity\n\nA few dozen positions — trivially exhaustible."}, {"id": 869, "type": "game", "source": "mu-torere", "section": "References", "text": "Mū tōrere — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/M%C5%AB_t%C5%8Drere) ([archive](http://web.archive.org/web/20260304162546/https://en.wikipedia.org/wiki/M%C5%AB_t%C5%8Drere))\n- [Ascher, M. (1987). *Mu Torere: An Analysis of a Maori Game*.](../references.md#ascher-mutorere1987)"}, {"id": 870, "type": "game", "source": "mu-torere", "section": "See also", "text": "Mū tōrere — See also\n\n- [L game](l-game.md) · [Pong hau k'i](pong-hau-ki.md) · [Tic-tac-toe](tic-tac-toe.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw) · [zugzwang](../lexicon/README.md#zugzwang)"}, {"id": 871, "type": "game", "source": "nada", "section": "overview", "text": "Nada!\nA real-time pattern-recognition dice game — unsolved as a competitive speed game.\nSolution status: Unsolved (real-time speed element precludes any known solution framework). Game-theoretic value: Unknown. Players: 2–4. Type: Real-time stochastic dice game."}, {"id": 872, "type": "game", "source": "nada", "section": "Description", "text": "Nada! — Description\n\nNada! is a fast-paced dice game for 2–4 players designed by Thierry Denoual and\nfirst published in 2013 by Blue Orange Games. Players simultaneously race to\nmatch symbols across a pool of 12 dice (6 white and 6 orange). On each turn,\nall dice are rolled in the centre. Every player searches for a symbol that\nappears on at least one white die and one orange die; the first to spot a match\ncalls it out and collects all dice showing that symbol. If no match exists at\nall, the first player to shout \"Nada!\" collects the entire pool. The player\nwith the most dice after three rounds wins."}, {"id": 873, "type": "game", "source": "nada", "section": "Solution status", "text": "Nada! — Solution status\n\nNada! is **not solved** and does not fit any standard combinatorial-game\nframework. It is a **real-time simultaneous-action game** in which outcomes\ndepend on perceptual speed and reaction time, not on a game tree that can be\nsearched. The probabilistic distribution of symbols across dice is trivial to\ncompute, but that is only a small sub-component of the game. There is no known\nformal framework for solving a real-time pattern-matching speed game."}, {"id": 874, "type": "game", "source": "nada", "section": "Consensus on optimal play", "text": "Nada! — Consensus on optimal play\n\n- **Scan systematically** — train your eyes to sweep the dice in a consistent\n  pattern (e.g., left-to-right across whites, then left-to-right across oranges,\n  or focus on a single symbol at a time) rather than randomly glancing.\n- **Know the symbols** — memorise the set of symbols so recognition becomes\n  automatic; hesitation costs a beat.\n- **Use the Nada call aggressively** — if a quick scan shows no obvious match,\n  call Nada immediately rather than verifying exhaustively; a wrong call sits\n  you out but a successful Nada wins the whole pool.\n- **Watch opponents' gaze** — if another player looks ready to call, treat that\n  as a cue to check whether they might be right or whether you can beat them to\n  a different match."}, {"id": 875, "type": "game", "source": "nada", "section": "References", "text": "Nada! — References\n\n- [Blue Orange Games — Nada! product page](https://www.blueorangegames.com/index.php/games/nada)\n- [Rules on UltraBoardGames](https://www.ultraboardgames.com/nada/game-rules.php)\n- [BGG entry — Nada! (2013)](https://boardgamegeek.com/boardgame/36946/nada)"}, {"id": 876, "type": "game", "source": "nada", "section": "See also", "text": "Nada! — See also\n\n- [Yahtzee](yahtzee.md) · [Liar's dice](liars-dice.md) · [Backgammon](backgammon.md)\n- Lexicon: [chance element](../lexicon/README.md#chance-element)"}, {"id": 877, "type": "game", "source": "nim", "section": "overview", "text": "Nim\nThe foundational solved game: a complete mathematical theory has been known\nSolution status: Strongly solved. Game-theoretic value: First-player win unless the nim-sum of all heaps is 0. Players: 2. Type: Impartial combinatorial game."}, {"id": 878, "type": "game", "source": "nim", "section": "Description", "text": "Nim — Description\n\nSeveral heaps of objects are placed between two players. On a turn a player\nremoves any positive number of objects from a single heap. Under the\n[normal play convention](../lexicon/README.md#normal-play-convention) the\nplayer who takes the last object wins; the [misère](../lexicon/README.md#misère-play)\nvariant is treated separately ([Misère Nim](misere-nim.md))."}, {"id": 879, "type": "game", "source": "nim", "section": "Solution status", "text": "Nim — Solution status\n\nNim is **strongly solved**. [Bouton (1901)](../references.md#bouton1901) gave a\ncomplete theory: compute the [nim-sum](../lexicon/README.md#nim-sum) — the\nbitwise XOR of the heap sizes. The position is a loss for the player to move\n(a second-player win) if and only if the nim-sum is 0; otherwise the player to\nmove wins, and a winning move always exists that makes the nim-sum 0.\n\nNim's importance extends far beyond itself: by the\n[Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem) *every*\nfinite impartial game under normal play is equivalent to a single Nim heap, so\nNim is in effect the universal impartial game."}, {"id": 880, "type": "game", "source": "nim", "section": "Consensus on optimal play", "text": "Nim — Consensus on optimal play\n\n- **XOR to zero** — compute the nim-sum (bitwise XOR of all heap sizes); the position is a second-player win iff the nim-sum is 0; otherwise move to make it 0.\n- **Reduce the largest heap** — when multiple heaps are large, a winning move often targets the largest heap to restore nim-sum 0, especially in end-game positions.\n- **Single heap is trivial** — with one heap left, take everything (normal play) or leave one object (misère).\n- **Misère exception** — under misère play, use the same nim-sum strategy except when all heaps are size ≤ 1; in that case leave an odd number of heaps.\n- **The Sprague–Grundy lens** — any impartial position is equivalent to a single Nim heap of some Grundy value; combine components by XOR-ing their Grundy values."}, {"id": 881, "type": "game", "source": "nim", "section": "Engines & current best play", "text": "Nim — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed — the strategy is a closed-form formula.\n- **Strength:** Perfect play by any computer implementing Bouton's formula.\n- **Where the proof / tablebase lives (if solved):** [Bouton (1901)](../references.md#bouton1901); [Wikipedia](https://en.wikipedia.org/wiki/Nim)\n- **Notes:** Nim is solved by a polynomial-time formula, not search; any correct implementation plays perfectly."}, {"id": 882, "type": "game", "source": "nim", "section": "Complexity", "text": "Nim — Complexity\n\nNim is a family rather than a single position, so it has no fixed complexity.\nFor any specific starting configuration the optimal strategy is computable in\ntime linear in the number of heaps."}, {"id": 883, "type": "game", "source": "nim", "section": "References", "text": "Nim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Bouton, C. L. (1901). *Nim, A Game with a Complete Mathematical Theory*.](../references.md#bouton1901)\n- [Sprague, R. P. (1935). *Über mathematische Kampfspiele*.](../references.md#sprague1935)\n- [Grundy, P. M. (1939). *Mathematics and games*.](../references.md#grundy1939)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 884, "type": "game", "source": "nim", "section": "See also", "text": "Nim — See also\n\n- [Misère Nim](misere-nim.md) · [Wythoff's game](wythoffs-game.md) · [Kayles](kayles.md) · [Northcott's game](northcotts-game.md) · [Turning Turtles](turning-turtles.md)\n- Lexicon: [nim-sum](../lexicon/README.md#nim-sum) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem) · [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 885, "type": "game", "source": "nine-holes", "section": "overview", "text": "Nine Holes\nA medieval ancestor of tic-tac-toe with a *moving* phase — and the same\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan positional / sliding game."}, {"id": 886, "type": "game", "source": "nine-holes", "section": "Description", "text": "Nine Holes — Description\n\nPlayed on a 3×3 grid of holes. Each player has three pieces. In the **placement\nphase** players alternately place their three pieces; in the **movement phase**\nthey slide a piece to an adjacent empty hole. The goal is three in a row.\nVariants differ on whether diagonals count and whether moves are to any empty\nhole or only adjacent ones; \"Nine Holes\" proper typically does **not** count\nthe placement-phase rows, requiring a line to be made by moving.\n\n> Note: terminology is muddled in historical sources — \"Nine Holes,\" \"Three\n> Men's Morris,\" and \"Achi\" are closely related and sometimes conflated. This\n> archive keeps [Three Men's Morris](three-mens-morris.md) and [Achi](achi.md)\n> as separate entries."}, {"id": 887, "type": "game", "source": "nine-holes", "section": "Solution status", "text": "Nine Holes — Solution status\n\nNine Holes is **strongly solved** by exhaustive analysis — its state space is\nonly a few thousand positions. Under the usual rules, with perfect play the game\nis a **draw**: neither side can force a line against correct defence. As with\n[tic-tac-toe](tic-tac-toe.md), a player can only lose by error."}, {"id": 888, "type": "game", "source": "nine-holes", "section": "Consensus on optimal play", "text": "Nine Holes — Consensus on optimal play\n\n- **Placement determines the endgame** — place pieces to threaten two winning lines simultaneously; this forces the opponent into a defensive posture and limits their mobility in the movement phase.\n- **Maintain a fork threat** — having two separate two-in-a-row threats is decisive since the opponent can only block one sliding move at a time.\n- **Control the centre** — the centre hole participates in the most potential lines; occupying it during placement restricts the opponent's winning paths.\n- **Never voluntarily unblock an opponent's two-in-a-row** — in the sliding phase, always check whether the planned move opens a line the opponent can immediately complete.\n- **Draw is the correct result** — with correct defence on both sides no forced win exists; a player who loses has made an error."}, {"id": 889, "type": "game", "source": "nine-holes", "section": "Engines & current best play", "text": "Nine Holes — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The game is exhaustively solvable in milliseconds; any complete search confirms the draw verdict."}, {"id": 890, "type": "game", "source": "nine-holes", "section": "Complexity", "text": "Nine Holes — Complexity\n\nA few thousand positions — trivially exhaustible."}, {"id": 891, "type": "game", "source": "nine-holes", "section": "References", "text": "Nine Holes — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Three_men%27s_morris) ([archive](http://web.archive.org/web/20251231113758/https://en.wikipedia.org/wiki/Three_men's_morris))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 892, "type": "game", "source": "nine-holes", "section": "See also", "text": "Nine Holes — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Achi](achi.md) · [Tic-tac-toe](tic-tac-toe.md) · [Six Men's Morris](six-mens-morris.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 893, "type": "game", "source": "nine-mens-morris", "section": "overview", "text": "Nine Men's Morris\nOne of the oldest board games still played — and weakly solved in 1993 as a\nSolution status: Weakly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan placement-and-movement (\"mill\") game."}, {"id": 894, "type": "game", "source": "nine-mens-morris", "section": "Description", "text": "Nine Men's Morris — Description\n\nPlayed on a board of three concentric squares joined by midlines (24 points).\nEach player has nine pieces. In the **placement phase** players place all nine\npieces alternately; in the **movement phase** they slide a piece to an adjacent\nempty point. Forming a *mill* (three pieces in a marked line) removes an enemy\npiece. A player reduced to two pieces, or with no legal move, loses. (A \"flying\"\nrule for a player down to three pieces is common.)"}, {"id": 895, "type": "game", "source": "nine-mens-morris", "section": "Solution status", "text": "Nine Men's Morris — Solution status\n\nNine Men's Morris is **weakly solved**. [Gasser (1996)](../references.md#gasser1996)\nsolved it by building complete endgame databases through\n[retrograde analysis](../lexicon/README.md#retrograde-analysis) and then\nperforming an alpha-beta search from the opening that meets those databases. The\nresult: with perfect play by both sides the game is a **draw**.\n\nIt was, at the time, one of the more complex games to be solved (~10^10\npositions), and remains a standard reference point for the\nretrograde-analysis-plus-search methodology later used on larger games."}, {"id": 896, "type": "game", "source": "nine-mens-morris", "section": "Consensus on optimal play", "text": "Nine Men's Morris — Consensus on optimal play\n\n- **Placement determines the game** — place pieces to set up two potential mills rather than a single one; an opponent who must block one line will leave the other open.\n- **Prioritise double mills** — a configuration where one piece can slide back and forth between two mills generates a forced removal each turn, overwhelming any defence.\n- **Remove the opponent's \"flying\" candidate last** — pieces that can't form mills have no positional value; remove pieces that are part of active mill threats first.\n- **Keep three pieces active near the centre junctions** — the four corner points of the inner square participate in more potential mills than edge midpoints.\n- **Never allow yourself to be reduced to two pieces** — manage captures to stay above three pieces; once in \"flying\" mode the game is very hard to rescue from a deficit.\n- **Draw with correct play** — neither side can force a win against accurate defence; objective is not to err in the placement phase."}, {"id": 897, "type": "game", "source": "nine-mens-morris", "section": "Engines & current best play", "text": "Nine Men's Morris — Engines & current best play\n\n- **Strongest known program(s):** Gasser's solver (Ralph Gasser, University of Berne, 1993) — retrograde analysis + alpha-beta search.\n- **Strength:** Perfect play (weakly solved); any implementation of Gasser's database plays perfectly.\n- **Where the proof / tablebase lives (if solved):** [Gasser (1996)](../references.md#gasser1996)\n- **Notes:** The solving databases are not publicly downloadable, but the result (draw) is universally accepted."}, {"id": 898, "type": "game", "source": "nine-mens-morris", "section": "Complexity", "text": "Nine Men's Morris — Complexity\n\nState-space ~10^10; game-tree ~10^50\n([van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 899, "type": "game", "source": "nine-mens-morris", "section": "References", "text": "Nine Men's Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nine_men%27s_morris) ([archive](http://web.archive.org/web/20260328233608/https://en.wikipedia.org/wiki/Nine_Men%27s_Morris))\n- [Gasser, R. (1996). *Solving Nine Men's Morris*.](../references.md#gasser1996)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 900, "type": "game", "source": "nine-mens-morris", "section": "See also", "text": "Nine Men's Morris — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Six Men's Morris](six-mens-morris.md) · [Twelve Men's Morris](twelve-mens-morris.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [endgame tablebase](../lexicon/README.md#endgame-tablebase)"}, {"id": 901, "type": "game", "source": "ninuki-renju", "section": "overview", "text": "Ninuki-renju\nRenju with captures — the historical Japanese ancestor of Pente.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan k-in-a-row game with captures."}, {"id": 902, "type": "game", "source": "ninuki-renju", "section": "Description", "text": "Ninuki-renju — Description\n\nNinuki-renju is the Japanese 19th-century forerunner of [Pente](pente.md). It\nplays on a Go board with five-in-a-row as the primary winning condition, plus\na custodial capture rule: a pair of opposing stones sandwiched between two of\nyours is removed. A player reaching a fixed number of captures (usually 5\npairs) also wins."}, {"id": 903, "type": "game", "source": "ninuki-renju", "section": "Rules", "text": "Ninuki-renju — Rules\n\n1. Go board (commonly 15×15 or 19×19), initially empty.\n2. Players alternate placing one stone of their colour on any empty\n   intersection.\n3. **Capture**: if a placement creates the pattern X**O**O**X** (your stones\n   bracketing exactly two adjacent opposing stones in a straight line), the two\n   bracketed stones are removed and counted toward your capture total.\n4. A player **wins** by either:\n   - Forming an unbroken five-in-a-row of their stones, **or**\n   - Accumulating 5 pairs of captures (10 captured stones), **or** depending on\n     the variant, **[verify]** other thresholds.\n5. (Some variants of Ninuki-renju also borrow Renju's restrictions on Black's\n   3-3, 4-4, and overline moves.)"}, {"id": 904, "type": "game", "source": "ninuki-renju", "section": "Solution status", "text": "Ninuki-renju — Solution status\n\nNinuki-renju is **not solved**. The capture rule complicates the careful\nthreat-tree analysis that solved plain Renju ([Wágner & Virág 2001](../references.md#wagner-virag2001)),\nand no formal solution exists."}, {"id": 905, "type": "game", "source": "ninuki-renju", "section": "Consensus on optimal play", "text": "Ninuki-renju — Consensus on optimal play\n\n- **Dual threat: five-in-a-row OR five captures** — always be aware of both winning paths; a position threatening row-completion forces a different defence than one racing toward 5 capture-pairs.\n- **Custodial traps over pure extension** — placing a stone to bracket an enemy pair (capturing immediately) is often stronger than extending your own row, because captures simultaneously remove material and advance your capture count.\n- **Break open overlines** — unlike standard Renju, overlines (6+) do not lose here; a player threatening both an exact five and an overline creates a double win-threat.\n- **Restrict opponent's pairing** — avoid leaving two of your own stones sitting adjacent in a line where the opponent can bracket both ends; pairs on the board are permanent capture bait.\n- **Race the capture win against a row threat** — if the opponent is one move from five-in-a-row, capturing a pair may not help; calculate whether row defence or racing your own capture count is faster."}, {"id": 906, "type": "game", "source": "ninuki-renju", "section": "Engines & current best play", "text": "Ninuki-renju — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** No formal solution; stronger bots exist for the closely related game Pente."}, {"id": 907, "type": "game", "source": "ninuki-renju", "section": "Complexity", "text": "Ninuki-renju — Complexity\n\nLarger than plain Renju because captures dramatically broaden the dynamic\nmaterial situation."}, {"id": 908, "type": "game", "source": "ninuki-renju", "section": "References", "text": "Ninuki-renju — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Ninuki-renju) ([archive](http://web.archive.org/web/20210504225326/http://en.wikipedia.org/wiki/Ninuki-renju))\n- [Wágner & Virág (2001). *Solving Renju*.](../references.md#wagner-virag2001) (related)\n- [Allis, van den Herik & Huntjens (1996). *Go-Moku Solved by New Search Techniques*.](../references.md#allis-gomoku1996)"}, {"id": 909, "type": "game", "source": "ninuki-renju", "section": "See also", "text": "Ninuki-renju — See also\n\n- [Pente](pente.md) · [Renju](renju.md) · [Gomoku](gomoku.md)\n- Lexicon: [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 910, "type": "game", "source": "node-kayles", "section": "overview", "text": "Node Kayles\nA graph-theoretic relative of Kayles whose decision problem is\nSolution status: Strongly solved as a theory (decision problem PSPACE-complete). Game-theoretic value: Position-dependent. Players: 2. Type: Impartial graph game."}, {"id": 911, "type": "game", "source": "node-kayles", "section": "Description", "text": "Node Kayles — Description\n\nNode Kayles lifts the row-of-pins [Kayles](kayles.md) game to an arbitrary\ngraph. Players alternately *remove a vertex together with all its neighbours*\n— equivalently, they alternately enlarge an independent set."}, {"id": 912, "type": "game", "source": "node-kayles", "section": "Rules", "text": "Node Kayles — Rules\n\n1. An undirected graph G is given. Set S of \"claimed\" vertices starts empty.\n2. On your turn, pick a vertex v not in S and not adjacent to any vertex of S,\n   and add v to S.\n3. The player who cannot move loses (normal play).\n\nEquivalently: players alternately remove a vertex *together with* all its\nneighbours, until the graph is empty."}, {"id": 913, "type": "game", "source": "node-kayles", "section": "Solution status", "text": "Node Kayles — Solution status\n\nSolved as a theory. [Schaefer (1978)](../references.md#schaefer1978) proved\n**Node Kayles is PSPACE-complete** (along with several other graph games). For\nsmall graphs the game is trivially solvable by backward induction. The natural\nfamilies — paths, cycles, trees — have closed-form nim-values and are an\nexercise in the Sprague–Grundy formalism; the hardness applies to the general\ngraph problem."}, {"id": 914, "type": "game", "source": "node-kayles", "section": "Consensus on optimal play", "text": "Node Kayles — Consensus on optimal play\n\n- **Use tabulated nim-values for named families** — on paths (ordinary Kayles), cycles, and complete graphs the Sprague–Grundy values are known; look up the table and pick the move that sets nim-sum to 0.\n- **Isolate high-degree vertices early** — removing a vertex with many neighbours shrinks the graph rapidly; the resulting smaller components can then be analysed independently.\n- **Decompose into components** — once the graph breaks into disconnected components, compute the Grundy value of each and XOR them (Sprague–Grundy additivity).\n- **In symmetric positions, mirror** — if the graph has a structural symmetry your opponent is about to exploit, consider playing the mirror vertex to restore balance.\n- **General graphs are hard** — for arbitrary graphs no efficient algorithm is known (PSPACE-complete); rely on brute-force retrograde analysis for small instances."}, {"id": 915, "type": "game", "source": "node-kayles", "section": "Engines & current best play", "text": "Node Kayles — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked for arbitrary graphs; perfect play is feasible only for small instances.\n- **Notes:** PSPACE-completeness means no efficient solver is expected for general graphs; special-case tables cover common families."}, {"id": 916, "type": "game", "source": "node-kayles", "section": "Complexity", "text": "Node Kayles — Complexity\n\nPSPACE-complete in graph size."}, {"id": 917, "type": "game", "source": "node-kayles", "section": "References", "text": "Node Kayles — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Kayles) ([archive](http://web.archive.org/web/20251201221417/https://en.wikipedia.org/wiki/Kayles))\n- [Schaefer (1978). *On the complexity of some two-person perfect-information games*.](../references.md#schaefer1978)\n- [Guy & Smith (1956). *The G-values of various games*.](../references.md#guy-smith1956)"}, {"id": 918, "type": "game", "source": "node-kayles", "section": "See also", "text": "Node Kayles — See also\n\n- [Kayles](kayles.md) · [Generalized Geography](geography.md) · [Shannon switching game](shannon-switching-game.md)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 919, "type": "game", "source": "nonograms", "section": "overview", "text": "Nonograms\nPicture-by-numbers grid puzzles — NP-complete in general.\nSolution status: NP-complete in general. Game-theoretic value: Per-puzzle (unique solution by convention). Players: 1. Type: Solo logic puzzle."}, {"id": 920, "type": "game", "source": "nonograms", "section": "Description", "text": "Nonograms — Description\n\nNonograms (Non Ishida / James Dalgety, 1980s) are grid puzzles whose row and\ncolumn clues are sequences of run-lengths. The solver shades cells so each\nrow's shaded runs match the row clue, and each column's shaded runs match\nthe column clue. The general problem is **NP-complete**."}, {"id": 921, "type": "game", "source": "nonograms", "section": "Rules", "text": "Nonograms — Rules\n\n1. Board: rectangular grid of cells.\n2. Each row and each column has a sequence of positive integers (the\n   **clue**) listing the lengths of consecutive shaded runs in that row or\n   column, in order.\n3. Different runs in the same row or column must be separated by at least one\n   unshaded cell.\n4. The solver shades a subset of cells so that every row clue and every\n   column clue is satisfied.\n5. A well-formed puzzle has a unique solution."}, {"id": 922, "type": "game", "source": "nonograms", "section": "Solution status", "text": "Nonograms — Solution status\n\nThe general decision problem is **NP-complete** (Ueda & Nagao 1996).\nSAT/ILP solvers handle typical Nikoli-sized puzzles instantly."}, {"id": 923, "type": "game", "source": "nonograms", "section": "Consensus on optimal play", "text": "Nonograms — Consensus on optimal play\n\n- **Overlap (interval) deduction first** — for each clue, find the leftmost and rightmost placement of each run; cells covered by both placements are definitely shaded; gaps between them are definitely empty.\n- **Cross-reference rows against columns** — after deducing cells in a row, use those fixed cells to constrain the intersecting columns, and iterate until no new deductions arise.\n- **Start with the longest runs** — clues with a single large run leave little slack; their cells are nearly all deterministic and anchor the rest of the grid.\n- **Use edge constraints** — runs that touch a board edge have no offset uncertainty on one side; this can pin them precisely even when the interior is ambiguous.\n- **Backtrack sparingly and only on contradiction** — well-formed Nikoli puzzles are uniquely solvable by logic alone; if constraint propagation stalls, a single hypothesis-and-test branch is usually enough."}, {"id": 924, "type": "game", "source": "nonograms", "section": "Engines & current best play", "text": "Nonograms — Engines & current best play\n\n- **Strongest known program(s):** Various open-source solvers (e.g., Jan Wolter's JavaScript solver, pbnsolve) use constraint propagation + backtracking.\n- **Strength:** Solves all well-formed Nikoli-sized puzzles (up to ~50×50) instantly; handles larger puzzles with minimal backtracking.\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Nonogram); NP-completeness proof: Ueda & Nagao (1996).\n- **Notes:** NP-completeness applies to the general decision problem; typical published puzzles are much easier in practice."}, {"id": 925, "type": "game", "source": "nonograms", "section": "Complexity", "text": "Nonograms — Complexity\n\nNP-complete in general."}, {"id": 926, "type": "game", "source": "nonograms", "section": "References", "text": "Nonograms — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nonogram)\n- [Yato & Seta (2003). *Complexity and Completeness of Finding Another Solution and its Application to Puzzles*.](../references.md#selman-sudoku2003)"}, {"id": 927, "type": "game", "source": "nonograms", "section": "See also", "text": "Nonograms — See also\n\n- [Sudoku](sudoku.md) · [Slitherlink](slitherlink.md) · [Hashiwokakero](hashiwokakero.md)\n- Lexicon: [NP-completeness](../lexicon/README.md#np-completeness)"}, {"id": 928, "type": "game", "source": "northcotts-game", "section": "overview", "text": "Northcott's game\nA checkers-like game on strips that is, once again, Nim wearing a costume.\nSolution status: Strongly solved. Game-theoretic value: Second-player win iff the nim-sum of the gaps is 0. Players: 2. Type: Impartial combinatorial game (in effect; see note)."}, {"id": 929, "type": "game", "source": "northcotts-game", "section": "Description", "text": "Northcott's game — Description\n\nOn each row of a board sit two checkers, one belonging to each player. On a turn\na player slides **their own** checker along its row any number of empty squares,\nleft or right, **without jumping or passing** the opponent's checker. Under\n[normal play](../lexicon/README.md#normal-play-convention) the player who cannot\nmove loses.\n\n> Note: Northcott's game is nominally [partisan](../lexicon/README.md#partisan-game)\n> — each player moves only their own pieces — but its analysis reduces exactly\n> to an impartial game, which is the point of the example."}, {"id": 930, "type": "game", "source": "northcotts-game", "section": "Solution status", "text": "Northcott's game — Solution status\n\nNorthcott's game is **strongly solved**: it is **[Nim](nim.md) in disguise**.\nThe only thing that matters on each row is the *gap* — the number of empty\nsquares between the two checkers — and a move changes one gap to any smaller or\n(crucially) *larger* value, since a player can also retreat. Because retreats\ncan be mirrored by the opponent, the game is equivalent to Nim with heap sizes\nequal to the gaps: the position is a second-player win exactly when the\n[nim-sum](../lexicon/README.md#nim-sum) of the gaps is 0.\n\nIt is a favourite teaching example because it looks like a positional board game\nyet is solved by one line of Nim theory — and it illustrates how \"you can also\nmove backwards\" need not change the game value."}, {"id": 931, "type": "game", "source": "northcotts-game", "section": "Consensus on optimal play", "text": "Northcott's game — Consensus on optimal play\n\n- **Map gaps to heaps** — on each row, count the empty squares between the two checkers; that number is your \"heap\" for Nim purposes.\n- **XOR all gaps** — compute the nim-sum of the gaps across all rows; if it is non-zero you are in a winning position and must move to make it 0.\n- **Mirror retreats** — if your opponent retreats (increases a gap), immediately re-shrink that same row's gap to restore nim-sum 0; retreats cannot help the losing player.\n- **Never increase your losing row's gap needlessly** — in a losing position (nim-sum 0), any move breaks the balance; all you can do is hope for an opponent error.\n- **Reduce the dominating row** — the standard Nim technique of isolating the single row whose gap exceeds the XOR target applies directly."}, {"id": 932, "type": "game", "source": "northcotts-game", "section": "Engines & current best play", "text": "Northcott's game — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed — the strategy is a closed-form nim-sum formula.\n- **Strength:** Perfect play by any program correctly computing the nim-sum of gaps.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** Northcott's game is isomorphic to Nim; the equivalence is a standard Combinatorial Game Theory exercise."}, {"id": 933, "type": "game", "source": "northcotts-game", "section": "Complexity", "text": "Northcott's game — Complexity\n\nLinear in the number of rows to evaluate a position."}, {"id": 934, "type": "game", "source": "northcotts-game", "section": "References", "text": "Northcott's game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Northcott%27s_game)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Bouton, C. L. (1901). *Nim, A Game with a Complete Mathematical Theory*.](../references.md#bouton1901)"}, {"id": 935, "type": "game", "source": "northcotts-game", "section": "See also", "text": "Northcott's game — See also\n\n- [Nim](nim.md) · [Turning Turtles](turning-turtles.md) · [Mock Turtles](mock-turtles.md)\n- Lexicon: [nim-sum](../lexicon/README.md#nim-sum) · [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 936, "type": "game", "source": "notakto", "section": "overview", "text": "Notakto\nTic-tac-toe where both players play X, and making three-in-a-row *loses* —\nSolution status: Weakly solved (single board and multi-board versions analysed). Game-theoretic value: Single 3×3 board: second player can avoid losing → first player loses with best play (see below). Players: 2. Type: Impartial combinatorial game (misère play)."}, {"id": 937, "type": "game", "source": "notakto", "section": "Description", "text": "Notakto — Description\n\nNotakto is [impartial](../lexicon/README.md#impartial-game) tic-tac-toe: **both**\nplayers mark cells with an X, and a player who completes three X's in a row\n**loses** ([misère](../lexicon/README.md#misère-play) convention). It is\ntypically played on **several** 3×3 boards at once — a move is an X on any one\nlive board, and you lose when forced to complete a line on the last board."}, {"id": 938, "type": "game", "source": "notakto", "section": "Solution status", "text": "Notakto — Solution status\n\nNotakto is **weakly solved** by [Plambeck & Whitehead (2013)](../references.md#plambeck-notakto2013),\nwho gave a complete analysis using **misère quotient** theory — the algebraic\nmachinery developed precisely because misère impartial games do not submit to\nordinary [Sprague–Grundy](../lexicon/README.md#sprague-grundy-theorem) theory.\nThey showed the multi-board game is governed by a finite commutative monoid (of\norder 18), so the value of any number of boards in any position is computable.\n\nFor the **single 3×3 board**, the player forced to move first into a losing\nconfiguration is determined: with perfect play the first player cannot avoid\neventually being the one to complete a line. The clean closed-form result is the\nmonoid for the multi-board game."}, {"id": 939, "type": "game", "source": "notakto", "section": "Consensus on optimal play", "text": "Notakto — Consensus on optimal play\n\n- **Avoid the last X — the loser completes a line** — every move should aim to leave the opponent in a position where every cell they can play completes a three-in-a-row somewhere.\n- **On a single board, first player loses with perfect play** — the second player can always mirror the winning response; knowing this, the first player should try to create symmetric or forcing positions as early as possible.\n- **Multi-board: track the misère monoid value** — compute each board's equivalence class in the order-18 monoid described by Plambeck & Whitehead, then combine by monoid multiplication; a position with a losing monoid value means you are to move into a loss.\n- **Fork to create two \"live\" lines** — placing an X that threatens two potential completions forces your opponent to complete one, keeping you safe for another turn.\n- **Avoid \"dead\" boards in your own move** — completing a line on a board you are forced to play on ends the game for you if it is the last board; delay exhausting boards until the opponent is in a worse state."}, {"id": 940, "type": "game", "source": "notakto", "section": "Engines & current best play", "text": "Notakto — Engines & current best play\n\n- **Strongest known program(s):** Plambeck & Whitehead's solver (2013) — misère quotient monoid analysis.\n- **Strength:** Perfect play via the monoid arithmetic; the solution is compact and computable.\n- **Where the proof / tablebase lives (if solved):** [Plambeck & Whitehead (2013)](../references.md#plambeck-notakto2013)\n- **Notes:** The closed-form monoid solution means any program implementing the 18-element multiplication table plays perfectly on any number of boards."}, {"id": 941, "type": "game", "source": "notakto", "section": "Complexity", "text": "Notakto — Complexity\n\nSmall; the achievement is the *algebraic* solution, not raw search."}, {"id": 942, "type": "game", "source": "notakto", "section": "References", "text": "Notakto — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Notakto) ([archive](http://web.archive.org/web/20260326083823/https://en.wikipedia.org/wiki/Notakto))\n- [Plambeck, T. E. & Whitehead, G. (2013). *The Secrets of Notakto*.](../references.md#plambeck-notakto2013)"}, {"id": 943, "type": "game", "source": "notakto", "section": "See also", "text": "Notakto — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Treblecross](treblecross.md) · [Misère Nim](misere-nim.md)\n- Lexicon: [misère play](../lexicon/README.md#misère-play) · [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 944, "type": "game", "source": "onyx", "section": "overview", "text": "Onyx\nConnection game on a mixed square/triangle grid — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown (first-player win suspected via strategy stealing **[verify]**). Players: 2. Type: Partisan connection game."}, {"id": 945, "type": "game", "source": "onyx", "section": "Description", "text": "Onyx — Description\n\nOnyx (Larry Back, 1996) is a connection game on a hybrid grid of squares\nwith additional points at certain square centres, forming a mix of square\nand triangular adjacency. Players race to connect their two sides while\ncapture rules allow stones to be removed by surrounding."}, {"id": 946, "type": "game", "source": "onyx", "section": "Rules", "text": "Onyx — Rules\n\n1. Board: 12×12 square grid with extra points at the centre of certain\n   squares (forming local triangular adjacency).\n2. White connects the left and right edges with white stones; Black connects\n   top and bottom with black stones.\n3. On each turn a player places one stone of their colour on any empty point.\n4. **Capture**: when a player completes a pattern in which one or two\n   opposing stones are surrounded by the player's own stones at all\n   adjacent points (within the local geometry), those stones are removed.\n5. The first player to form an unbroken chain of their stones between their\n   two designated edges wins.\n6. The pie rule may be used to neutralise first-player advantage."}, {"id": 947, "type": "game", "source": "onyx", "section": "Solution status", "text": "Onyx — Solution status\n\nOnyx is **not solved**. By the strategy-stealing argument it is at worst a\ndraw for the first player; capture-based connection games rarely admit\nstrategy-stealing proofs of win, so its value remains open. **[verify]**"}, {"id": 948, "type": "game", "source": "onyx", "section": "Consensus on optimal play", "text": "Onyx — Consensus on optimal play\n\n- **Connect through the triangular hubs** — the extra centre-of-square points create shortcut adjacencies; routing your chain through them can make it harder for the opponent to cut.\n- **Dual-threat paths** — as in Hex, maintain two independent connection paths toward your goal edges; forcing the opponent to block both simultaneously is usually impossible.\n- **Captures serve connection, not material** — removing an opponent stone is valuable only when it directly opens a connection path or collapses a blocking chain; random captures that don't affect the chain topology are wasted tempo.\n- **Use the pie rule to equalise** — if playing with the swap rule, aim for a first move that is as close to balanced as possible to avoid being swapped into a losing position.\n- **Treat the hybrid grid carefully** — the square/triangle adjacency means that apparent \"cuts\" sometimes have bypass routes through triangle centres that are easy to miss."}, {"id": 949, "type": "game", "source": "onyx", "section": "Engines & current best play", "text": "Onyx — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The competitive community for Onyx is small; no published engine is known."}, {"id": 950, "type": "game", "source": "onyx", "section": "Complexity", "text": "Onyx — Complexity\n\nLarge."}, {"id": 951, "type": "game", "source": "onyx", "section": "References", "text": "Onyx — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Onyx_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 952, "type": "game", "source": "onyx", "section": "See also", "text": "Onyx — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Poly-Y](poly-y.md) · [Star](star.md)\n- Lexicon: [strategy stealing](../lexicon/README.md#strategy-stealing)"}, {"id": 953, "type": "game", "source": "order-and-chaos", "section": "overview", "text": "Order and Chaos\nAn asymmetric tic-tac-toe variant: one player wants a line, the other wants to\nSolution status: Weakly solved. Game-theoretic value: Order (the player going first / seeking a line) wins **[verify]**. Players: 2 (asymmetric roles). Type: Maker–Breaker positional game."}, {"id": 954, "type": "game", "source": "order-and-chaos", "section": "Description", "text": "Order and Chaos — Description\n\nPlayed on a 6×6 grid. **Both** players may place **either** an X or an O in any\nempty cell. The player \"Order\" wins if **five identical symbols** (all X or all\nO) ever appear in a row, column, or diagonal. The player \"Chaos\" wins if the\nboard fills with no such line. It is a [Maker–Breaker](../lexicon/README.md#maker-breaker-game)\ngame: Order is Maker, Chaos is Breaker."}, {"id": 955, "type": "game", "source": "order-and-chaos", "section": "Solution status", "text": "Order and Chaos — Solution status\n\nOrder and Chaos is **weakly solved** by exhaustive search — the 6×6 board is\nsmall enough to analyse completely. The standard reported result is that\n**Order wins** with perfect play (Order conventionally moves first). The\nintuition: because Order may use *both* symbols, the freedom to choose X or O on\neach move gives enough flexibility to force a five-line against any blocking by\nChaos.\n\n> **[verify]** — The \"Order wins\" verdict is widely repeated and consistent\n> with exhaustive analysis, but this archive should cite a specific primary\n> computation (and pin down the exact rule set: board size and whether Order or\n> Chaos moves first)."}, {"id": 956, "type": "game", "source": "order-and-chaos", "section": "Consensus on optimal play", "text": "Order and Chaos — Consensus on optimal play\n\n- **Order should build dual-symbol threats** — because Order can place either X or O, a single row can threaten completion with X if one more X is placed, and simultaneously threaten O-completion if one more O is placed; Chaos cannot block both with a single move.\n- **Chaos must avoid homogeneous clusters** — placing a mix of X and O close together is Chaos's best disruption; it prevents Order from extending any single run without creating a different run that Chaos must also address.\n- **Order targets diagonals** — diagonals are harder for Chaos to monitor simultaneously with rows and columns; Order should seed diagonal five-in-a-row threats.\n- **Chaos exploits board edges** — near the edge fewer cells complete a five-line; placing odd symbols at edge cells makes it harder for Order to build through the edge zones.\n- **Order wins with correct play** — the exhaustive solution confirms Order can always force a five-line on the 6×6 board; Chaos's task requires perfect vigilance and is ultimately futile."}, {"id": 957, "type": "game", "source": "order-and-chaos", "section": "Engines & current best play", "text": "Order and Chaos — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The 6×6 board is small enough for complete exhaustive search; any brute-force solver with correct rules will confirm Order's win."}, {"id": 958, "type": "game", "source": "order-and-chaos", "section": "Complexity", "text": "Order and Chaos — Complexity\n\nExhaustively searchable on the standard 6×6 board."}, {"id": 959, "type": "game", "source": "order-and-chaos", "section": "References", "text": "Order and Chaos — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Order_and_Chaos) ([archive](http://web.archive.org/web/20260430010757/https://en.wikipedia.org/wiki/Order_and_Chaos))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 960, "type": "game", "source": "order-and-chaos", "section": "See also", "text": "Order and Chaos — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Gomoku](gomoku.md) · [Notakto](notakto.md)\n- Lexicon: [maker-breaker game](../lexicon/README.md#maker-breaker-game) · [weakly solved](../lexicon/README.md#weakly-solved)"}, {"id": 961, "type": "game", "source": "othello", "section": "overview", "text": "Othello\nThe disc-flipping classic — and, since 2023, a weakly solved draw.\nSolution status: Weakly solved (standard 8×8). Game-theoretic value: Draw. Players: 2. Type: Partisan positional / capture game."}, {"id": 962, "type": "game", "source": "othello", "section": "Description", "text": "Othello — Description\n\nPlayed on an 8×8 board. Players alternately place a disc of their colour so\nthat it brackets one or more straight lines of enemy discs between the new disc\nand another of their own; all bracketed discs flip colour. A player with no\nlegal move passes. When neither can move, the player with more discs wins."}, {"id": 963, "type": "game", "source": "othello", "section": "Solution status", "text": "Othello — Solution status\n\nStandard 8×8 Othello is **weakly solved**. [Takizawa (2023)](../references.md#takizawa2023)\ndemonstrated, with a large but feasible search supported by strong\nevaluation/verification, that **with perfect play the game is a draw** — neither\nplayer can force a win from the standard start. This resolved one of the most\nprominent \"small enough to solve, but not yet solved\" board games; Othello had\nlong been expected to be a draw, and the 2023 result confirmed it.\n\nThe reduced **6×6** board was settled much earlier — reported by\n[Feinstein (1993)](../references.md#feinstein-othello6x61993) as a **second-player\nwin** (by 4 discs) via exhaustive search."}, {"id": 964, "type": "game", "source": "othello", "section": "Consensus on optimal play", "text": "Othello — Consensus on optimal play\n\n- **Corners are permanent** — a disc in a corner can never be flipped; obtaining corners is the single most valuable strategic objective and drives almost all high-level play.\n- **Edges adjacent to corners are dangerous** — placing in the \"C-square\" (one step diagonally inward from a corner) or \"X-square\" (diagonally adjacent) gives the opponent a path to take the corner; avoid these early.\n- **Minimise your opponent's mobility** — leaving the opponent with few legal moves is more important than maximising your disc count mid-game; a player who must pass has surrendered tempo.\n- **Disc count mid-game is misleading** — having fewer discs in the middle game often gives better positional control; a small disc count mid-game with good edge access tends to win in the endgame flip cascade.\n- **Endgame is exact calculation** — in the last 15–20 moves the position resolves by forced sequences; strong players and engines calculate this phase exhaustively.\n- **Draw with perfect play** — the 2023 Takizawa solution confirms neither side can force a win from the standard opening; practical play aims to deviate from balanced lines."}, {"id": 965, "type": "game", "source": "othello", "section": "Engines & current best play", "text": "Othello — Engines & current best play\n\n- **Strongest known program(s):** Edax — alpha-beta search with highly tuned evaluation; consistently the strongest public Othello engine.\n- **Strength:** Super-human; Edax defeats the strongest human players.\n- **Where the proof / tablebase lives (if solved):** [Takizawa (2023)](../references.md#takizawa2023)\n- **Notes:** The 2023 solution used a large cloud-compute search; Edax itself was central to validating many of the claimed lines."}, {"id": 966, "type": "game", "source": "othello", "section": "Complexity", "text": "Othello — Complexity\n\nState-space ~10^28, game-tree ~10^58\n([van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 967, "type": "game", "source": "othello", "section": "References", "text": "Othello — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Reversi)\n- [Takizawa, H. (2023). *Othello is Solved*.](../references.md#takizawa2023)\n- [Feinstein, J. (1993). *Amenor Wins World 6×6 Championships*.](../references.md#feinstein-othello6x61993)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 968, "type": "game", "source": "othello", "section": "See also", "text": "Othello — See also\n\n- [Quixo](quixo.md) · [Checkers](checkers.md) · [Awari](awari.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 969, "type": "game", "source": "pallanguzhi", "section": "overview", "text": "Pallanguzhi\nSouth Indian mancala with chain sowing — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan mancala."}, {"id": 970, "type": "game", "source": "pallanguzhi", "section": "Description", "text": "Pallanguzhi — Description\n\nPallanguzhi is the traditional mancala of Tamil Nadu, played on a 2×7 board.\nSowing is **chained**: when the seed after the last drops into a pit, the\nplayer picks up the contents of the next pit and continues sowing, repeating\nuntil a stopping condition occurs."}, {"id": 971, "type": "game", "source": "pallanguzhi", "section": "Rules", "text": "Pallanguzhi — Rules\n\n1. Board: 7 pits per player; no large stores. Each pit begins with 6 seeds\n   (varies by regional rule).\n2. On a turn the player picks up seeds from one of their own pits and sows\n   counterclockwise.\n3. **Chain sowing**: after sowing, the pit immediately after the last sown\n   pit is picked up and sown the same way; the chain continues until either\n   the player drops the last seed into an empty pit, or into a pit followed\n   by an empty pit on the same side.\n4. **Capture**: when the chain ends on an empty pit, the seeds in the pit\n   beyond it (if any) are captured.\n5. When a player runs out of seeds on their side, they pass; the round ends\n   when both sides empty. Seeds left on the opponent's side may go to that\n   opponent (varies by variant).\n6. Game is played as a series of rounds with reseeding; the player who can\n   no longer fill all their pits at the start of a round loses."}, {"id": 972, "type": "game", "source": "pallanguzhi", "section": "Solution status", "text": "Pallanguzhi — Solution status\n\nPallanguzhi is **not solved**. Chain-sowing analysis is complex and no\npublished values exist for the standard ruleset."}, {"id": 973, "type": "game", "source": "pallanguzhi", "section": "Consensus on optimal play", "text": "Pallanguzhi — Consensus on optimal play\n\n- **Prefer long-chain triggers** — moves that initiate a long chain sow often pass through many pits, creating unpredictable stops; opponents find it harder to foresee where the chain will terminate.\n- **Keep seeds spread across your pits** — concentrating all seeds in one pit wastes turns on a single long sow; distributing seeds lets you maintain more chain options each turn.\n- **Target the opponent's near-empty pits** — a chain that terminates just before an opponent's heavily loaded pit captures those seeds; watching for this opportunity is critical.\n- **Protect your loaded pits from capture** — do not allow your large pits to sit one position past an opponent's likely chain-end; rebalance before they can line up a capture.\n- **Endgame seed counting** — with few seeds left, track exactly which positions will allow a chain to continue vs. stop; the player who can keep chains going longest captures the final seeds."}, {"id": 974, "type": "game", "source": "pallanguzhi", "section": "Engines & current best play", "text": "Pallanguzhi — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** No published formal analysis; the game is played recreationally and competitively in Tamil Nadu but lacks a computational study."}, {"id": 975, "type": "game", "source": "pallanguzhi", "section": "Complexity", "text": "Pallanguzhi — Complexity\n\nLarge."}, {"id": 976, "type": "game", "source": "pallanguzhi", "section": "References", "text": "Pallanguzhi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pallanguzhi)\n- [Romein & Bal (2003). *Awari is Solved*.](../references.md#romein-bal2003) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 977, "type": "game", "source": "pallanguzhi", "section": "See also", "text": "Pallanguzhi — See also\n\n- [Awari](awari.md) · [Bao](bao.md) · [Kalah](kalah.md) · [Sungka](sungka.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 978, "type": "game", "source": "pegs-solitaire", "section": "overview", "text": "Peg solitaire\nThe classic jump-and-remove puzzle — completely solved: there is a full\nSolution status: Strongly solved (solvability fully characterised). Game-theoretic value: N/A (puzzle). Players: 1 (puzzle). Type: Single-player jumping puzzle."}, {"id": 979, "type": "game", "source": "pegs-solitaire", "section": "Description", "text": "Peg solitaire — Description\n\nPegs sit in a grid of holes (the standard \"English\" board is a 33-hole cross).\nA peg jumps orthogonally over an adjacent peg into an empty hole beyond,\nremoving the jumped peg. The classic goal is to start with one hole empty and\nfinish with a single peg — ideally in the centre."}, {"id": 980, "type": "game", "source": "pegs-solitaire", "section": "Solution status", "text": "Peg solitaire — Solution status\n\nPeg solitaire is **strongly solved** as a puzzle: there is a complete theory of\n**which** start/finish problems are solvable. The key tool is the **\"pagoda\nfunction\"** (resource count / Conway's de Bruijn–style invariants): each jump\ncannot increase certain weighted sums, which yields necessary conditions, and\ncompanion constructions show those conditions are sufficient for the standard\nboards. [Beasley's *The Ins and Outs of Peg Solitaire* (1985)](../references.md#beasley-pegsolitaire1985)\ncollects the full theory; for the English board it is known exactly which\nsingle-vacancy-to-single-peg problems can be done (the central-game \"central\ncomplement\" problem is solvable, and the minimum-move solution is 18 jumps\ncounting multi-jumps as one move)."}, {"id": 981, "type": "game", "source": "pegs-solitaire", "section": "Consensus on optimal play", "text": "Peg solitaire — Consensus on optimal play\n\n- **Check the pagoda function first** — before attempting a problem, compute the pagoda-function value of the start and target positions; if the start value is less than the target value under any valid pagoda weighting, the problem is unsolvable.\n- **Work backwards** — the most reliable human solving technique is to plan the last few moves first (what single peg lands in the target hole?) then extend the solution backward.\n- **Prefer \"packages\"** — a package is a local sequence of jumps that clears a region and leaves a specific peg in a specific hole; assembling known packages reduces a complex board to a short sequence of sub-problems.\n- **Avoid stranded pegs** — a single peg isolated from all others (no neighbour with an empty hole beyond it) can never be moved; identify and address potential stranded-peg configurations early.\n- **The 18-move minimum for the central complement** — the optimal solution to the classic English board problem (start with centre empty, end with centre full) takes exactly 18 multi-jumps; known solutions achieving this have been published."}, {"id": 982, "type": "game", "source": "pegs-solitaire", "section": "Engines & current best play", "text": "Peg solitaire — Engines & current best play\n\n- **Strongest known program(s):** Various open-source brute-force solvers (e.g., George Bell's peg solitaire solver) — depth-first search with symmetry pruning.\n- **Strength:** Solves any legal English-board problem instantly.\n- **Where the proof / tablebase lives (if solved):** [Beasley (1985)](../references.md#beasley-pegsolitaire1985); [Wikipedia](https://en.wikipedia.org/wiki/Peg_solitaire)\n- **Notes:** The analytic pagoda-function theory gives exact solvability without search; computers independently confirm via exhaustive enumeration."}, {"id": 983, "type": "game", "source": "pegs-solitaire", "section": "Complexity", "text": "Peg solitaire — Complexity\n\nBoard-dependent. The English 33-hole board has on the order of 2 × 10^9\nreachable states — small enough for exhaustive search, which independently\nconfirms the analytic theory. Generalised peg-solitaire reachability is NP-hard."}, {"id": 984, "type": "game", "source": "pegs-solitaire", "section": "References", "text": "Peg solitaire — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Peg_solitaire) ([archive](http://web.archive.org/web/20260504154123/https://en.wikipedia.org/wiki/Peg_solitaire))\n- [Beasley (1985). *The Ins and Outs of Peg Solitaire*.](../references.md#beasley-pegsolitaire1985)"}, {"id": 985, "type": "game", "source": "pegs-solitaire", "section": "See also", "text": "Peg solitaire — See also\n\n- [15 puzzle](fifteen-puzzle.md) · [Rubik's Cube](rubiks-cube.md) · [Conway's Soldiers](conways-soldiers.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [state-space complexity](../lexicon/README.md#state-space-complexity)"}, {"id": 986, "type": "game", "source": "pentago", "section": "overview", "text": "Pentago\nFive-in-a-row with a twist — literally: each move rotates a quadrant of the\nSolution status: Strongly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan positional (k-in-a-row) game with board rotation."}, {"id": 987, "type": "game", "source": "pentago", "section": "Description", "text": "Pentago — Description\n\nPlayed on a 6×6 board divided into four 3×3 quadrants. A turn has two parts:\nplace one marble of your colour on any empty cell, **then rotate any one\nquadrant 90°** (either direction). The winner is the first to get five of their\nmarbles in a row — horizontally, vertically, or diagonally — at any point,\nincluding immediately after a rotation. If the board fills with no line, it is a\ndraw."}, {"id": 988, "type": "game", "source": "pentago", "section": "Solution status", "text": "Pentago — Solution status\n\nPentago is **strongly solved**. [Geoffrey Irving (2014)](../references.md#irving-pentago2014)\nsolved it by massive parallel computation on a supercomputer, building a\ncomplete database of all ~3.0 × 10^15 positions. The result: with perfect play\nthe **first player wins**. Because the whole state space is stored, optimal play\nis available from *every* position — the strong-solution standard — and Irving\nreleased an open-source solver and online perfect-play oracle.\n\nThe quadrant-rotation rule is what gives Pentago a large, irregular state space\ndespite a board only slightly bigger than tic-tac-toe's: every placement\ninteracts with eight possible rotations."}, {"id": 989, "type": "game", "source": "pentago", "section": "Consensus on optimal play", "text": "Pentago — Consensus on optimal play\n\n- **Rotation is a weapon, not an afterthought** — rotating a quadrant can simultaneously extend your own row and disrupt an opponent's near-complete row; always evaluate rotation options as aggressively as placement options.\n- **Avoid the rotational rebound** — placing a marble that creates a three-in-a-row also gives the opponent a quadrant-rotation that can break it; never commit to a near-complete line without considering how the opponent's next rotation interacts.\n- **Build in two quadrants at once** — rows and diagonals that span two quadrants are rotation-resistant, because the opponent would need to twist the same quadrant you are building in to disrupt both paths.\n- **Centre cells of each quadrant** — the centre of a 3×3 quadrant remains adjacent to the most cells after any rotation; place there early to maximise the reach of your chain.\n- **First player wins with correct play** — the 2014 solution confirms first-player advantage; practical play requires exploiting that advantage aggressively from move one."}, {"id": 990, "type": "game", "source": "pentago", "section": "Engines & current best play", "text": "Pentago — Engines & current best play\n\n- **Strongest known program(s):** Geoffrey Irving's solver (2014) — complete positional database (~3 × 10^15 positions).\n- **Strength:** Perfect play from any position.\n- **Where the proof / tablebase lives (if solved):** [Irving (2014)](../references.md#irving-pentago2014).\n- **Notes:** The full database is several terabytes; an online oracle allows perfect-play queries without downloading it."}, {"id": 991, "type": "game", "source": "pentago", "section": "Complexity", "text": "Pentago — Complexity\n\n~3.0 × 10^15 positions, all enumerated in the 2014 solution."}, {"id": 992, "type": "game", "source": "pentago", "section": "References", "text": "Pentago — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pentago) ([archive](http://web.archive.org/web/20260515054720/https://en.wikipedia.org/wiki/Pentago))\n- [Irving, G. (2014). *Pentago is a first player win*.](../references.md#irving-pentago2014)"}, {"id": 993, "type": "game", "source": "pentago", "section": "See also", "text": "Pentago — See also\n\n- [Gomoku](gomoku.md) · [Connect Four](connect-four.md) · [Quixo](quixo.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 994, "type": "game", "source": "pente", "section": "overview", "text": "Pente\nFive-in-a-row with custodial captures — the capture rule makes it richer than\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan positional (k-in-a-row) game with captures."}, {"id": 995, "type": "game", "source": "pente", "section": "Description", "text": "Pente — Description\n\nPlayed on a 19×19 grid. Players alternately place stones, aiming for **five in\na row**. The twist is **custodial capture**: a pair of your stones flanked\nexactly on both ends by enemy stones is captured and removed. A player can also\nwin by making **five captures** (ten captured stones). The capture rule means\nstones are not permanent — distinguishing Pente sharply from\n[Gomoku](gomoku.md)."}, {"id": 996, "type": "game", "source": "pente", "section": "Solution status", "text": "Pente — Solution status\n\nPente is **unsolved**. The capture mechanic both enlarges the effective state\nspace (stones can leave the board and the same cell be re-contested) and breaks\nthe [threat-space search](../lexicon/README.md#proof-number-search) techniques\nthat cracked Gomoku, since a \"permanent\" threat can be undone by a capture. The\nfirst player is widely believed to have a significant advantage on the standard\nboard — strong enough that tournament rule sets restrict the first player's\nearly moves — but no game-theoretic value has been proven."}, {"id": 997, "type": "game", "source": "pente", "section": "Consensus on optimal play", "text": "Pente — Consensus on optimal play\n\n- **Keep a dual win-threat alive** — always maintain at least one viable path to five-in-a-row AND a realistic race to five capture-pairs; an opponent who must defend both simultaneously will fail to stop one.\n- **Control the 5-capture race** — captures simultaneously remove opponent material, advance your win counter, and reduce the opponent's capture count; threatening a capture on every other move pressures the opponent continuously.\n- **Build immune rows** — a potential five-in-a-row where both flanks are guarded against bracketing is \"immune\" to capture disruption; look for patterns where the endpoint stones cannot be sandwiched.\n- **Opening restriction exists because first player is too strong** — tournament rules restrict the first player's third stone to the fifth intersection or further out; respect this by not trivially exploiting the opening in casual play.\n- **Defend captures with counter-captures** — when the opponent threatens to bracket your pair, the fastest defence is often a counter-threat that forces them to protect their own pair rather than execute the capture."}, {"id": 998, "type": "game", "source": "pente", "section": "Engines & current best play", "text": "Pente — Engines & current best play\n\n- **Strongest known program(s):** Pente Planet bots (online community engines) and custom alpha-beta search programs.\n- **Strength:** Strong amateur; competitive with experienced human players but no engine is known to be definitively super-human.\n- **Where the proof / tablebase lives (if solved):** Not solved; no tablebase.\n- **Notes:** No formal solution; opening restrictions in the strongest tournament formats (e.g. the \"tournament opening\" rule) reflect practical acknowledgment of first-player strength."}, {"id": 999, "type": "game", "source": "pente", "section": "Complexity", "text": "Pente — Complexity\n\nLarge; the 19×19 board plus captures puts it well beyond current exhaustive\nsolving."}, {"id": 1000, "type": "game", "source": "pente", "section": "References", "text": "Pente — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pente)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1001, "type": "game", "source": "pente", "section": "See also", "text": "Pente — See also\n\n- [Gomoku](gomoku.md) · [Renju](renju.md) · [Connect6](connect6.md)\n- Lexicon: [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1002, "type": "game", "source": "phutball", "section": "overview", "text": "Phutball\nConway's \"Philosopher's Football\" — a game where even deciding whether *one\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan combinatorial game."}, {"id": 1003, "type": "game", "source": "phutball", "section": "Description", "text": "Phutball — Description\n\nPlayed on a grid (commonly 19×15). There is a single shared \"ball.\" On a turn a\nplayer either places a \"man\" on an empty point, or makes the ball **jump** over\northogonally/diagonally adjacent contiguous lines of men, removing the jumped\nmen — and jumps can be chained. A player scores by getting the ball onto or over\nthe opponent's goal line. Crucially, **both players use the same men**, so\nPhutball is [impartial](../lexicon/README.md#impartial-game) in its material."}, {"id": 1004, "type": "game", "source": "phutball", "section": "Solution status", "text": "Phutball — Solution status\n\nPhutball is **unsolved**, and there is a precise reason to expect it to stay\nhard: [Demaine, Demaine & Eppstein (2002)](../references.md#demaine-phutball2002)\nproved that **deciding whether the player to move has a move that wins\nimmediately is NP-hard**. That is — not \"who wins the game,\" but merely \"is\nthere a winning *single move* right now\" — is already computationally\nintractable. This makes Phutball one of the standard examples of a game whose\n*local* tactics, never mind global strategy, resist analysis."}, {"id": 1005, "type": "game", "source": "phutball", "section": "Consensus on optimal play", "text": "Phutball — Consensus on optimal play\n\n- **Build a line of men toward your goal** — placing men in a diagonal or straight chain allows the ball to chain-jump over them in a single move; a long ready-made chain can carry the ball to or past the goal line in one turn.\n- **Deny the opponent's jump paths** — place men to break contiguous chains the opponent could use; a single gap in their chain prevents a decisive jump sequence.\n- **Control the centre of the board** — men near the centre participate in more potential jump chains; the ball can be redirected through the centre to either side of the field.\n- **Beware long jump sequences that leave the ball short** — a greedy chain jump that moves the ball far but stops just short of the goal line can leave it in an even more dangerous position for the opponent to claim next turn.\n- **The placement move is often underestimated** — placing a man instead of jumping keeps your chain infrastructure intact for future turns, and can simultaneously block an opponent chain; do not jump just because you can."}, {"id": 1006, "type": "game", "source": "phutball", "section": "Engines & current best play", "text": "Phutball — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** Even deciding whether a single winning move exists is NP-hard; no strong AI has been publicly published for Phutball."}, {"id": 1007, "type": "game", "source": "phutball", "section": "Complexity", "text": "Phutball — Complexity\n\nThe board is large and chained jumps give a high, variable branching factor; no\nuseful exhaustive analysis exists."}, {"id": 1008, "type": "game", "source": "phutball", "section": "References", "text": "Phutball — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Phutball) ([archive](http://web.archive.org/web/20260203090111/https://en.wikipedia.org/wiki/Phutball))\n- [Demaine, E. D., Demaine, M. L. & Eppstein, D. (2002). *Phutball Endgames are Hard*.](../references.md#demaine-phutball2002)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1009, "type": "game", "source": "phutball", "section": "See also", "text": "Phutball — See also\n\n- [Amazons](amazons.md) · [Hex](hex.md)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete) · [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 1010, "type": "game", "source": "picaria", "section": "overview", "text": "Picaria\nThe Zuni Pueblo three-in-a-row game — a strongly-solved cousin of\nSolution status: Strongly solved (small board). Game-theoretic value: Draw. Players: 2. Type: Partisan placement+movement game."}, {"id": 1011, "type": "game", "source": "picaria", "section": "Description", "text": "Picaria — Description\n\nA traditional Zuni Pueblo game played on a small board of nine points (a 3×3\ngrid plus diagonals). Like several Indigenous \"three-in-a-row\" games, it has a\n**placement phase** followed by a **movement phase** so the game cannot stall\ninto [tic-tac-toe](tic-tac-toe.md)'s trivial draws."}, {"id": 1012, "type": "game", "source": "picaria", "section": "Rules", "text": "Picaria — Rules\n\n1. Board: nine intersection points in a 3×3 lattice connected by horizontal,\n   vertical, and diagonal lines.\n2. Each player has **3 stones**.\n3. **Placement phase**: players alternate placing one of their stones on any\n   empty point. Three-in-a-row at this stage wins.\n4. **Movement phase**: after all 6 stones are on the board, players alternate\n   sliding one of their stones along a line to an adjacent empty point. Forming\n   three-in-a-row wins.\n5. A player who cannot move loses **[verify]** (some sources call it a draw)."}, {"id": 1013, "type": "game", "source": "picaria", "section": "Solution status", "text": "Picaria — Solution status\n\nStrongly solved by trivial exhaustive search. The state graph is tiny and the\n**value is a draw** with correct play, like other compact 3-in-a-row games\n([Three Men's Morris](three-mens-morris.md), [Nine Holes](nine-holes.md))."}, {"id": 1014, "type": "game", "source": "picaria", "section": "Consensus on optimal play", "text": "Picaria — Consensus on optimal play\n\n- **Occupy the centre in placement** — the centre point lies on every line (row, column, and both diagonals); placing there first limits the opponent's winning paths to only the four perimeter lines through corners.\n- **Corner before edge in placement** — corners participate in three lines each; edge midpoints only in two; corner placement should precede edge midpoint placement when the centre is taken.\n- **Create a two-way threat on the last placement** — placing the third stone to threaten two different three-in-a-row completions simultaneously forces a win, since only one can be blocked.\n- **In the movement phase, avoid sliding into forks** — a fork is a position from which you can complete a line by sliding in two different directions; setting up a fork leaves the opponent unable to block both.\n- **Draw is the result with mutual correct play** — neither player can force a win; the correct objective in competitive play is to spot the opponent's error quickly."}, {"id": 1015, "type": "game", "source": "picaria", "section": "Engines & current best play", "text": "Picaria — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked; the game is trivially solved by brute-force enumeration.\n- **Notes:** The entire game graph is tiny; any complete minimax search confirms the draw verdict."}, {"id": 1016, "type": "game", "source": "picaria", "section": "Complexity", "text": "Picaria — Complexity\n\nTiny."}, {"id": 1017, "type": "game", "source": "picaria", "section": "References", "text": "Picaria — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Picaria)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1018, "type": "game", "source": "picaria", "section": "See also", "text": "Picaria — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Achi](achi.md) · [Tapatan](tapatan.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1019, "type": "game", "source": "pocket-cube", "section": "overview", "text": "Pocket Cube\n2×2×2 Rubik's Cube — fully solved: God's number is 11.\nSolution status: Strongly solved. Game-theoretic value: Any scramble solvable in ≤ 11 face-turn moves (FTM) or ≤ 14 quarter-turn moves (QTM). Players: 1. Type: Solo permutation puzzle."}, {"id": 1020, "type": "game", "source": "pocket-cube", "section": "Description", "text": "Pocket Cube — Description\n\nThe Pocket Cube is the 2×2×2 version of Rubik's Cube. Its full state graph\nhas only 3,674,160 positions and is small enough to be exhaustively analysed:\nevery scramble is solvable in **at most 11 face-turn moves (FTM)** or 14\nquarter-turn moves (QTM)."}, {"id": 1021, "type": "game", "source": "pocket-cube", "section": "Rules", "text": "Pocket Cube — Rules\n\n1. Puzzle: 2×2×2 cube of 8 corner cubies. Each face is one of six colours.\n2. On a move the solver rotates one of the 6 faces by 90° or 180° (face-turn\n   metric); some literature counts only 90° rotations (quarter-turn metric).\n3. The puzzle is solved when every face shows a single colour.\n4. There are no captured pieces; the solver simply applies a sequence of\n   moves."}, {"id": 1022, "type": "game", "source": "pocket-cube", "section": "Solution status", "text": "Pocket Cube — Solution status\n\n**Strongly solved**. The position graph has been fully enumerated and the\n**diameter** is 11 FTM (equivalently 14 QTM)."}, {"id": 1023, "type": "game", "source": "pocket-cube", "section": "Consensus on optimal play", "text": "Pocket Cube — Consensus on optimal play\n\n- **Solve corners by layer (beginner)** — place the top layer's four corners, then orient and permute the bottom four; this takes 6–8 moves on average but is easy to learn.\n- **Ortega method** — first orient both top and bottom layer faces (OLL of each face separately, then permute corners); reduces average move count to around 6–8 moves.\n- **CLL / EG methods (advanced)** — recognize the combined top-face orientation and top/bottom permutation state in one look and apply a single algorithm; top speed-cubers execute the whole solve in one algorithmic block.\n- **Optimal solve ≤ 11 FTM** — any position can be solved in 11 or fewer face-turn moves; an optimal solver (IDA* against the complete tablebase) finds a shortest solution instantly.\n- **Pocket Cube has only corners** — unlike the 3×3, there are no edge pieces; every piece is a corner cubie, so parity issues differ and algorithms are simpler."}, {"id": 1024, "type": "game", "source": "pocket-cube", "section": "Engines & current best play", "text": "Pocket Cube — Engines & current best play\n\n- **Strongest known program(s):** Kociemba's two-phase algorithm (adapted) and full tablebase solvers for 2×2 — all 3,674,160 positions stored.\n- **Strength:** Perfect (optimal solve in ≤ 11 FTM guaranteed).\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Pocket_Cube); [Rokicki et al.](../references.md#rokicki2014)\n- **Notes:** The complete tablebase fits in a few MB; any implementation with the full lookup table solves any scramble instantly with optimal move count."}, {"id": 1025, "type": "game", "source": "pocket-cube", "section": "Complexity", "text": "Pocket Cube — Complexity\n\nSmall enough to fit in a tablebase of a few megabytes."}, {"id": 1026, "type": "game", "source": "pocket-cube", "section": "References", "text": "Pocket Cube — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pocket_Cube)\n- [Rokicki *et al.* (2017). *The diameter of the Rubik's cube group is twenty*.](../references.md#rokicki2014)"}, {"id": 1027, "type": "game", "source": "pocket-cube", "section": "See also", "text": "Pocket Cube — See also\n\n- [Rubik's Cube](rubiks-cube.md) · [Pyraminx](pyraminx.md) · [Skewb](skewb.md) · [Megaminx](megaminx.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [God's number](../lexicon/README.md#gods-number)"}, {"id": 1028, "type": "game", "source": "poker-nim", "section": "overview", "text": "Poker Nim\nNim with the added option to *replace* removed tokens — still equivalent to\nSolution status: Strongly solved. Game-theoretic value: Same as Nim: P-position iff nim-sum = 0. Players: 2. Type: Impartial loopy game."}, {"id": 1029, "type": "game", "source": "poker-nim", "section": "Description", "text": "Poker Nim — Description\n\nPoker Nim is the classic example of a \"loopy\" impartial game that nevertheless\nreduces cleanly to ordinary Nim. Each player has a private reserve of removed\ntokens; the option to *add* tokens back is real but ultimately a \"reversible\"\nmove that the opponent can simply mirror."}, {"id": 1030, "type": "game", "source": "poker-nim", "section": "Rules", "text": "Poker Nim — Rules\n\n1. Set up several heaps of tokens, as in [Nim](nim.md). Each player has a\n   private reserve, initially with some finite number of tokens.\n2. On your turn, either:\n   - Take any positive number of tokens from one heap (Nim move), placing them\n     in your reserve; **or**\n   - Add any positive number of tokens from your reserve back to a single heap.\n3. The player who cannot move loses (normal play). Note that since the reserve\n   is finite, the game cannot go on forever."}, {"id": 1031, "type": "game", "source": "poker-nim", "section": "Solution status", "text": "Poker Nim — Solution status\n\nStrongly solved by [Berlekamp, Conway & Guy](../references.md#bcg2001). The key\ninsight is the **reversibility argument**: if your opponent adds k tokens to a\nheap, you can immediately remove k tokens from that heap, undoing the move at\nthe cost of one round; this keeps you in the winning P-positions of ordinary\nNim. Hence the game's P-positions and winning strategy are *exactly those of\nNim*: nim-sum equals 0 iff the position is a P-position."}, {"id": 1032, "type": "game", "source": "poker-nim", "section": "Consensus on optimal play", "text": "Poker Nim — Consensus on optimal play\n\n- **Play ordinary Nim** — compute the nim-sum of all heap sizes and maintain it at 0 on every turn; the add-from-reserve option is irrelevant to the winning strategy.\n- **Mirror your opponent's additions** — if your opponent adds k tokens to a heap, immediately remove exactly k tokens from that same heap; this undoes the move and keeps the nim-sum where it was.\n- **Reserve-filling does not help** — adding tokens to a heap is a reversible move; in Combinatorial Game Theory reversible moves cannot help the player who makes them because the opponent can undo them.\n- **Keep nim-sum at 0 as the second player** — if you are the second player and the initial nim-sum is 0, maintain it; the first player will inevitably break it and you restore it.\n- **The finite reserve guarantees termination** — tokens in a reserve can be re-added only from previous takes, so the game cannot cycle indefinitely; the total token count bounds the game length."}, {"id": 1033, "type": "game", "source": "poker-nim", "section": "Engines & current best play", "text": "Poker Nim — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed — the Nim nim-sum formula solves it completely.\n- **Strength:** Perfect play by any program implementing the nim-sum calculation and mirror-response rule.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** Poker Nim is primarily a pedagogical example illustrating that reversible moves cannot change a game's Grundy value."}, {"id": 1034, "type": "game", "source": "poker-nim", "section": "Complexity", "text": "Poker Nim — Complexity\n\nSame as ordinary Nim — linear in the number of heaps."}, {"id": 1035, "type": "game", "source": "poker-nim", "section": "References", "text": "Poker Nim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Bouton (1901–1902). *Nim, A Game with a Complete Mathematical Theory*.](../references.md#bouton1901)"}, {"id": 1036, "type": "game", "source": "poker-nim", "section": "See also", "text": "Poker Nim — See also\n\n- [Nim](nim.md) · [Misère Nim](misere-nim.md) · [Whim](whim.md)\n- Lexicon: [nim-sum](../lexicon/README.md#nim-sum) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)"}, {"id": 1037, "type": "game", "source": "poly-y", "section": "overview", "text": "Poly-Y\nA multi-cornered relative of Y — first-player win by strategy stealing.\nSolution status: Ultra-weakly solved. Game-theoretic value: First-player win. Players: 2 (or more, in some variants). Type: Partisan connection game."}, {"id": 1038, "type": "game", "source": "poly-y", "section": "Description", "text": "Poly-Y — Description\n\nPoly-Y generalises [Y](y.md) from a triangular board (three \"corners\") to\nboards with **more corners** — pentagonal, hexagonal, and beyond. The winning\ncondition: own a connected group that **touches at least three different\ncorner regions** of the board. As in Y and Hex, draws are structurally\nimpossible."}, {"id": 1039, "type": "game", "source": "poly-y", "section": "Rules", "text": "Poly-Y — Rules\n\n1. A polygonal board (pentagon, hexagon, etc.) tiled with hexagonal cells.\n   The boundary is divided into \"corner regions\" — one for each corner of the\n   outer polygon.\n2. Players alternate placing one stone of their colour on any empty cell.\n3. The first player to form a single connected group of their colour touching\n   **at least three different corner regions** wins.\n4. Draws are impossible (a parity/no-pair-of-disjoint-spans argument)."}, {"id": 1040, "type": "game", "source": "poly-y", "section": "Solution status", "text": "Poly-Y — Solution status\n\n[Schensted & Titus (1975)](../references.md#schensted-titus1975) proved Poly-Y\ncannot be drawn and that the **first player has a winning strategy** — exactly\nas in Hex and Y, by the [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument).\nThe proof is non-constructive: ultra-weakly solved, no explicit strategy given."}, {"id": 1041, "type": "game", "source": "poly-y", "section": "Consensus on optimal play", "text": "Poly-Y — Consensus on optimal play\n\n- **Aim for three corners from the start** — since you need to touch at least three distinct corner regions, plan your overall path to branch toward corners rather than building a single straight connection.\n- **Multi-corner threats beat single-path play** — forcing the opponent to defend two or more of your potential corner-touches simultaneously is the key tactical objective.\n- **Occupy the board centre to preserve routing flexibility** — central cells are equidistant to multiple corners; stones placed there can be incorporated into paths heading in any direction.\n- **Block opponent corner approaches** — a group touching three corners wins immediately; defending one of their potential third corners is as urgent as advancing your own.\n- **Draws are impossible** — unlike many board games, there is no need to consider a drawing defence; every game is decided, which simplifies the calculation of whether a position is winning or losing."}, {"id": 1042, "type": "game", "source": "poly-y", "section": "Engines & current best play", "text": "Poly-Y — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The first-player win is proved by strategy-stealing; no explicit winning strategy is known."}, {"id": 1043, "type": "game", "source": "poly-y", "section": "Complexity", "text": "Poly-Y — Complexity\n\nComparable in size to Hex and Y on equivalent boards."}, {"id": 1044, "type": "game", "source": "poly-y", "section": "References", "text": "Poly-Y — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Y_(game))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975)"}, {"id": 1045, "type": "game", "source": "poly-y", "section": "See also", "text": "Poly-Y — See also\n\n- [Y](y.md) · [Hex](hex.md) · [*Star](star.md)\n- Lexicon: [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved) · [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 1046, "type": "game", "source": "pong-hau-ki", "section": "overview", "text": "Pong hau k'i\nA tiny traditional blocking game — small enough to solve completely by hand;\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan blocking game."}, {"id": 1047, "type": "game", "source": "pong-hau-ki", "section": "Description", "text": "Pong hau k'i — Description\n\nPlayed on a board of 5 points connected by lines (the classic shape has a square\nof 4 points with a diagonal, plus a 5th apex point). Each player has 2 pieces;\none point is empty. Players alternate sliding a piece along a line into the empty\npoint. A player who cannot move loses (their pieces are blocked)."}, {"id": 1048, "type": "game", "source": "pong-hau-ki", "section": "Solution status", "text": "Pong hau k'i — Solution status\n\nPong hau k'i is **strongly solved** — trivially. The entire game graph has only\na handful of distinct positions, so every position's value can be enumerated by\nhand. With correct play **neither side can force a win**: the game is a **draw**\n(in practice, an endless cycle, since a player simply avoids ever being blocked).\nIt is a standard classroom example of a game whose full state graph fits on a\nsingle page."}, {"id": 1049, "type": "game", "source": "pong-hau-ki", "section": "Consensus on optimal play", "text": "Pong hau k'i — Consensus on optimal play\n\n- **Never slide into a corner where both exits are blocked** — the only way to lose is to allow your two pieces to simultaneously occupy points with no shared empty neighbour; always check that at least one of your pieces has an exit.\n- **Use the apex point to prevent opponent blockade** — the 5th apex point is adjacent to more connections than the four square corners; controlling it gives your pieces more routing options.\n- **Mirror the opponent's move when possible** — if the board has a symmetry your opponent just exploited, sliding the mirrored piece preserves your own mobility and denies theirs.\n- **Draw by cycling** — both players can maintain the cycle indefinitely; a player at risk of being blocked should immediately retreat to the safe cycle of positions rather than trying to trap the opponent.\n- **The game is decided entirely by blocking** — there is no scoring or capture; the sole goal is to retain at least one legal move at all times."}, {"id": 1050, "type": "game", "source": "pong-hau-ki", "section": "Engines & current best play", "text": "Pong hau k'i — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine needed — the full position graph (≈10 states) is trivially enumerable by hand.\n- **Strength:** Perfect play by any correct minimax implementation of the ~10 position graph.\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Pong_hau_ki)\n- **Notes:** Among the smallest games in any catalogue; a draw with mutual correct play."}, {"id": 1051, "type": "game", "source": "pong-hau-ki", "section": "Complexity", "text": "Pong hau k'i — Complexity\n\nAbout ten reachable positions — among the smallest non-trivial games in this\narchive."}, {"id": 1052, "type": "game", "source": "pong-hau-ki", "section": "References", "text": "Pong hau k'i — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pong_hau_ki)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1053, "type": "game", "source": "pong-hau-ki", "section": "See also", "text": "Pong hau k'i — See also\n\n- [Mū tōrere](mu-torere.md) · [Three Men's Morris](three-mens-morris.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1054, "type": "game", "source": "punct", "section": "overview", "text": "PÜNCT\nA connection game on a small hexagonal board with stackable pieces of three\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 1055, "type": "game", "source": "punct", "section": "Description", "text": "PÜNCT — Description\n\nPÜNCT (Kris Burm, 2005) is the fifth GIPF-project game. It is a **connection\ngame**: each player has pieces of three different shapes, each occupying a\ndifferent number of cells, and the goal is to link two opposite sides with a\ncontiguous chain of your pieces."}, {"id": 1056, "type": "game", "source": "punct", "section": "Rules", "text": "PÜNCT — Rules\n\n1. Board: small hexagonal grid (37 cells).\n2. Each player has pieces of three sizes (1, 2, and 3 cells in different\n   linear arrangements), totalling a fixed small set per player.\n3. On a turn, a player either:\n   - **Place** a new piece flat on the board, covering the appropriate empty\n     cells; **or**\n   - **Move/stack** an existing piece on top of another piece, advancing it\n     toward the connection.\n4. The first player to form a chain of connected pieces of their colour from\n   their starting edge to the opposite edge wins."}, {"id": 1057, "type": "game", "source": "punct", "section": "Solution status", "text": "PÜNCT — Solution status\n\nPÜNCT is **not solved**. The mix of shapes and stacking gives a unique\nevaluation problem; no published solution."}, {"id": 1058, "type": "game", "source": "punct", "section": "Consensus on optimal play", "text": "PÜNCT — Consensus on optimal play\n\n- **Use larger pieces for bridging** — the 2- and 3-cell pieces span more distance per move and can leap over single-cell gaps; prioritise them for advancing your connection path.\n- **Stack to bypass opponent blockers** — moving a piece on top of an opponent's piece both advances your chain and removes the opponent's piece from its blocking location; stacking is often the decisive manoeuvre.\n- **Build the connection through the board's shortest diameter** — on the small hexagonal board some diagonal paths are shorter than straight paths; route your chain along the minimum-distance axis.\n- **Threaten two routes simultaneously** — if your pieces create two separate partial chains that each need one more bridging move to complete, the opponent cannot block both.\n- **Defend with 1-cell pieces** — small single-cell pieces are cheap blockers; place them in the opponent's direct path to force them to stack and spend tempo."}, {"id": 1059, "type": "game", "source": "punct", "section": "Engines & current best play", "text": "PÜNCT — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** PÜNCT is a commercial GIPF-project game with a modest competitive community; no published computational analysis is known."}, {"id": 1060, "type": "game", "source": "punct", "section": "Complexity", "text": "PÜNCT — Complexity\n\nModerate."}, {"id": 1061, "type": "game", "source": "punct", "section": "References", "text": "PÜNCT — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/P%C3%9CNCT) ([archive](http://web.archive.org/web/20260130062407/https://en.wikipedia.org/wiki/P%C3%9CNCT))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1062, "type": "game", "source": "punct", "section": "See also", "text": "PÜNCT — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [Hex](hex.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1063, "type": "game", "source": "push", "section": "overview", "text": "Push\nA small one-dimensional partisan game from *Winning Ways* — a teaching\nSolution status: Strongly solved as a theory. Game-theoretic value: Position-dependent (switches and integers). Players: 2. Type: Partisan combinatorial game."}, {"id": 1064, "type": "game", "source": "push", "section": "Description", "text": "Push — Description\n\nPush is a partisan one-row game in the same spirit as [Toads and Frogs](toads-and-frogs.md)\nand [Shove](shove.md): pieces of two colours slide along a track, with each\nplayer controlling pieces of one colour. Its compact rules produce a rich\ncatalogue of CGT values."}, {"id": 1065, "type": "game", "source": "push", "section": "Rules", "text": "Push — Rules\n\n1. A row of squares, with some squares occupied by blue or red checkers.\n2. **Left** (blue) moves: slide a blue piece **one square right**, pushing any\n   contiguous run of pieces ahead of it (including red pieces) over by one;\n   the rightmost piece of any such run that would fall off the end is removed.\n3. **Right** (red) moves: mirror-image — slide a red piece **one square left**\n   under the same pushing rule.\n4. The player unable to move loses (normal play)."}, {"id": 1066, "type": "game", "source": "push", "section": "Solution status", "text": "Push — Solution status\n\nStrongly solved as a theory in [*Winning Ways*](../references.md#bcg2001): each\nPush position has an explicit surreal-number / switch value, computed by\nrecursively evaluating Left's and Right's best moves. Sums of independent Push\npositions add by ordinary CGT arithmetic."}, {"id": 1067, "type": "game", "source": "push", "section": "Consensus on optimal play", "text": "Push — Consensus on optimal play\n\n- **Compute the CGT value of each component** — each independent stretch of the row has a well-defined surreal-number or switch value; calculate it by evaluating Left's and Right's best moves recursively.\n- **Combine components by CGT addition** — a sum of independent Push positions has value equal to the sum of their individual values; play in the component with the most temperature (hottest game first).\n- **Push to eliminate rather than to advance** — using the push-and-remove rule to remove an opponent's piece is usually worth more than gaining a square, since it permanently reduces their move count.\n- **Exploit switches** — a position with value {a | b} is a \"switch\"; the player who moves there gains temperature (a − b)/2; left should move in positive-value switches, right in negative-value ones.\n- **Pass to your opponent when the sum is fuzzy** — in a sum of hot games, forcing the opponent to move when all remaining games are positive for them is sometimes the correct \"Nim-like\" endgame strategy."}, {"id": 1068, "type": "game", "source": "push", "section": "Engines & current best play", "text": "Push — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine; CGT analysis by hand or with a computer algebra system suffices.\n- **Strength:** Perfect play via the CGT formula for any given row configuration.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** Push is a pedagogical CGT example; it is not played competitively but is a standard reference for partisan game values."}, {"id": 1069, "type": "game", "source": "push", "section": "Complexity", "text": "Push — Complexity\n\nSmall per position; tractable to enumerate exhaustively for any reasonable row\nlength."}, {"id": 1070, "type": "game", "source": "push", "section": "References", "text": "Push — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Push_(game))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Conway (1976). *On Numbers and Games*.](../references.md#conway1976)"}, {"id": 1071, "type": "game", "source": "push", "section": "See also", "text": "Push — See also\n\n- [Shove](shove.md) · [Toads and Frogs](toads-and-frogs.md) · [Domineering](domineering.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [surreal number](../lexicon/README.md#surreal-number)"}, {"id": 1072, "type": "game", "source": "pylos", "section": "overview", "text": "Pylos\nA 4×4 stacking pyramid game — reported weakly solved as a first-player win\nSolution status: Weakly solved **[verify]**. Game-theoretic value: First-player win **[verify]**. Players: 2. Type: Partisan stacking game."}, {"id": 1073, "type": "game", "source": "pylos", "section": "Description", "text": "Pylos — Description\n\nPylos (David Parlett, 1990s) is played on a 4×4 base of indentations that\nsupport a pyramid; each layer up has one fewer row and column. Players\nalternately place balls of their colour on empty positions; balls placed on\n**four square-adjacent same-level balls** can be played on top, stacking up\nthe pyramid. The player forced to place the apex ball loses."}, {"id": 1074, "type": "game", "source": "pylos", "section": "Rules", "text": "Pylos — Rules\n\n1. Board: 4×4 base; balls form a stable pyramid as 4×4 + 3×3 + 2×2 + 1×1 = 30\n   positions.\n2. Each player has 15 balls of their colour, held off-board in reserve.\n3. On a turn, a player either:\n   - **Place** a new ball on any empty base position; **or**\n   - **Place** a new ball on top of any 2×2 square of balls (regardless of\n     colour) that is fully filled — *promotion*; **or**\n   - **Move up**: take one of your own balls from a *lower* level (where its\n     removal doesn't undermine an upper ball) and replay it onto a 2×2 square\n     it can be promoted to.\n4. **Bonus**: forming a row of four same-coloured balls (horizontally or\n   vertically) on any layer lets the player remove one or two of their own\n   balls back to reserve.\n5. The player who places the **apex** ball loses (it is the 30th ball, and the\n   loser is the one who has no playable move except the apex)."}, {"id": 1075, "type": "game", "source": "pylos", "section": "Solution status", "text": "Pylos — Solution status\n\nA distributed retrograde-analysis solve of Pylos was reported circa 2010,\nfinding a **first-player win** with perfect play. **[verify]** the canonical\nsolver attribution; this archive author has not located a peer-reviewed\nwrite-up."}, {"id": 1076, "type": "game", "source": "pylos", "section": "Consensus on optimal play", "text": "Pylos — Consensus on optimal play\n\n- **Avoid being the one to complete a 2×2 square the opponent can promote from** — promoting is free tempo for the opponent; completing the last side of a 2×2 square only to hand promotion rights to the opponent is a common beginner error.\n- **Earn row bonuses aggressively** — forming a four-in-a-row on any layer allows you to reclaim one or two balls; this conserves your reserve and extends your options, especially in the endgame when the apex approaches.\n- **Control the upper layers** — placing on higher levels early locks in the pyramid structure; the player who places at level 2 and 3 has earlier visibility of the apex position and can manoeuvre to force the opponent to fill it.\n- **Reclaim before the apex is forced** — use bonus reclaims to avoid running out of balls just as the apex becomes the only legal move.\n- **The loser places the apex ball** — therefore force the opponent to take the last available non-apex slot; count remaining empty positions carefully in the endgame."}, {"id": 1077, "type": "game", "source": "pylos", "section": "Engines & current best play", "text": "Pylos — Engines & current best play\n\n- **Strongest known program(s):** Distributed retrograde-analysis solver (attributed to Nievergelt et al., c. 2010) — full position database of ~10^8 positions. **[verify attribution]**\n- **Strength:** Perfect play if the solver database is confirmed correct.\n- **Where the proof / tablebase lives (if solved):** Not publicly available; no peer-reviewed write-up located by this cataloguer.\n- **Notes:** The first-player win result should be treated as preliminary until a verified publication is found."}, {"id": 1078, "type": "game", "source": "pylos", "section": "Complexity", "text": "Pylos — Complexity\n\nModerate: roughly 10^8 reachable positions."}, {"id": 1079, "type": "game", "source": "pylos", "section": "References", "text": "Pylos — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pylos_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1080, "type": "game", "source": "pylos", "section": "See also", "text": "Pylos — See also\n\n- [Connect Four](connect-four.md) · [Score Four](score-four.md) · [GIPF](gipf.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 1081, "type": "game", "source": "pyraminx", "section": "overview", "text": "Pyraminx\nTetrahedral twist puzzle — fully solved: God's number is 11 (or 6 ignoring tips).\nSolution status: Strongly solved. Game-theoretic value: Any scramble solvable in ≤ 11 moves (≤ 6 ignoring tips). Players: 1. Type: Solo permutation puzzle."}, {"id": 1082, "type": "game", "source": "pyraminx", "section": "Description", "text": "Pyraminx — Description\n\nThe Pyraminx (Uwe Mèffert, 1981) is a tetrahedral twist puzzle. Its\nstate-graph is small (under 10^8) and has been fully analysed: any scramble\nsolves in at most **11 moves**, or **6 moves** ignoring the four trivial\ncorner tips."}, {"id": 1083, "type": "game", "source": "pyraminx", "section": "Rules", "text": "Pyraminx — Rules\n\n1. Puzzle: tetrahedral puzzle with 4 corner tips, 4 axial \"trivial\" pieces,\n   and 6 edges.\n2. On a move the solver rotates one of the 4 axes by 120° or 240°.\n3. Each tip is on its own axis and trivially rotates independently — solving\n   them is essentially a free operation.\n4. The puzzle is solved when every face shows a single colour."}, {"id": 1084, "type": "game", "source": "pyraminx", "section": "Solution status", "text": "Pyraminx — Solution status\n\n**Strongly solved**: any scramble can be solved in **≤ 11 moves** (≤ 6\nignoring tips)."}, {"id": 1085, "type": "game", "source": "pyraminx", "section": "Consensus on optimal play", "text": "Pyraminx — Consensus on optimal play\n\n- **Fix tips last (or first — they're free)** — the four corner tips each have a trivial independent axis; orient them at any point without affecting the rest of the puzzle; many speedcubers fix tips last as a final trivial step.\n- **V method (solve edges in a V-shape on one face)** — place three edge pieces on the bottom face first, then solve the top cap; this is faster than a strict layer-by-layer approach.\n- **Keyhole / L4E methods** — reduce the remaining pieces to a known lookup case and apply a single short algorithm; top speedcubers often reduce the solve to one or two algorithm applications.\n- **Optimal solve ≤ 11 moves** — any scramble is within 11 axis rotations of solved; an IDA* search over the small state space finds optimal solutions instantly.\n- **Only 75 million positions** — the full state graph is smaller than many games' opening books; brute-force optimal lookup is practical."}, {"id": 1086, "type": "game", "source": "pyraminx", "section": "Engines & current best play", "text": "Pyraminx — Engines & current best play\n\n- **Strongest known program(s):** Complete lookup-table solvers (small enough for exhaustive enumeration).\n- **Strength:** Perfect (optimal solve in ≤ 11 moves, ≤ 6 ignoring tips).\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Pyraminx)\n- **Notes:** God's number of 11 (with tips) / 6 (without) is verified by complete enumeration of all ~75 million positions."}, {"id": 1087, "type": "game", "source": "pyraminx", "section": "Complexity", "text": "Pyraminx — Complexity\n\nSmall."}, {"id": 1088, "type": "game", "source": "pyraminx", "section": "References", "text": "Pyraminx — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Pyraminx) ([archive](http://web.archive.org/web/20260323231722/https://en.wikipedia.org/wiki/Pyraminx))\n- [Rokicki *et al.* (2017). *The diameter of the Rubik's cube group is twenty*.](../references.md#rokicki2014) (related)"}, {"id": 1089, "type": "game", "source": "pyraminx", "section": "See also", "text": "Pyraminx — See also\n\n- [Rubik's Cube](rubiks-cube.md) · [Pocket Cube](pocket-cube.md) · [Skewb](skewb.md) · [Megaminx](megaminx.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [God's number](../lexicon/README.md#gods-number)"}, {"id": 1090, "type": "game", "source": "quarto", "section": "overview", "text": "Quarto\nA four-in-a-row game with a devious twist — *your opponent* chooses the piece\nSolution status: Weakly solved **[verify]**. Game-theoretic value: Draw **[verify]**. Players: 2. Type: Partisan positional game with opponent-chosen pieces."}, {"id": 1091, "type": "game", "source": "quarto", "section": "Description", "text": "Quarto — Description\n\nPlayed on a 4×4 board with 16 distinct pieces, each having four binary\nattributes (tall/short, light/dark, round/square, solid/hollow). The twist: on\nyour turn you place the piece **your opponent hands you**, then hand them the\npiece they must place next. A player wins by completing a line of four pieces\nthat **share at least one attribute**. (A common variant also counts 2×2\nsquares.)"}, {"id": 1092, "type": "game", "source": "quarto", "section": "Solution status", "text": "Quarto — Solution status\n\nQuarto is small — a 4×4 board and 16 pieces give only a few million reachable\npositions — and is **reported to be weakly solved by exhaustive search, with\nthe standard game a draw**: with perfect play neither player can force a\nshared-attribute line, because the opponent always retains a safe piece to hand\nover.\n\n> **[verify]** — The draw verdict is the commonly cited result and is\n> consistent with exhaustive analysis of a game this size, but this archive has\n> not confirmed a single canonical primary citation. The rule variant matters:\n> adding the 2×2-square winning condition changes the analysis, and a solving\n> source should specify which rules it used."}, {"id": 1093, "type": "game", "source": "quarto", "section": "Consensus on optimal play", "text": "Quarto — Consensus on optimal play\n\n- **Never hand over a \"quarto-completing\" piece** — before handing the opponent their next piece, check all partial lines of three with a shared attribute; if any such line exists and the piece completes it, find a different piece to hand over.\n- **Build lines that require rare attributes** — a partial row of three \"tall round\" pieces needs one more tall round piece to complete it; if only one such piece remains unplaced, control of that piece is decisive.\n- **Force the opponent to hand you a dangerous piece** — by holding the board in a state where almost every remaining piece completes some line, you corner the opponent into handing over a winner.\n- **The 2×2 square variant is harder to defend** — if playing with the 2×2 square win condition, also track partial 2×2 groups; the extra winning conditions sharply limit the \"safe\" pieces to hand over.\n- **Midgame piece selection matters as much as placement** — placing optimally but handing over a game-losing piece is the same as playing a losing move; the hand-over decision is half the game."}, {"id": 1094, "type": "game", "source": "quarto", "section": "Engines & current best play", "text": "Quarto — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked publicly; the game is small enough for any complete search to play perfectly.\n- **Where the proof / tablebase lives (if solved):** No canonical primary citation confirmed; draw verdict widely reported.\n- **Notes:** The draw result should be treated as preliminary until a verified publication is located; the rule variant (with/without 2×2 squares) must be specified."}, {"id": 1095, "type": "game", "source": "quarto", "section": "Complexity", "text": "Quarto — Complexity\n\nA few million positions — exhaustively searchable."}, {"id": 1096, "type": "game", "source": "quarto", "section": "References", "text": "Quarto — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Quarto_(board_game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1097, "type": "game", "source": "quarto", "section": "See also", "text": "Quarto — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Teeko](teeko.md) · [Connect Four](connect-four.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1098, "type": "game", "source": "qubic", "section": "overview", "text": "Qubic\nThree-dimensional tic-tac-toe on a 4×4×4 cube — and a first-player win.\nSolution status: Weakly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan positional (k-in-a-row) game."}, {"id": 1099, "type": "game", "source": "qubic", "section": "Description", "text": "Qubic — Description\n\nPlayed in a 4×4×4 cube of 64 cells. Players alternately claim cells; the winner\nis the first to claim four cells in a straight line — along any of the cube's\nrows, columns, pillars, or the 2-D and 3-D diagonals (76 winning lines in all)."}, {"id": 1100, "type": "game", "source": "qubic", "section": "Solution status", "text": "Qubic — Solution status\n\nQubic is **weakly solved**. [Patashnik (1980)](../references.md#patashnik1980)\nproved it a **first-player win**, using a carefully structured proof combining\nhuman strategic reasoning with computer verification of the resulting case\ntree — an early landmark in computer-assisted game solving. The result was later\nre-derived independently by [Allis (1994)](../references.md#allis1994) using\nproof-number search, confirming the first-player win.\n\nUnlike flat [tic-tac-toe](tic-tac-toe.md) (a draw), the extra dimension gives the\nfirst player enough overlapping threats to force a win against any defence."}, {"id": 1101, "type": "game", "source": "qubic", "section": "Consensus on optimal play", "text": "Qubic — Consensus on optimal play\n\n- **First player must threaten multiple lines simultaneously** — with 76 winning lines in a 4×4×4 cube, the winning strategy depends on building multiple overlapping threats; a single line is easily blocked.\n- **3-D diagonals are hard to visualise and defend** — the four long space-diagonals of the cube (corner to opposite corner) are frequently overlooked by defenders; build through them early.\n- **Centre layers are more valuable than faces** — cells in the two inner layers (z=2, z=3) participate in more winning lines than surface cells; prioritise them in the opening.\n- **The 2×2 threat cluster** — claiming the four corners of any face of a sub-cube creates multiple simultaneous winning-line seeds; the opponent cannot address all resulting threats.\n- **First player wins with correct play** — the Patashnik/Allis result is definitive; as second player your only hope is an error by the first player in the intricate winning tree."}, {"id": 1102, "type": "game", "source": "qubic", "section": "Engines & current best play", "text": "Qubic — Engines & current best play\n\n- **Strongest known program(s):** Patashnik's 1980 computer-aided solver; Allis's proof-number search implementation (1994).\n- **Strength:** Both solve perfectly from the start position; neither is packaged as a downloadable competitive program.\n- **Where the proof / tablebase lives (if solved):** [Patashnik (1980)](../references.md#patashnik1980); [Allis (1994)](../references.md#allis1994)\n- **Notes:** The first-player win is double-confirmed by two independent methods; the full winning tree is intricate but finite."}, {"id": 1103, "type": "game", "source": "qubic", "section": "Complexity", "text": "Qubic — Complexity\n\nA 64-cell board with 76 winning lines; the solution tree was large for 1980 but\ntractable with the structured approach used."}, {"id": 1104, "type": "game", "source": "qubic", "section": "References", "text": "Qubic — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/3D_tic-tac-toe) ([archive](http://web.archive.org/web/20260311160137/https://en.wikipedia.org/wiki/3D_tic-tac-toe))\n- [Patashnik, O. (1980). *Qubic: 4×4×4 Tic-Tac-Toe*.](../references.md#patashnik1980)\n- [Allis, V. (1994). *Searching for Solutions in Games and Artificial Intelligence*.](../references.md#allis1994)"}, {"id": 1105, "type": "game", "source": "qubic", "section": "See also", "text": "Qubic — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Score Four](score-four.md) · [Gomoku](gomoku.md) · [Connect Four](connect-four.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [proof-number search](../lexicon/README.md#proof-number-search)"}, {"id": 1106, "type": "game", "source": "quixo", "section": "overview", "text": "Quixo\nA sliding-cube tic-tac-toe-like on a 5×5 board — solved in 2020 as a draw.\nSolution status: Weakly solved (standard 5×5). Game-theoretic value: Draw. Players: 2. Type: Partisan positional / sliding game."}, {"id": 1107, "type": "game", "source": "quixo", "section": "Description", "text": "Quixo — Description\n\nPlayed with a 5×5 grid of cubes, each face blank, marked X, or marked O. On a\nturn a player takes a blank or own-marked cube **from the border**, sets it to\ntheir mark, and pushes it back into a row or column from one end — sliding the\nother cubes along. The winner is the first to form a line of five of their\nmarks. (A move that would complete a line of *both* marks counts for the\nopponent, discouraging some pushes.)"}, {"id": 1108, "type": "game", "source": "quixo", "section": "Solution status", "text": "Quixo — Solution status\n\nQuixo on the standard **5×5** board is **weakly solved**.\n[Tanaka, Bonnet, Tixeuil & Tamura (2020)](../references.md#tanaka-quixo2020)\nsolved it using [retrograde analysis](../lexicon/README.md#retrograde-analysis):\nwith perfect play the game is a **draw** — the first player cannot force a win.\nThe same work also reported results for smaller boards (e.g. first-player wins\non some reduced sizes), and noted that without a rule capping repetition the\ngame could in principle continue indefinitely, which the analysis accounts for."}, {"id": 1109, "type": "game", "source": "quixo", "section": "Consensus on optimal play", "text": "Quixo — Consensus on optimal play\n\n- **Only take from the border** — pieces can only be drawn from the perimeter; keep your inner pieces (which cannot be moved) in positions that resist line completion for the opponent.\n- **Use push direction to disrupt opponent lines** — inserting from one end of a row slides all existing pieces one step; calculate whether the insertion breaks an opponent's near-complete line or, worse, completes one.\n- **Avoid completing lines for both players simultaneously** — the rule that a move completing both players' five-in-a-row scores for the opponent is a critical trap; always check the pushed-row result for unintended opponent wins.\n- **Claim blank border cubes before the opponent** — a blank cube can be turned to either player's mark; securing border blanks early gives flexibility and denies conversion opportunities.\n- **Draw is the correct result** — with perfect play neither side wins; practical play exploits small inaccuracies rather than trying to find a theoretical forced win."}, {"id": 1110, "type": "game", "source": "quixo", "section": "Engines & current best play", "text": "Quixo — Engines & current best play\n\n- **Strongest known program(s):** Tanaka et al. 2020 retrograde solver — complete 5×5 database.\n- **Strength:** Perfect play (weakly solved).\n- **Where the proof / tablebase lives (if solved):** [Tanaka, Bonnet, Tixeuil & Tamura (2020)](../references.md#tanaka-quixo2020)\n- **Notes:** The solved database is the result of the 2020 paper; no standalone downloadable engine for Quixo is widely available."}, {"id": 1111, "type": "game", "source": "quixo", "section": "Complexity", "text": "Quixo — Complexity\n\nThe border-only move rule and cube-pushing keep the reachable state space within\nreach of retrograde analysis, despite the 25-cell board."}, {"id": 1112, "type": "game", "source": "quixo", "section": "References", "text": "Quixo — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Quixo)\n- [Tanaka, S., Bonnet, F., Tixeuil, S. & Tamura, Y. (2020). *Quixo Is Solved*.](../references.md#tanaka-quixo2020)"}, {"id": 1113, "type": "game", "source": "quixo", "section": "See also", "text": "Quixo — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Othello](othello.md) · [Pentago](pentago.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 1114, "type": "game", "source": "quoridor", "section": "overview", "text": "Quoridor\nA race game where you also build walls to slow your opponent — solved only on\nSolution status: Partially solved (small boards). Game-theoretic value: Unknown for the standard 9×9 board. Players: 2 (also a 4-player variant). Type: Partisan race / blocking game."}, {"id": 1115, "type": "game", "source": "quoridor", "section": "Description", "text": "Quoridor — Description\n\nPlayed on a 9×9 grid of cells. Each player has a pawn starting on opposite edges\nand a supply of wall pieces (10 each in the two-player game). On a turn a player\neither moves their pawn one cell or places a wall segment between cells. Walls\nblock movement but **may never completely seal a player off** from their goal\nedge. The first pawn to reach the opposite edge wins."}, {"id": 1116, "type": "game", "source": "quoridor", "section": "Solution status", "text": "Quoridor — Solution status\n\nQuoridor is **partially solved**. Reduced boards (e.g. small *n*×*n* grids with\ncorrespondingly few walls) have been solved exhaustively —\n[Glendenning's (2005)](../references.md#glendenning-quoridor2005) thesis and\nlater work analyse such cases — and small-board results generally favour the\nfirst player. The standard **9×9** board, with its huge branching factor from\nwall placements, is **unsolved**. The \"no full blockade\" rule also makes legal\nmove generation non-trivial, since each candidate wall must be checked for\npath-preservation.\n\n> **[verify]** — The ~10^42 state-space figure for the 9×9 board is an\n> order-of-magnitude estimate that should be checked against a primary source."}, {"id": 1117, "type": "game", "source": "quoridor", "section": "Consensus on optimal play", "text": "Quoridor — Consensus on optimal play\n\n- **Advance while you have the shorter path** — count shortest path to your goal vs. the opponent's at every turn; as long as your path is shorter, just move and don't waste walls.\n- **Walls are a finite resource — save them for pivotal moments** — 10 walls per player run out quickly; placing walls to gain only one or two steps over the pawn-move alternative wastes this resource.\n- **Use walls to lengthen the opponent's path, not just to block** — a well-placed wall can add 3–4 moves to the opponent's shortest path; do the pathfinding calculation (BFS) before placing.\n- **Never let the opponent's pawn get ahead without using walls** — if the opponent's pawn is closer to their goal and you have walls remaining, this is the time to spend them; waiting too long is fatal.\n- **Horizontal walls near the opponent's starting side are usually stronger** — early walls placed deep in the opponent's territory are hard for them to route around and force long detours."}, {"id": 1118, "type": "game", "source": "quoridor", "section": "Engines & current best play", "text": "Quoridor — Engines & current best play\n\n- **Strongest known program(s):** Various open-source bots (e.g. Quoridor-specific minimax programs with BFS-based heuristics); no dominant publicly benchmarked engine.\n- **Strength:** Strong amateur; competitive with experienced human players.\n- **Where the proof / tablebase lives (if solved):** [Glendenning (2005)](../references.md#glendenning-quoridor2005) (small boards only)\n- **Notes:** The 9×9 board is unsolved; first-player advantage is widely assumed but unproven."}, {"id": 1119, "type": "game", "source": "quoridor", "section": "Complexity", "text": "Quoridor — Complexity\n\nStandard board ~10^42 positions **[verify]**; the wall-placement branching makes\nthe game tree very wide."}, {"id": 1120, "type": "game", "source": "quoridor", "section": "References", "text": "Quoridor — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Quoridor) ([archive](http://web.archive.org/web/20260512183438/https://en.wikipedia.org/wiki/Quoridor))\n- [Glendenning, L. (2005). *Mastering Quoridor*.](../references.md#glendenning-quoridor2005)"}, {"id": 1121, "type": "game", "source": "quoridor", "section": "See also", "text": "Quoridor — See also\n\n- [Breakthrough](breakthrough.md) · [Hex](hex.md)\n- Lexicon: [partially solved](../lexicon/README.md#solved-game) · [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1122, "type": "game", "source": "red-blue-green-hackenbush", "section": "overview", "text": "Red-Blue-Green Hackenbush\nHackenbush with three edge colours — a partisan game whose values introduce\nSolution status: Strongly solved as a theory (each position has a CGT value). Game-theoretic value: Each position has a surreal-number / switch value. Players: 2 (Blue and Red, with neutral Green). Type: Partisan combinatorial game."}, {"id": 1123, "type": "game", "source": "red-blue-green-hackenbush", "section": "Description", "text": "Red-Blue-Green Hackenbush — Description\n\nHackenbush is played on a graph attached to \"the ground.\" In RBG Hackenbush\neach edge is coloured **blue** (Left's), **red** (Right's), or **green**\n(either player's). Removing an edge causes any subgraph no longer connected to\nthe ground to fall off. The partisan moves on differently-coloured edges give a\nmuch richer value space than the impartial Green-only [Hackenbush](hackenbush.md)."}, {"id": 1124, "type": "game", "source": "red-blue-green-hackenbush", "section": "Rules", "text": "Red-Blue-Green Hackenbush — Rules\n\n1. A graph with one or more *ground* vertices and edges coloured blue, red, or\n   green is given.\n2. Left moves: remove a blue or green edge. Right moves: remove a red or green\n   edge.\n3. After removing an edge, any subgraph no longer connected to the ground is\n   removed too.\n4. The player unable to move loses (normal play)."}, {"id": 1125, "type": "game", "source": "red-blue-green-hackenbush", "section": "Solution status", "text": "Red-Blue-Green Hackenbush — Solution status\n\nStrongly solved as a theory. Each RBG-Hackenbush position has a well-defined\n**game value** in Conway's surreal-number system; the value of a string is\ncomputed by the \"sign-expansion\" / Berlekamp's algorithm, and disjoint\ncomponents add as games. Green edges contribute *star* (a nimber); switches and\ninfinitesimals appear naturally in mixed positions. See\n[*Winning Ways* / On Numbers and Games](../references.md#bcg2001)."}, {"id": 1126, "type": "game", "source": "red-blue-green-hackenbush", "section": "Consensus on optimal play", "text": "Red-Blue-Green Hackenbush — Consensus on optimal play\n\n- **Compute each component's surreal value separately** — disjoint subgraphs add as games; analyse each connected component in isolation and combine by CGT addition.\n- **Blue/Red edges on strings have dyadic-rational values** — a Blue-Red path from ground gives value equal to the \"sign-expansion\" of the edge sequence; memorise short strings and use Berlekamp's algorithm for longer ones.\n- **Green edges add nimbers (stars)** — a single green edge contributes *1 (star); groups of green edges in a tree produce larger nimbers by XOR; add these to the Blue-Red value of the position.\n- **Play the hottest component first** — in a sum, the move with the highest temperature (half the gap between Left and Right game values) is usually the correct \"hot game\" choice.\n- **Switches require careful timing** — a switch {a | b} should be taken by Left when a > b and it is your turn; converting a switch too early when the rest of the sum is hotter wastes opportunity."}, {"id": 1127, "type": "game", "source": "red-blue-green-hackenbush", "section": "Engines & current best play", "text": "Red-Blue-Green Hackenbush — Engines & current best play\n\n- **Strongest known program(s):** No competitive game-engine; CGT computation tools (e.g. Aaron Siegel's CGSuite) compute positions analytically.\n- **Strength:** Exact optimal play via CGT formula for any position whose value can be computed.\n- **Where the proof / tablebase lives (if solved):** [Conway (1976)](../references.md#conway1976); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** RBG Hackenbush is primarily a theoretical framework; no competitive human game has an established engine."}, {"id": 1128, "type": "game", "source": "red-blue-green-hackenbush", "section": "Complexity", "text": "Red-Blue-Green Hackenbush — Complexity\n\nExponential in general; specific structured families (Hackenbush \"strings,\"\n\"flowers\") have polynomial value-formulas."}, {"id": 1129, "type": "game", "source": "red-blue-green-hackenbush", "section": "References", "text": "Red-Blue-Green Hackenbush — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Hackenbush)\n- [Conway (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1130, "type": "game", "source": "red-blue-green-hackenbush", "section": "See also", "text": "Red-Blue-Green Hackenbush — See also\n\n- [Hackenbush](hackenbush.md) · [Domineering](domineering.md) · [Toads and Frogs](toads-and-frogs.md)\n- Lexicon: [surreal number](../lexicon/README.md#surreal-number) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1131, "type": "game", "source": "renju", "section": "overview", "text": "Renju\nProfessional five-in-a-row: handicaps on the first player tame Gomoku's\nSolution status: Weakly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan positional (k-in-a-row) game with first-player restrictions."}, {"id": 1132, "type": "game", "source": "renju", "section": "Description", "text": "Renju — Description\n\nRenju is [Gomoku](gomoku.md) on a 15×15 board with rules that **handicap the\nfirst player (Black)**: Black is forbidden to make \"double-three,\" \"double-four,\"\nor \"overline\" (six or more) — these are all losing fouls for Black — while White\nhas no such restrictions. The handicaps were introduced precisely because\nfree-style Gomoku is a clear first-player win."}, {"id": 1133, "type": "game", "source": "renju", "section": "Solution status", "text": "Renju — Solution status\n\nRenju is **weakly solved**. [Wágner & Virág (2001)](../references.md#wagner-virag2001)\nproved that, even with the forbidden-move handicaps, **the first player still\nwins** with perfect play (under the rule set without opening restrictions such\nas the modern swap/opening protocols). They used\n[proof-number search](../lexicon/README.md#proof-number-search) and threat-based\nsearch techniques in the lineage of Allis's Gomoku solution.\n\nThe result is a notable cautionary tale about game balancing: the Renju fouls\nsubstantially narrow Black's options but do not eliminate the first-player win —\nwhich is why modern competitive Renju also layers on opening-move protocols."}, {"id": 1134, "type": "game", "source": "renju", "section": "Consensus on optimal play", "text": "Renju — Consensus on optimal play\n\n- **Avoid Black's forbidden patterns** — as Black, never place a stone that creates a double-three (two open threes simultaneously), double-four (two open fours simultaneously), or overline (six or more in a row); these are immediate fouls.\n- **White should force Black into foul situations** — as White, direct play toward positions where every Black winning move is also a foul; this \"forbidden trap\" strategy is uniquely available in Renju.\n- **Five-in-a-row beats the foul for Black** — if Black can form exactly five-in-a-row in the same move that would create a forbidden pattern, the five-in-a-row wins; this requires precise counting.\n- **Threat-space search drives strong play** — both humans and programs build winning strategies by iteratively discovering forced-win tree paths (VCF = victory by consecutive fours, VCT = victory by consecutive threats).\n- **Opening protocols matter in practice** — competitive play adds swap2 or other opening neutralisations; prepare specific openings that steer into positions the Wágner–Virág winning tree covers."}, {"id": 1135, "type": "game", "source": "renju", "section": "Engines & current best play", "text": "Renju — Engines & current best play\n\n- **Strongest known program(s):** Renju-specific solvers built on threat-space search (Wágner & Virág's 2001 system); community programs such as Gomoku/Renju engines available in online play.\n- **Strength:** Super-human on the solved opening; beats top professionals from the starting position.\n- **Where the proof / tablebase lives (if solved):** [Wágner & Virág (2001)](../references.md#wagner-virag2001)\n- **Notes:** The proof covers the basic Renju rule set without swap/opening protocols; competitive engines additionally handle modern tournament rule sets."}, {"id": 1136, "type": "game", "source": "renju", "section": "Complexity", "text": "Renju — Complexity\n\nState-space ~10^105, game-tree ~10^70 (same board as Gomoku;\n[van den Herik et al., 2002](../references.md#vandenherik2002))."}, {"id": 1137, "type": "game", "source": "renju", "section": "References", "text": "Renju — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Renju) ([archive](http://web.archive.org/web/20260311032313/https://en.wikipedia.org/wiki/Renju))\n- [Wágner, J. & Virág, I. (2001). *Solving Renju*.](../references.md#wagner-virag2001)\n- [Allis, van den Herik & Huntjens (1996). *Go-Moku Solved by New Search Techniques*.](../references.md#allis-gomoku1996)"}, {"id": 1138, "type": "game", "source": "renju", "section": "See also", "text": "Renju — See also\n\n- [Gomoku](gomoku.md) · [Pente](pente.md) · [Connect6](connect6.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1139, "type": "game", "source": "rock-paper-scissors", "section": "overview", "text": "Rock-paper-scissors\nThe textbook example of a game with no pure-strategy solution — its unique\nSolution status: Solved (unique Nash equilibrium known). Game-theoretic value: Draw — value 0 under the equilibrium mixed strategy. Players: 2. Type: Simultaneous-move zero-sum game."}, {"id": 1140, "type": "game", "source": "rock-paper-scissors", "section": "Description", "text": "Rock-paper-scissors — Description\n\nBoth players simultaneously choose rock, paper, or scissors. Rock beats scissors,\nscissors beats paper, paper beats rock; identical choices draw. It is the\ncanonical **simultaneous-move zero-sum game** — there is no \"first player,\" and\nbecause moves are revealed at once it is technically a game of imperfect\ninformation."}, {"id": 1141, "type": "game", "source": "rock-paper-scissors", "section": "Solution status", "text": "Rock-paper-scissors — Solution status\n\nRock-paper-scissors is **solved** by classical game theory. It has **no\npure-strategy equilibrium** — for any deterministic choice there is a winning\ncounter — but by [von Neumann's minimax theorem](../lexicon/README.md#nash-equilibrium)\nit has a value in **mixed strategies**. The unique [Nash equilibrium](../lexicon/README.md#nash-equilibrium)\nis for each player to choose **uniformly at random (⅓, ⅓, ⅓)**; this guarantees\neach player an expected payoff of 0 regardless of what the opponent does, so the\ngame's value is a **draw**. This is the standard introductory example of why\noptimal play can require randomisation."}, {"id": 1142, "type": "game", "source": "rock-paper-scissors", "section": "Consensus on optimal play", "text": "Rock-paper-scissors — Consensus on optimal play\n\n- **Play uniformly at random (⅓, ⅓, ⅓)** — this is the unique Nash equilibrium; it guarantees expected payoff 0 against any opponent and cannot be exploited.\n- **Any deviation from uniform is exploitable** — if you play rock even slightly more than ⅓ of the time, an opponent who detects this can profitably shift toward paper; the uniform strategy is the only strategy with no counter.\n- **Against humans, exploit pattern biases** — people are notoriously non-random; studies consistently show that humans throw rock most often after a loss, repeat wins, and cycle R→P→S; pattern-exploitation beats equilibrium play against imperfect opponents.\n- **After a loss, most humans switch** — if an opponent just threw rock and lost, they are statistically less likely to throw rock again; updating on this prior can give an edge in competitive human play.\n- **Competitive RPS is a psychology game** — at the highest level (e.g. World RPS Society tournaments) players attempt to \"level\" each other's meta-reasoning, making the equilibrium a baseline to deviate from rather than a target to achieve."}, {"id": 1143, "type": "game", "source": "rock-paper-scissors", "section": "Engines & current best play", "text": "Rock-paper-scissors — Engines & current best play\n\n- **Strongest known program(s):** RoShamBo programming competition bots — agents that exploit opponent patterns via statistical learning (e.g. frequency analysis, history-based prediction).\n- **Strength:** Human-pattern-exploiting bots beat human players; equilibrium bots guarantee ≥ 0 expected payoff against any opponent.\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Rock_paper_scissors); classical minimax theorem (von Neumann 1928).\n- **Notes:** The uniform mixed strategy is the complete theoretical solution; practical competitive play adds meta-game psychology on top."}, {"id": 1144, "type": "game", "source": "rock-paper-scissors", "section": "Complexity", "text": "Rock-paper-scissors — Complexity\n\nTrivial: three actions per player, a single simultaneous move. Its interest is\nconceptual — it is the simplest game whose solution is irreducibly mixed."}, {"id": 1145, "type": "game", "source": "rock-paper-scissors", "section": "References", "text": "Rock-paper-scissors — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Rock_paper_scissors) ([archive](http://web.archive.org/web/20260506060951/https://en.wikipedia.org/wiki/Rock_paper_scissors))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1146, "type": "game", "source": "rock-paper-scissors", "section": "See also", "text": "Rock-paper-scissors — See also\n\n- [Sim](sim.md) · [Liar's dice](liars-dice.md)\n- Lexicon: [Nash equilibrium](../lexicon/README.md#nash-equilibrium) · [draw](../lexicon/README.md#draw) · [zero-sum game](../lexicon/README.md#zero-sum-game)"}, {"id": 1147, "type": "game", "source": "rubiks-cube", "section": "overview", "text": "Rubik's Cube\nThe most famous mechanical puzzle — strongly solved in the sense that \"God's\nSolution status: Strongly solved (God's Number = 20, half-turn metric). Game-theoretic value: N/A (puzzle) — every position is solvable; diameter = 20. Players: 1 (puzzle). Type: Single-player permutation puzzle."}, {"id": 1148, "type": "game", "source": "rubiks-cube", "section": "Description", "text": "Rubik's Cube — Description\n\nA 3×3×3 cube whose six faces can each be rotated. The 26 visible \"cubies\" can be\nscrambled into about 4.3 × 10^19 distinct reachable configurations; the goal is\nto return every face to a single colour. As a single-player puzzle it has no\ngame-theoretic value — the meaningful question is the **diameter** of its\nconfiguration graph: the largest number of moves ever needed to solve a position\noptimally."}, {"id": 1149, "type": "game", "source": "rubiks-cube", "section": "Solution status", "text": "Rubik's Cube — Solution status\n\nThe Rubik's Cube is **strongly solved** in the puzzle sense.\n[Rokicki, Kociemba, Davidson & Dethridge (2014)](../references.md#rokicki2014)\nproved that **every** configuration can be solved in at most **20** moves in the\nhalf-turn metric, and that some positions (\"superflip\" and others) genuinely\nrequire 20 — so the diameter, \"**God's Number**,\" is exactly 20. The proof\npartitioned all 4.3 × 10^19 states into ~2 billion cosets and searched them with\nmassive donated Google computing time. (In the quarter-turn metric the\ncorresponding number is 26.)"}, {"id": 1150, "type": "game", "source": "rubiks-cube", "section": "Consensus on optimal play", "text": "Rubik's Cube — Consensus on optimal play\n\n- **CFOP (Fridrich method) for beginners to advanced** — solve the cross (bottom layer edges), then four first-layer corners, then the second layer edges, then orient and permute the top layer; most speedcubers use this 4-phase approach and average 50–60 moves.\n- **Roux method reduces move count** — build two 1×2×3 blocks on left and right, then finish the top with M-slice and last-six-edges algorithms; requires fewer moves than CFOP but is harder to learn.\n- **Kociemba's two-phase algorithm (computers)** — reduce to a subgroup using phase 1 (≤20 moves), then solve the reduced position in phase 2; finds near-optimal solutions (usually ≤22 moves) in milliseconds.\n- **Optimal IDA* solver** — search with the Korf (1997) IDA* algorithm using pattern-database heuristics; finds a provably minimal-move solution for any position but can take seconds for deep scrambles.\n- **God's Number is 20 (half-turn metric)** — no position requires more than 20 moves; any solver claiming more than 20 moves is sub-optimal."}, {"id": 1151, "type": "game", "source": "rubiks-cube", "section": "Engines & current best play", "text": "Rubik's Cube — Engines & current best play\n\n- **Strongest known program(s):** Kociemba's two-phase algorithm (open source at [https://kociemba.org/cube.htm](https://kociemba.org/cube.htm)); optimal IDA* solvers based on Korf (1997).\n- **Strength:** Perfect (optimal in ≤ 20 HTM moves for any scramble).\n- **Where the proof / tablebase lives (if solved):** [Rokicki et al. (2014)](../references.md#rokicki2014); [Wikipedia](https://en.wikipedia.org/wiki/Rubik%27s_Cube)\n- **Notes:** The 2010/2014 proof required ~35 CPU-years on donated Google resources; no single machine holds the full tablebase."}, {"id": 1152, "type": "game", "source": "rubiks-cube", "section": "Complexity", "text": "Rubik's Cube — Complexity\n\n4.3 × 10^19 reachable states; configuration-graph diameter proven to be 20\n(half-turn metric). Generalised n×n×n cube solving is known to be NP-hard."}, {"id": 1153, "type": "game", "source": "rubiks-cube", "section": "References", "text": "Rubik's Cube — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Rubik%27s_Cube) ([archive](http://web.archive.org/web/20260513023211/https://en.wikipedia.org/wiki/Rubik%27s_Cube))\n- [Rokicki, Kociemba, Davidson & Dethridge (2014). *The Diameter of the Rubik's Cube Group Is Twenty*.](../references.md#rokicki2014)"}, {"id": 1154, "type": "game", "source": "rubiks-cube", "section": "See also", "text": "Rubik's Cube — See also\n\n- [15 puzzle](fifteen-puzzle.md) · [Peg solitaire](pegs-solitaire.md)\n- Lexicon: [God's number](../lexicon/README.md#gods-number) · [strongly solved](../lexicon/README.md#strongly-solved) · [state-space complexity](../lexicon/README.md#state-space-complexity)"}, {"id": 1155, "type": "game", "source": "ruler-game", "section": "overview", "text": "Ruler game\nA textbook octal game whose nim-sequence is the \"ruler\" function — a clean\nSolution status: Strongly solved. Game-theoretic value: Determined by nim-sum of pile nim-values. Players: 2. Type: Impartial take-and-break game."}, {"id": 1156, "type": "game", "source": "ruler-game", "section": "Description", "text": "Ruler game — Description\n\nPlayed on heaps of tokens. The game is named for its nim-sequence, which equals\nthe **ruler function** — the largest power of 2 dividing the heap size — so the\nnim-values look like markings on a ruler: 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3,..."}, {"id": 1157, "type": "game", "source": "ruler-game", "section": "Rules", "text": "Ruler game — Rules\n\n1. One or more heaps of tokens.\n2. A move picks one heap of size n and removes between 1 and n tokens, but the\n   exact allowed removals are given by the octal code .07 — concretely, you\n   may remove any positive amount and optionally split the remainder according\n   to a fixed rule.\n3. The player who cannot move loses (normal play).\n\nIn practice the game is most often described directly by its nim-value table\nrather than by the octal rule."}, {"id": 1158, "type": "game", "source": "ruler-game", "section": "Solution status", "text": "Ruler game — Solution status\n\nStrongly solved by [Guy & Smith (1956)](../references.md#guy-smith1956). The\nnim-value of a single pile of size n is the **2-adic valuation** of n (the\nlargest k such that 2^k divides n) plus 1 — the \"ruler sequence.\" Combine\nseveral piles by [nim-sum](../lexicon/README.md#nim-sum)."}, {"id": 1159, "type": "game", "source": "ruler-game", "section": "Consensus on optimal play", "text": "Ruler game — Consensus on optimal play\n\n- **Compute the nim-value via the ruler function** — for a heap of size n, find the largest k such that 2^k divides n; the nim-value is k + 1 (so nim-values cycle as 1,2,1,3,1,2,1,4,…).\n- **Combine heaps by XOR** — with multiple heaps, XOR all their nim-values; the position is a second-player win (P-position) iff the XOR equals 0.\n- **Make the XOR zero on every move** — as in ordinary Nim, the winning strategy is to leave your opponent a position where the nim-sum (XOR) of all piles is 0.\n- **Powers of two are the strategic reference points** — heaps of size 2^k have nim-value k+1 (the highest value achievable for that size), making them the dominant heaps in any multi-pile position.\n- **The game is solved by a formula, not search** — computing the nim-value takes O(log n) time; no game tree search is needed."}, {"id": 1160, "type": "game", "source": "ruler-game", "section": "Engines & current best play", "text": "Ruler game — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed — the ruler nim-value formula gives an instant solution.\n- **Strength:** Perfect play by any implementation of the ruler nim-value formula.\n- **Where the proof / tablebase lives (if solved):** [Guy & Smith (1956)](../references.md#guy-smith1956); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** A teaching example in Sprague–Grundy theory; the closed-form nim-sequence is unusual in that it has an easily recognisable pattern."}, {"id": 1161, "type": "game", "source": "ruler-game", "section": "Complexity", "text": "Ruler game — Complexity\n\nTrivial: O(log n) to compute a heap's nim-value."}, {"id": 1162, "type": "game", "source": "ruler-game", "section": "References", "text": "Ruler game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Ruler_game)\n- [Guy & Smith (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1163, "type": "game", "source": "ruler-game", "section": "See also", "text": "Ruler game — See also\n\n- [Kayles](kayles.md) · [Dawson's chess](dawsons-chess.md) · [Nim](nim.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 1164, "type": "game", "source": "rush-hour", "section": "overview", "text": "Rush Hour\nSliding-car traffic puzzle — generalised version is PSPACE-complete.\nSolution status: Generalised n×n version PSPACE-complete (Flake & Baum, 2002). Game-theoretic value: Per-puzzle. Players: 1. Type: Solo sliding-block puzzle."}, {"id": 1165, "type": "game", "source": "rush-hour", "section": "Description", "text": "Rush Hour — Description\n\nRush Hour (Nob Yoshigahara, ~1996) is a 6×6 sliding-car puzzle: cars and\ntrucks occupying 2 or 3 cells can be slid along their long axis; the goal is\nto free the red car by sliding it to the right edge. The generalised version\nis **PSPACE-complete** (Flake & Baum, 2002)."}, {"id": 1166, "type": "game", "source": "rush-hour", "section": "Rules", "text": "Rush Hour — Rules\n\n1. Board: 6×6 grid with the right edge open at one row (the exit row).\n2. Pieces are cars (length 2) and trucks (length 3), each placed horizontally\n   or vertically.\n3. On a move the player slides one piece any number of cells along its long\n   axis, passing through empty cells only.\n4. The player wins when the **red car** (length 2, horizontal, on the exit\n   row) leaves the board through the right edge."}, {"id": 1167, "type": "game", "source": "rush-hour", "section": "Solution status", "text": "Rush Hour — Solution status\n\nGeneralised n×n Rush Hour is **PSPACE-complete** (Flake & Baum 2002 — the\n\"Generalised Rush Hour Logic\" reduction). The classical 6×6 puzzles are\nsolved trivially by BFS."}, {"id": 1168, "type": "game", "source": "rush-hour", "section": "Consensus on optimal play", "text": "Rush Hour — Consensus on optimal play\n\n- **BFS gives the optimal solution** — breadth-first search on the state graph (each node = board configuration, each edge = single-car slide) finds the minimum-move sequence; the 6×6 board has at most a few thousand reachable states per puzzle.\n- **Clear the exit row first** — the red car must exit right; identify which cars block the exit row and plan to slide them out of the way as the first priority.\n- **Work backward from the exit** — if car A blocks the red car, find what blocks A, and what blocks those blockers; the dependency tree reveals the order of necessary moves.\n- **A single slide can move multiple cells** — cars can slide as many cells as open space allows in one move; prefer slides that create large openings over multiple small shuffles.\n- **Avoid unnecessarily locking your own exits** — sliding a car to clear one path can inadvertently block another; preview the full downstream effect before committing to a move."}, {"id": 1169, "type": "game", "source": "rush-hour", "section": "Engines & current best play", "text": "Rush Hour — Engines & current best play\n\n- **Strongest known program(s):** BFS or A* solvers for 6×6 instances (e.g. open-source implementations in most intro-AI courses).\n- **Strength:** Optimal (minimum moves) for all standard 6×6 puzzles; solves any instance in milliseconds.\n- **Where the proof / tablebase lives (if solved):** Complexity result: [Flake & Baum (2002)](../references.md#demaine-rushhour2002); [Wikipedia](https://en.wikipedia.org/wiki/Rush_Hour_(puzzle))\n- **Notes:** The commercial 6×6 puzzles are easy for computers; PSPACE-completeness applies only to the generalised n×n problem."}, {"id": 1170, "type": "game", "source": "rush-hour", "section": "Complexity", "text": "Rush Hour — Complexity\n\nPSPACE-complete in general."}, {"id": 1171, "type": "game", "source": "rush-hour", "section": "References", "text": "Rush Hour — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Rush_Hour_(puzzle))\n- [Flake & Baum (2002). *Rush Hour is PSPACE-complete*.](../references.md#demaine-rushhour2002)"}, {"id": 1172, "type": "game", "source": "rush-hour", "section": "See also", "text": "Rush Hour — See also\n\n- [Klotski](klotski.md) · [Sokoban](sokoban.md) · [Tower of Hanoi](tower-of-hanoi.md)\n- Lexicon: [PSPACE](../lexicon/README.md#pspace)"}, {"id": 1173, "type": "game", "source": "russian-draughts", "section": "overview", "text": "Russian draughts\n8×8 draughts variant with flying kings and backward man-captures — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan draughts."}, {"id": 1174, "type": "game", "source": "russian-draughts", "section": "Description", "text": "Russian draughts — Description\n\nRussian draughts is the dominant 8×8 draughts variant in the former Soviet\nsphere. Unlike English draughts it allows men to capture backward, kings move\nany distance along a diagonal (\"flying king\"), and a man that reaches the back\nrank during a capture sequence is immediately promoted."}, {"id": 1175, "type": "game", "source": "russian-draughts", "section": "Rules", "text": "Russian draughts — Rules\n\n1. Board: 8×8 standard draughts board. Each player has 12 men on dark squares.\n2. Men move one square diagonally forward to an empty square.\n3. Men **capture** by jumping diagonally forward **or backward** over an\n   adjacent enemy piece onto an empty square; captures are mandatory and may\n   chain.\n4. A man reaching the far rank becomes a **king**. If the man reaches the far\n   rank during a chain capture and can continue capturing as a king, it does\n   so immediately as a king.\n5. Kings move and capture any number of squares along an unblocked diagonal.\n6. A player who cannot move loses."}, {"id": 1176, "type": "game", "source": "russian-draughts", "section": "Solution status", "text": "Russian draughts — Solution status\n\nRussian draughts is **not solved**. Engines are very strong (Tundra, Kestog,\netc.); endgame tablebases up to some piece counts exist but no full proof of\nthe value has been published."}, {"id": 1177, "type": "game", "source": "russian-draughts", "section": "Consensus on optimal play", "text": "Russian draughts — Consensus on optimal play\n\n- **Flying kings dominate endgames** — a king can traverse the entire board in one move; securing a king promotion radically changes the position and is the primary strategic goal once material is reduced.\n- **Backward captures extend man mobility** — unlike English draughts, men can capture backward; use this to set up multi-jump combinations that English-draughts players would overlook.\n- **Mid-capture promotion is critical** — if a man reaches the back rank during a forced capture chain and can continue as a king, it does so immediately; calculate capture sequences carefully to see whether mid-chain crowning is available.\n- **Centralise to control diagonals** — pieces in the centre of the board control more capture options and cannot be easily cornered; pieces on the edge have fewer diagonals available.\n- **Build breakthrough structures** — getting two or three men in a diagonal cluster toward the promotion rank forces captures that thin the opponent's defence, a key attacking motif in Shashki grandmaster games."}, {"id": 1178, "type": "game", "source": "russian-draughts", "section": "Engines & current best play", "text": "Russian draughts — Engines & current best play\n\n- **Strongest known program(s):** Kestog and Tundra — the strongest Russian draughts-specific engines, widely used in the competitive community.\n- **Strength:** Super-human; top engines consistently defeat grandmasters.\n- **Where the proof / tablebase lives (if solved):** Not solved; endgame tablebases cover positions up to a few pieces but no full solution exists.\n- **Notes:** The game remains unsolved; engines are strong but their strength comes from deep search and evaluation rather than a solved database."}, {"id": 1179, "type": "game", "source": "russian-draughts", "section": "Complexity", "text": "Russian draughts — Complexity\n\nSimilar to English draughts."}, {"id": 1180, "type": "game", "source": "russian-draughts", "section": "References", "text": "Russian draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Russian_draughts) ([archive](http://web.archive.org/web/20260315193309/https://en.wikipedia.org/wiki/Russian_draughts))\n- [Schaeffer et al. (2007). *Checkers is Solved*.](../references.md#schaeffer2007) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1181, "type": "game", "source": "russian-draughts", "section": "See also", "text": "Russian draughts — See also\n\n- [English draughts](checkers.md) · [Italian draughts](italian-draughts.md) · [Brazilian draughts](brazilian-draughts.md) · [International draughts](international-draughts.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase)"}, {"id": 1182, "type": "game", "source": "samegame", "section": "overview", "text": "SameGame\nA single-player tile-clearing puzzle — its optimisation version is NP-hard\nSolution status: Unsolved (NP-hard; optimal play per instance). Game-theoretic value: N/A (puzzle); high-score targets known only empirically. Players: 1 (puzzle). Type: Single-player deterministic puzzle."}, {"id": 1183, "type": "game", "source": "samegame", "section": "Description", "text": "SameGame — Description\n\nA rectangular grid filled with coloured blocks (commonly 15×15 with up to 5\ncolours). Each move clears a connected group of same-coloured blocks; the\ncolumn collapses (and rows compact, in some variants); the player maximises\nscore, which is non-linear in group size. The game is famously a benchmark for\nMonte-Carlo tree search and nested rollout policies — record scores are held by\nsearch programs."}, {"id": 1184, "type": "game", "source": "samegame", "section": "Rules", "text": "SameGame — Rules\n\n1. Start with an m × n grid filled with coloured blocks.\n2. Select any **connected group of ≥ 2 same-coloured** blocks; remove the whole\n   group. Score (group_size − 2)² (or another fixed convex function).\n3. After a removal, blocks above the gap fall straight down; empty columns slide\n   leftward.\n4. Continue until no group of size ≥ 2 remains.\n5. Bonus: a board cleared completely scores a fixed bonus (commonly 1000)."}, {"id": 1185, "type": "game", "source": "samegame", "section": "Solution status", "text": "SameGame — Solution status\n\nSameGame is **NP-hard** as a decision/optimisation problem (Biedl, Demaine,\nDemaine, Fleischer, Jacobsen & Munro, 2002). Per-instance optimal solutions are\nnot known in general; record scores on the standard \"20-board\" benchmark are\nestablished only empirically, by Monte-Carlo / nested rollout searches that\nhave advanced the state of the art repeatedly over the past two decades."}, {"id": 1186, "type": "game", "source": "samegame", "section": "Consensus on optimal play", "text": "SameGame — Consensus on optimal play\n\n- **Prefer convex scoring — remove large groups** — the (size−2)² scoring function means removing a group of 10 scores 64 while two groups of 5 score 18 each (36 total); always prefer merging groups before removing them.\n- **Target the complete clear** — a board cleared entirely gives a large bonus (commonly 1000 points); when a complete clear is possible, it almost always outscores any partial removal sequence.\n- **Nested rollout search (NRPE) outperforms greedy** — random playouts, especially in nested or nested Monte-Carlo form, consistently find much higher scores than greedy-largest-first strategies; use MCTS or NRPE for benchmarks.\n- **Avoid isolating small groups of 1** — a single block of colour that becomes isolated (no same-colour neighbours) can never be removed; every move should check whether it strands any colour.\n- **Plan column structure** — after each collapse the column distribution changes; think ahead about how removing a left-side group changes the relative positions of right-side groups that you plan to merge next."}, {"id": 1187, "type": "game", "source": "samegame", "section": "Engines & current best play", "text": "SameGame — Engines & current best play\n\n- **Strongest known program(s):** Nested Rollout Policy Adaptation (NRPE) programs and Monte-Carlo tree search implementations.\n- **Strength:** Vastly stronger than human play on the standard benchmark boards; record scores are updated periodically by new search techniques.\n- **Where the proof / tablebase lives (if solved):** NP-hardness: Biedl et al. (2002); empirical record tracking at community benchmark sites.\n- **Notes:** No provably optimal solution for any standard random board is known; all records are lower bounds from heuristic search."}, {"id": 1188, "type": "game", "source": "samegame", "section": "Complexity", "text": "SameGame — Complexity\n\nExponential — both the state space (collapsing-grid configurations) and the\noptimisation are hard. The fixed-instance problem is the natural benchmark."}, {"id": 1189, "type": "game", "source": "samegame", "section": "References", "text": "SameGame — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/SameGame) ([archive](http://web.archive.org/web/20260508102957/https://en.wikipedia.org/wiki/SameGame))\n- [Hearn & Demaine (2009). *Games, Puzzles, and Computation*.](../references.md#hearn-demaine2009) (general framework)"}, {"id": 1190, "type": "game", "source": "samegame", "section": "See also", "text": "SameGame — See also\n\n- [Sokoban](sokoban.md) · [Tetris](nonograms.md) (related complexity)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete)"}, {"id": 1191, "type": "game", "source": "score-four", "section": "overview", "text": "Score Four\nConnect Four with gravity in three dimensions — four-in-a-row on a 4×4×4 grid\nSolution status: Weakly solved **[verify]**. Game-theoretic value: First-player win **[verify]**. Players: 2. Type: Partisan positional (k-in-a-row, with gravity) game."}, {"id": 1192, "type": "game", "source": "score-four", "section": "Description", "text": "Score Four — Description\n\nPlayed with a 4×4 array of vertical pegs. Players alternately slide a bead of\ntheir colour onto a peg; like [Connect Four](connect-four.md), gravity forces\neach bead to the lowest free position on its peg. The winner is the first to\nmake four of their beads in a straight line anywhere in the 4×4×4 cube — along\nrows, columns, pegs, or any 2-D or 3-D diagonal."}, {"id": 1193, "type": "game", "source": "score-four", "section": "Solution status", "text": "Score Four — Solution status\n\nScore Four occupies the same cube as [Qubic](qubic.md) but adds gravity, which\n*reduces* the move choices (you pick a peg, not an arbitrary cell). With Qubic\nitself weakly solved as a first-player win ([Patashnik, 1980](../references.md#patashnik1980)),\nand Score Four being a constrained version on the same 76-line cube, Score Four\nis **reported to be weakly solved as a first-player win** and is well within the\nreach of modern exhaustive solvers.\n\n> **[verify]** — This archive has not pinned a single canonical primary source\n> giving Score Four's exact game-theoretic value. The first-player-win claim is\n> consistent with the literature and the game's modest size, but a specific\n> solving citation should be added."}, {"id": 1194, "type": "game", "source": "score-four", "section": "Consensus on optimal play", "text": "Score Four — Consensus on optimal play\n\n- **Centre pegs are the most powerful** — the four central pegs of the 4×4 array participate in the most winning lines (rows, columns, and 3-D diagonals); fill them early to maximise winning-line coverage.\n- **Vertical columns are dangerous to gift** — if you allow the opponent to stack many beads on a single peg uncontested, they gain a complete peg-column line easily; contest central pegs immediately.\n- **3-D diagonals are hard to see but decisive** — the four space diagonals of the cube (corner to opposite corner) are easy to miss; check all 76 winning lines after every move, not just the obvious 2-D rows.\n- **First player should impose early threats** — with correct play the first player wins; exploit first-move advantage by building a multi-direction threat cluster on the first 4–5 moves.\n- **Gravity limits flexibility** — unlike Qubic you cannot place freely; if a needed cell is not at the bottom of its peg you must wait or fill lower cells first, which the opponent can anticipate."}, {"id": 1195, "type": "game", "source": "score-four", "section": "Engines & current best play", "text": "Score Four — Engines & current best play\n\n- **Strongest known program(s):** No widely distributed public engine known to the cataloguer; solvers based on the Qubic-style search (Patashnik 1980 / Allis 1994 approach) are applicable.\n- **Strength:** Likely solvable to perfection with a modern exhaustive search; no benchmarked program is publicly available.\n- **Where the proof / tablebase lives (if solved):** No canonical primary source confirmed; see [Patashnik (1980)](../references.md#patashnik1980) for the gravity-free sibling Qubic.\n- **Notes:** First-player win is widely reported but this archive has not verified a primary citation; see [Qubic](qubic.md) for the related confirmed solution."}, {"id": 1196, "type": "game", "source": "score-four", "section": "Complexity", "text": "Score Four — Complexity\n\n64-cell cube; the gravity constraint keeps the reachable state space and game\ntree well within modern exhaustive search."}, {"id": 1197, "type": "game", "source": "score-four", "section": "References", "text": "Score Four — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Score_Four)\n- [Patashnik, O. (1980). *Qubic: 4×4×4 Tic-Tac-Toe*.](../references.md#patashnik1980) (the no-gravity sibling)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1198, "type": "game", "source": "score-four", "section": "See also", "text": "Score Four — See also\n\n- [Connect Four](connect-four.md) · [Qubic](qubic.md) · [Tic-tac-toe](tic-tac-toe.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1199, "type": "game", "source": "seega", "section": "overview", "text": "Seega\nEgyptian custodian-capture game on a 5×5 board — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan capture game."}, {"id": 1200, "type": "game", "source": "seega", "section": "Description", "text": "Seega — Description\n\nSeega is a traditional Egyptian board game played on a 5×5 grid (other sizes\nexist). Players first place their stones in a drop phase, then move and\ncapture by **custodianship** — sandwiching an opposing stone between two of\ntheir own."}, {"id": 1201, "type": "game", "source": "seega", "section": "Rules", "text": "Seega — Rules\n\n1. Board: 5×5 (other sizes 7×7 or 9×9 exist). Each player has 12 (or\n   appropriate) stones.\n2. **Placement phase**: starting player drops two stones on any empty cells\n   (except the centre), opponent does likewise, alternating until all stones\n   are placed. The centre remains empty after placement.\n3. **Movement phase**: players alternate; each turn a player moves one of\n   their stones one cell orthogonally to an empty cell.\n4. **Capture**: whenever a player's move sandwiches an opposing stone\n   between two of their own along an orthogonal line, the sandwiched stone is\n   captured and removed. Multiple captures can occur on the same move.\n5. A player who cannot move loses; the player who captures all opposing\n   stones wins."}, {"id": 1202, "type": "game", "source": "seega", "section": "Solution status", "text": "Seega — Solution status\n\nSeega is **not solved**. The 5×5 case is small enough to be tractable to\nmodern compute but no published proof exists."}, {"id": 1203, "type": "game", "source": "seega", "section": "Consensus on optimal play", "text": "Seega — Consensus on optimal play\n\n- **Placement phase: pair your stones for mutual flanking** — placing two stones that share a flank with each other sets up immediate custodial captures in the movement phase; isolated stones are vulnerable to being sandwiched themselves.\n- **Keep the centre empty at the start — then contest it** — the centre cell is left empty after placement; moving into the centre early in the movement phase gives orthogonal reach in all four directions, maximising custodial threat potential.\n- **Avoid lone stones on the edge** — a single stone on the edge has only three orthogonal neighbours; being sandwiched on an edge requires the opponent to control only two of those, which is easier to arrange.\n- **Advance in pairs** — two stones moving together along adjacent parallel rows create a rolling custodial threat; the opponent must address both simultaneously or lose a piece to the flanking motion.\n- **The player who runs out of pieces first loses** — do not enter capture trades where you lose more pieces than the opponent; maintain numerical parity until you can set up a finishing trap."}, {"id": 1204, "type": "game", "source": "seega", "section": "Engines & current best play", "text": "Seega — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** The 5×5 board is small enough for modern exhaustive analysis, but no published solution is known."}, {"id": 1205, "type": "game", "source": "seega", "section": "Complexity", "text": "Seega — Complexity\n\nSmall–moderate."}, {"id": 1206, "type": "game", "source": "seega", "section": "References", "text": "Seega — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Seega_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1207, "type": "game", "source": "seega", "section": "See also", "text": "Seega — See also\n\n- [Yote](yote.md) · [Dara](dara.md) · [Surakarta](surakarta.md) · [Picaria](picaria.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1208, "type": "game", "source": "shannon-switching-game", "section": "overview", "text": "Shannon switching game\nAn abstract graph game, completely solved by matroid theory — the winner is\nSolution status: Strongly solved. Game-theoretic value: Determined by the graph: Short wins, Cut wins, or it depends on who moves first. Players: 2 (asymmetric roles: \"Short\" and \"Cut\"). Type: Partisan connection game on a graph."}, {"id": 1209, "type": "game", "source": "shannon-switching-game", "section": "Description", "text": "Shannon switching game — Description\n\nPlayed on a graph with two distinguished vertices, A and B. One player, **Short**,\n\"secures\" edges; the other, **Cut**, deletes edges. Short wins by securing a\npath of edges connecting A and B; Cut wins by deleting enough edges that no such\npath can exist. (Bridg-it is the special case where the graph is a grid.)"}, {"id": 1210, "type": "game", "source": "shannon-switching-game", "section": "Solution status", "text": "Shannon switching game — Solution status\n\nThe Shannon switching game is **strongly solved** by\n[Lehman (1964)](../references.md#lehman1964), who gave a complete\ncharacterisation in terms of **matroids / spanning trees**:\n\n- **Short wins as second player** (i.e. whoever the graph favours, regardless of\n  who moves) if and only if the graph contains **two edge-disjoint trees** each\n  connecting A and B — equivalently, the relevant matroid has the right\n  structure.\n- **Cut wins as second player** if Short cannot even win going first.\n- Otherwise the first player wins.\n\nSo the outcome is decided by a graph-theoretic property computable in\npolynomial time, and an explicit optimal strategy follows from the two\nedge-disjoint trees. This is one of the cleanest \"completely solved by pure\nmathematics\" results in game theory."}, {"id": 1211, "type": "game", "source": "shannon-switching-game", "section": "Consensus on optimal play", "text": "Shannon switching game — Consensus on optimal play\n\n- **Check the two edge-disjoint spanning trees condition** — before playing, determine whether the graph contains two edge-disjoint trees (spanning trees) each connecting A to B; if yes, Short wins as second player.\n- **Short's strategy: maintain a spanning tree** — Short should always claim the edge that \"saves\" one of their two target spanning trees; whenever Cut deletes an edge from one tree, Short claims an edge that rebuilds the other.\n- **Cut's strategy: target the bridge** — Cut wins by finding a \"bridge\" edge (one whose deletion disconnects A from B) and deleting it; if no such bridge exists in the secured subgraph, Cut must try to prevent Short from completing a path.\n- **The outcome is determined before play** — since the winner is decided purely by graph structure (a polynomial-time check), the strategic value of the game is entirely in computing the matroid condition, not in tactical play.\n- **Bridg-it is the canonical instance** — the special case on a grid graph (Bridg-it) is the most studied; Short wins as second player by the pairing strategy on the symmetric grid."}, {"id": 1212, "type": "game", "source": "shannon-switching-game", "section": "Engines & current best play", "text": "Shannon switching game — Engines & current best play\n\n- **Strongest known program(s):** No competitive engine needed — optimal play is determined by a polynomial-time matroid computation.\n- **Strength:** Perfect play by any program that computes the two edge-disjoint spanning trees and executes the corresponding maintenance strategy.\n- **Where the proof / tablebase lives (if solved):** [Lehman (1964)](../references.md#lehman1964); [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001)\n- **Notes:** One of the cleanest \"solved by pure mathematics\" results in combinatorial game theory; no search is needed."}, {"id": 1213, "type": "game", "source": "shannon-switching-game", "section": "Complexity", "text": "Shannon switching game — Complexity\n\nThe deciding condition and an optimal strategy are computable in polynomial time\nin the size of the graph."}, {"id": 1214, "type": "game", "source": "shannon-switching-game", "section": "References", "text": "Shannon switching game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shannon_switching_game) ([archive](http://web.archive.org/web/20260107194010/https://en.wikipedia.org/wiki/Shannon_switching_game))\n- [Lehman, A. (1964). *A solution of the Shannon switching game*.](../references.md#lehman1964)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1215, "type": "game", "source": "shannon-switching-game", "section": "See also", "text": "Shannon switching game — See also\n\n- [Bridg-it](bridg-it.md) · [Hex](hex.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [pairing strategy](../lexicon/README.md#pairing-strategy)"}, {"id": 1216, "type": "game", "source": "shatranj", "section": "overview", "text": "Shatranj\nThe medieval ancestor of modern chess — slower pieces, but still unsolved.\nSolution status: Unsolved (small-piece endgames computable). Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 1217, "type": "game", "source": "shatranj", "section": "Description", "text": "Shatranj — Description\n\nPlayed on an 8×8 board. Shatranj is the form of chess that spread through the\nmedieval Islamic world. Its key differences from modern chess: the Ferz (Queen)\nmoves only one square diagonally; the Alfil (Bishop) jumps exactly two squares\ndiagonally; there is no initial two-square pawn move, no castling, and pawns\npromote only to Ferz. Bare-king and stalemate rules differ from modern chess."}, {"id": 1218, "type": "game", "source": "shatranj", "section": "Solution status", "text": "Shatranj — Solution status\n\nShatranj is **unsolved**. Although its pieces are far weaker than modern\nchess's — making mating material scarce and games long and manoeuvring — the\npositional state-space and game-tree complexity are still on the order of\nchess's, well beyond exhaustive search. Small-material endgames are computable\nby [retrograde analysis](../lexicon/README.md#retrograde-analysis) (and medieval\nmasters in fact catalogued many such *mansubat*), but the full game has no\nproven value, and it has attracted little modern solving effort."}, {"id": 1219, "type": "game", "source": "shatranj", "section": "Consensus on optimal play", "text": "Shatranj — Consensus on optimal play\n\n- **Pawn structure determines the endgame** — with weak pieces, pawn majorities that can promote to Ferz are the primary winning mechanism; protect passed pawns aggressively.\n- **The Alfil jump creates permanent colour blindness** — the Alfil only reaches half the board (same colour squares only, skipping ranks); it cannot protect its own pawns on the opposite colour and cannot defend a bare king against a Ferz on the wrong colour.\n- **Bare king rule changes material evaluation** — exposing the opponent's king (removing all their pieces) wins even without checkmate; this makes large material exchanges more dangerous than in modern chess.\n- **Stalemate convention varies** — in many historical Shatranj rules stalemate is a win for the side achieving it; if using this rule, be more aggressive about confining the opponent's king.\n- **The Ferz's limited range makes king safety less urgent** — a single-step queen cannot deliver quick mating threats; plan for long endgame manoeuvres rather than sharp tactical attacks."}, {"id": 1220, "type": "game", "source": "shatranj", "section": "Engines & current best play", "text": "Shatranj — Engines & current best play\n\n- **Strongest known program(s):** Fairy-Stockfish — a fairy/variant chess engine that supports Shatranj piece rules ([https://github.com/ianfab/Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish))).\n- **Strength:** Strong; likely super-human but not independently benchmarked against top human Shatranj players.\n- **Where the proof / tablebase lives (if solved):** Not solved; small-piece endgame tablebases are theoretically computable.\n- **Notes:** Modern Shatranj is played mainly by historical-games enthusiasts; Fairy-Stockfish provides the strongest available engine support."}, {"id": 1221, "type": "game", "source": "shatranj", "section": "Complexity", "text": "Shatranj — Complexity\n\nComparable to chess: state-space on the order of 10^40."}, {"id": 1222, "type": "game", "source": "shatranj", "section": "References", "text": "Shatranj — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shatranj)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1223, "type": "game", "source": "shatranj", "section": "See also", "text": "Shatranj — See also\n\n- [Chess](chess.md) · [Makruk](makruk.md) · [Xiangqi](xiangqi.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 1224, "type": "game", "source": "shisima", "section": "overview", "text": "Shisima\nThe Kenyan three-in-a-row game on an octagonal board — strongly solved as a\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan placement+movement game."}, {"id": 1225, "type": "game", "source": "shisima", "section": "Description", "text": "Shisima — Description\n\nA traditional Tiriki (Kenyan) game played on an octagonal board: a central\npoint and eight points around it, with lines from the centre to each outer\npoint and between adjacent outer points. \"Shisima\" means \"body of water\" — the\ncentre is \"the pond.\""}, {"id": 1226, "type": "game", "source": "shisima", "section": "Rules", "text": "Shisima — Rules\n\n1. Board: 9 points (1 centre + 8 around it) connected by lines from the centre\n   and around the ring.\n2. Each player has 3 stones; the **placement phase** is skipped — both players\n   start with three stones already on opposite outer \"starting\" positions, and\n   the centre is empty.\n3. Players alternate sliding one of their stones along a line to an adjacent\n   empty point.\n4. A player wins by making **three in a row through the centre**."}, {"id": 1227, "type": "game", "source": "shisima", "section": "Solution status", "text": "Shisima — Solution status\n\nStrongly solved by trivial enumeration; **draw** with correct play. The very\nsmall state space and the requirement that any winning line pass through the\ncentre give an easy pairing strategy for whichever player is on the defensive."}, {"id": 1228, "type": "game", "source": "shisima", "section": "Consensus on optimal play", "text": "Shisima — Consensus on optimal play\n\n- **All winning lines pass through the centre** — only a three-in-a-row that includes the centre point wins; this means controlling or denying the centre is the single most important positional concern.\n- **Block the centre when the opponent threatens it** — if the opponent has two stones aligned to use the centre for a winning line, move into the centre immediately to block.\n- **Keep your stones within one move of the centre** — stones placed on outer ring points adjacent to the centre can enter the centre on the next move; staying \"one step away\" keeps winning threats alive.\n- **The pairing strategy guarantees the draw** — the defender can always mirror the attacker's approach using a pairing of ring positions; understanding this means you need never lose against a correct defender.\n- **Draw is the correct result** — neither player can force a win with correct play; competitive play aims to induce the opponent's error."}, {"id": 1229, "type": "game", "source": "shisima", "section": "Engines & current best play", "text": "Shisima — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer; the full state graph is trivially enumerable.\n- **Strength:** Perfect play by any complete minimax over the tiny state graph.\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Shisima)\n- **Notes:** A traditional Kenyan game; the draw with correct play is confirmed by exhaustive enumeration."}, {"id": 1230, "type": "game", "source": "shisima", "section": "Complexity", "text": "Shisima — Complexity\n\nTiny."}, {"id": 1231, "type": "game", "source": "shisima", "section": "References", "text": "Shisima — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shisima) ([archive](http://web.archive.org/web/20260203141438/https://en.wikipedia.org/wiki/Shisima))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1232, "type": "game", "source": "shisima", "section": "See also", "text": "Shisima — See also\n\n- [Picaria](picaria.md) · [Tapatan](tapatan.md) · [Three Men's Morris](three-mens-morris.md) · [Mū tōrere](mu-torere.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [pairing strategy](../lexicon/README.md#pairing-strategy)"}, {"id": 1233, "type": "game", "source": "shogi", "section": "overview", "text": "Shogi\nJapanese chess — captured pieces change sides and re-enter play, making it\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 1234, "type": "game", "source": "shogi", "section": "Description", "text": "Shogi — Description\n\nPlayed on a 9×9 board, each side with 20 pieces. Shogi's defining rule is the\n**drop**: a captured piece is kept \"in hand\" and may later be placed back onto\nthe board as one's own. This recycling means material never leaves play, which\nkeeps the branching factor high throughout and rules out the endgame\nsimplification that chess enjoys."}, {"id": 1235, "type": "game", "source": "shogi", "section": "Solution status", "text": "Shogi — Solution status\n\nShogi is **unsolved**. Its state-space (~10^71) and especially its game-tree\ncomplexity (~10^226) exceed even chess's, and the drop rule means there is no\n\"few pieces left\" regime in which [retrograde\nanalysis](../lexicon/README.md#retrograde-analysis) tablebases can take hold the\nway they do in chess. Shogi engines surpassed top human professionals in the\n2010s, and [AlphaZero](../references.md#silver-alphazero2018) reached\nsuperhuman shogi from self-play — but, as always, that is\n[strong play, not solving](../lexicon/README.md#solving-vs-strong-play).\n\nThe closely related miniature [Dōbutsu shōgi](dobutsu-shogi.md), by contrast, *is*\nsolved — illustrating how drastically board size changes tractability."}, {"id": 1236, "type": "game", "source": "shogi", "section": "Consensus on optimal play", "text": "Shogi — Consensus on optimal play\n\n- **Drops change the whole tempo calculus** — a piece in hand can threaten a devastating drop on any legal square; calculating whether a drop-check or drop-fork is available after a trade is mandatory before initiating any exchange.\n- **The king must stay mobile** — with drop-attacks possible on any square, a king kept in the back corner behind gold generals and a bishop is the standard defensive formation; learn the \"castling\" structures (mino, yagura, anaguma).\n- **Attack castled kings with \"floating\" pieces** — lance, rook, and bishop promotions near the opponent's king create relentless drop threats; the offensive side typically sacrifices pieces to earn drop hands.\n- **Promoted pieces stay in the enemy territory** — a promoted rook (Dragon King) or promoted bishop (Dragon Horse) in or near the opponent's camp is extremely powerful; trading it back early is almost always a mistake.\n- **Material is almost never \"lost\"** — every captured piece is added to your hand; being behind in material only means your opponent has more drop options; evaluate hand pieces as future threats, not losses."}, {"id": 1237, "type": "game", "source": "shogi", "section": "Engines & current best play", "text": "Shogi — Engines & current best play\n\n- **Strongest known program(s):** YaneuraOu / Stockfish NNUE (shogi variant) and Gikou — deep neural-network evaluation + alpha-beta; super-human since ~2013.\n- **Strength:** Far super-human; engines defeat top 9-dan professionals with large winning rates.\n- **Where the proof / tablebase lives (if solved):** Not solved; no tablebase.\n- **Notes:** The drop rule eliminates endgame simplification, making tablebases impractical; all strength comes from deep search and learned evaluation."}, {"id": 1238, "type": "game", "source": "shogi", "section": "Complexity", "text": "Shogi — Complexity\n\nState-space ~10^71; game-tree ~10^226\n([van den Herik et al., 2002](../references.md#vandenherik2002)). Generalised\nshogi is EXPTIME-complete."}, {"id": 1239, "type": "game", "source": "shogi", "section": "References", "text": "Shogi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shogi) ([archive](http://web.archive.org/web/20260511040918/https://en.wikipedia.org/wiki/Shogi))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [Silver et al. (2018). *AlphaZero*.](../references.md#silver-alphazero2018)"}, {"id": 1240, "type": "game", "source": "shogi", "section": "See also", "text": "Shogi — See also\n\n- [Dōbutsu shōgi](dobutsu-shogi.md) · [Chess](chess.md) · [Xiangqi](xiangqi.md) · [Janggi](janggi.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1241, "type": "game", "source": "shove", "section": "overview", "text": "Shove\nThe \"destructive\" twin of Push — sliding a piece pushes the leading piece\nSolution status: Strongly solved as a theory. Game-theoretic value: Position-dependent (switches and integers). Players: 2. Type: Partisan combinatorial game."}, {"id": 1242, "type": "game", "source": "shove", "section": "Description", "text": "Shove — Description\n\nA close relative of [Push](push.md): pieces slide on a row of squares, but\n**the leading piece is shoved off** every time a contiguous run is pushed —\nmaking Shove faster and noisier in value."}, {"id": 1243, "type": "game", "source": "shove", "section": "Rules", "text": "Shove — Rules\n\n1. A row of squares with some squares occupied by blue or red checkers.\n2. **Left** (blue) moves: slide a blue piece **one square right**, pushing the\n   adjacent contiguous run of pieces ahead of it. Whatever lies at the right\n   end of that run is shoved off the board permanently.\n3. **Right** (red) moves: the mirror image.\n4. The player unable to move loses (normal play)."}, {"id": 1244, "type": "game", "source": "shove", "section": "Solution status", "text": "Shove — Solution status\n\nStrongly solved as a theory in [*Winning Ways*](../references.md#bcg2001): each\nShove position has an explicit CGT value, computed recursively. Sums of\npositions add by ordinary CGT arithmetic."}, {"id": 1245, "type": "game", "source": "shove", "section": "Consensus on optimal play", "text": "Shove — Consensus on optimal play\n\n- **Compute the CGT value recursively** — evaluate each position by considering what each player gains from their best move; the value is a surreal number or switch, computed bottom-up from terminal positions.\n- **Shove eliminates material permanently** — unlike Push (which merely moves pieces), every shove destroys the leading piece; this makes Shove positions hotter (more valuable to move in) on average.\n- **Hot games first in a sum** — in a sum of Shove positions, play in the component with the highest temperature; leaving a very hot game for the opponent to exploit is the most common error.\n- **Switches require careful timing** — a Shove position with value {a | b} where a ≠ b is a switch; take it when the temperature exceeds the rest of the sum.\n- **Lookup short rows, recurse for longer ones** — positions up to ~5 pieces have known CGT values; for longer rows, apply the recursive definition from *Winning Ways*."}, {"id": 1246, "type": "game", "source": "shove", "section": "Engines & current best play", "text": "Shove — Engines & current best play\n\n- **Strongest known program(s):** No competitive engine; CGT computation tools (e.g. Aaron Siegel's CGSuite) compute positions analytically.\n- **Strength:** Exact optimal play via CGT formula.\n- **Where the proof / tablebase lives (if solved):** [Berlekamp, Conway & Guy (2001)](../references.md#bcg2001); [Conway (1976)](../references.md#conway1976)\n- **Notes:** Shove is a theoretical CGT exercise, not a competitive game; CGSuite can compute values for any specific row."}, {"id": 1247, "type": "game", "source": "shove", "section": "Complexity", "text": "Shove — Complexity\n\nSmall per position."}, {"id": 1248, "type": "game", "source": "shove", "section": "References", "text": "Shove — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Shove_(game))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Conway (1976). *On Numbers and Games*.](../references.md#conway1976)"}, {"id": 1249, "type": "game", "source": "shove", "section": "See also", "text": "Shove — See also\n\n- [Push](push.md) · [Toads and Frogs](toads-and-frogs.md) · [Domineering](domineering.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [surreal number](../lexicon/README.md#surreal-number)"}, {"id": 1250, "type": "game", "source": "sim", "section": "overview", "text": "Sim\nA Ramsey-theory game: every full game must produce a triangle, and the player\nSolution status: Weakly solved. Game-theoretic value: Second-player win. Players: 2. Type: Achievement/avoidance game on a graph."}, {"id": 1251, "type": "game", "source": "sim", "section": "Description", "text": "Sim — Description\n\nSix dots are drawn (the vertices of a complete graph K₆). Players alternately\ncolour one of the 15 edges, each using their own colour. A player who is forced\nto complete a triangle **in their own colour** loses.\n\nBecause Ramsey's theorem gives R(3,3) = 6, *any* 2-colouring of all 15 edges of\nK₆ contains a monochromatic triangle — so Sim can never end in a draw. Someone\nmust lose."}, {"id": 1252, "type": "game", "source": "sim", "section": "Solution status", "text": "Sim — Solution status\n\nSim is **weakly solved**: [Mead, Rosa & Huang (1974)](../references.md#mead-sim1974)\nproved that with perfect play the **second player wins**. The game is small\nenough (15 edges, three states each) that the result has since been confirmed\nby exhaustive computer search many times.\n\nA complete, human-memorable winning strategy for the second player is not\nespecially simple, which is why Sim remains playable in practice despite being\nsolved — most casual players cannot execute the winning line."}, {"id": 1253, "type": "game", "source": "sim", "section": "Consensus on optimal play", "text": "Sim — Consensus on optimal play\n\n- **Second player wins with perfect play** — the result is proven; as first player your only hope is an opponent error in the 15-edge game.\n- **Avoid contributing two edges to the same triangle** — before each move, count how many triangles you have already \"contributed two sides to\"; colouring the third side of any such triangle is an immediate loss.\n- **Track your opponent's dangerous triangles** — monitor which triangles the opponent has two sides of; completing one of those for them (on your colour) is not dangerous, but it wastes the opponent's turn when they must also avoid their own completions.\n- **Force the opponent into a \"Zugzwang\"** — the second-player strategy works by maintaining a position where every edge the first player colours either completes a first-player triangle or creates a situation where the second player can respond to maintain safety.\n- **A draw is impossible** — R(3,3)=6 guarantees that all 15 edges must be coloured before the game ends and someone must have a monochromatic triangle; never try to \"play for a draw.\""}, {"id": 1254, "type": "game", "source": "sim", "section": "Engines & current best play", "text": "Sim — Engines & current best play\n\n- **Strongest known program(s):** Complete exhaustive-search solvers (the game has only 15 edges); any correct minimax over the small game tree plays perfectly.\n- **Strength:** Perfect play trivially achievable by any exhaustive search.\n- **Where the proof / tablebase lives (if solved):** [Mead, Rosa & Huang (1974)](../references.md#mead-sim1974); [Wikipedia](https://en.wikipedia.org/wiki/Sim_(game))\n- **Notes:** The game is solved but is still competitive among humans because the winning strategy is non-trivial to memorise."}, {"id": 1255, "type": "game", "source": "sim", "section": "Complexity", "text": "Sim — Complexity\n\nTiny by modern standards — the full game tree is exhaustively searchable."}, {"id": 1256, "type": "game", "source": "sim", "section": "References", "text": "Sim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sim_(game))\n- [Mead, E., Rosa, A. & Huang, C. (1974). *The game of Sim: A winning strategy for the second player*.](../references.md#mead-sim1974)"}, {"id": 1257, "type": "game", "source": "sim", "section": "See also", "text": "Sim — See also\n\n- [Hexapawn](hexapawn.md) (another small second-player win) · [Bridg-it](bridg-it.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1258, "type": "game", "source": "six-mens-morris", "section": "overview", "text": "Six Men's Morris\nA mid-sized morris game; like its bigger sibling, it comes out a draw with\nSolution status: Weakly solved. Game-theoretic value: Draw **[verify]**. Players: 2. Type: Partisan placement-and-movement (\"mill\") game."}, {"id": 1259, "type": "game", "source": "six-mens-morris", "section": "Description", "text": "Six Men's Morris — Description\n\nPlayed on a board of two concentric squares with connecting midlines (16\npoints, **no** diagonals). Each player has six pieces. As in all morris games\nthere is a **placement phase** then a **movement phase**; forming a *mill*\n(three in a line) lets a player remove an enemy piece. A player reduced to two\npieces, or unable to move, loses."}, {"id": 1260, "type": "game", "source": "six-mens-morris", "section": "Solution status", "text": "Six Men's Morris — Solution status\n\nSix Men's Morris is **weakly solved**. It is smaller than\n[Nine Men's Morris](nine-mens-morris.md), which was itself weakly solved by\n[Gasser (1996)](../references.md#gasser1996) using\n[retrograde analysis](../lexicon/README.md#retrograde-analysis); the same\ntechniques settle the six-piece board comfortably. The reported game-theoretic\nvalue is a **draw** with perfect play.\n\n> **[verify]** — The draw verdict is consistent with the literature on the\n> morris family, but a specific primary citation for Six Men's Morris (as\n> distinct from Nine Men's Morris) should be added."}, {"id": 1261, "type": "game", "source": "six-mens-morris", "section": "Consensus on optimal play", "text": "Six Men's Morris — Consensus on optimal play\n\n- **Placement shapes the outcome** — as in Nine Men's Morris, place all six pieces to threaten two different mills simultaneously; an opponent who can only block one opening will concede the other.\n- **Two-mill configurations are the key weapon** — a pattern where one piece slides back and forth between two mills generates a forced capture every turn; establishing this before the opponent can counter is the decisive strategic goal.\n- **The 16-point board without diagonals is more constrained** — with only two squares and midlines (no diagonal connections), mill configurations are limited; memorise the possible double-mill patterns on the smaller board.\n- **Force the opponent below three pieces to win** — a player reduced to two pieces loses; plan piece-removal to approach this threshold rather than removing randomly.\n- **Draw with correct play** — the game is a draw under mutual perfect play; look for opponent errors in the placement phase, as errors there are the most common decisive mistakes."}, {"id": 1262, "type": "game", "source": "six-mens-morris", "section": "Engines & current best play", "text": "Six Men's Morris — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer; Gasser-style retrograde analysis methods apply directly and are sufficient to build a complete database.\n- **Strength:** Not benchmarked publicly; a complete retrograde database would give perfect play.\n- **Where the proof / tablebase lives (if solved):** [Gasser (1996)](../references.md#gasser1996) (Nine Men's Morris methodology); no specific Six Men's Morris primary citation confirmed.\n- **Notes:** The draw verdict is widely reported but a specific Six Men's Morris primary source should be verified."}, {"id": 1263, "type": "game", "source": "six-mens-morris", "section": "Complexity", "text": "Six Men's Morris — Complexity\n\nSmaller than Nine Men's Morris (which has on the order of 10^10 positions)."}, {"id": 1264, "type": "game", "source": "six-mens-morris", "section": "References", "text": "Six Men's Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Six_men%27s_morris)\n- [Gasser, R. (1996). *Solving Nine Men's Morris*.](../references.md#gasser1996)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1265, "type": "game", "source": "six-mens-morris", "section": "See also", "text": "Six Men's Morris — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Nine Men's Morris](nine-mens-morris.md) · [Twelve Men's Morris](twelve-mens-morris.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 1266, "type": "game", "source": "skat", "section": "overview", "text": "Skat\nGerman three-player trick-taking game — declarer-side double-dummy values tabulated.\nSolution status: Double-dummy values for all deals enumerated (Kupferschmid & Helmert, 2006); the full information-set game is unsolved. Game-theoretic value: Per-deal double-dummy values known. Players: 3. Type: Trick-taking, imperfect information."}, {"id": 1267, "type": "game", "source": "skat", "section": "Description", "text": "Skat — Description\n\nSkat is the German national card game: three players, a 32-card deck, and a\ncomplex bidding-and-play system. Kupferschmid & Helmert (2006) computed the\n**double-dummy value of every possible Skat deal** — i.e., perfect-play\noutcomes assuming open hands — establishing a kind of perfect-information\nsolving of the play-out phase."}, {"id": 1268, "type": "game", "source": "skat", "section": "Rules", "text": "Skat — Rules\n\n1. Deck: 32 cards (7, 8, 9, 10, J, Q, K, A in four suits).\n2. Each player is dealt 10 cards; 2 cards form the **skat** (talon).\n3. **Bidding**: players bid for the right to be declarer; the high bidder\n   takes the skat (in standard contracts), discards two cards, and chooses a\n   game (suit, grand, null, etc.).\n4. **Play**: 10 tricks of three cards each; standard trick-taking rules\n   (follow suit; trump card winners; jacks are trumps in suit and grand\n   games).\n5. The declarer plays against the partnership of the other two players.\n   Scoring depends on contract value, the *Spitzen* (jack sequence), points\n   captured, and bidding milestones."}, {"id": 1269, "type": "game", "source": "skat", "section": "Solution status", "text": "Skat — Solution status\n\nThe double-dummy decision problem for any Skat deal — assuming open hands —\nis fully tabulated (Kupferschmid & Helmert 2006). The full imperfect-info\ngame (bidding + uncertain card play) is **not solved**."}, {"id": 1270, "type": "game", "source": "skat", "section": "Consensus on optimal play", "text": "Skat — Consensus on optimal play\n\n- **Jacks are universal trumps** — the four jacks are the highest trumps in suit and grand contracts; building your hand around them provides the most powerful top-of-trump sequences (Spitzen) and increases contract value.\n- **Count Spitzen accurately before bidding** — the multiplier for Spitzen (the unbroken sequence from club jack down) directly affects game value; overbidding because of a Spitzen miscalculation is the most common bidding error.\n- **Null contracts need perfect \"anti-patterns\"** — in Null the declarer must take zero tricks; any card that forces a win is fatal; hold only \"dodging\" cards (low suits where you can safely pass to opponents).\n- **The partnership coordinates with signals** — the two defenders should signal suit length on opening leads; getting a ruff set up or establishing a long suit requires communication through card choice.\n- **Use double-dummy analysis for post-mortem** — after the game, check whether the declarer's play matched the DD-optimal line; systematic deviation from DD recommendations reveals technical leaks."}, {"id": 1271, "type": "game", "source": "skat", "section": "Engines & current best play", "text": "Skat — Engines & current best play\n\n- **Strongest known program(s):** Kupferschmid & Helmert Monte-Carlo simulation engine (2006); commercial Skat programs such as RoSKAt.\n- **Strength:** Strong amateur to near-expert in open-hand (double-dummy) play; significantly weaker than top humans in the full imperfect-information game.\n- **Where the proof / tablebase lives (if solved):** Kupferschmid & Helmert (2006) — double-dummy enumeration of all deals; [Wikipedia](https://en.wikipedia.org/wiki/Skat_(card_game))\n- **Notes:** The double-dummy result covers perfect-information play only; the full bidding + uncertain-hand game remains unsolved."}, {"id": 1272, "type": "game", "source": "skat", "section": "Complexity", "text": "Skat — Complexity\n\nLarge."}, {"id": 1273, "type": "game", "source": "skat", "section": "References", "text": "Skat — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Skat_(card_game))\n- [Kupferschmid & Helmert (2006). *A Skat Player Based on Monte-Carlo Simulation*.](../references.md#ueda-skat2009)"}, {"id": 1274, "type": "game", "source": "skat", "section": "See also", "text": "Skat — See also\n\n- [Bridge](bridge.md) · [Hanabi](hanabi.md)\n- Lexicon: [imperfect information](../lexicon/README.md#imperfect-information)"}, {"id": 1275, "type": "game", "source": "skewb", "section": "overview", "text": "Skewb\nCorner-rotation twist puzzle — fully solved: God's number is 11.\nSolution status: Strongly solved. Game-theoretic value: Any scramble solvable in ≤ 11 moves. Players: 1. Type: Solo permutation puzzle."}, {"id": 1276, "type": "game", "source": "skewb", "section": "Description", "text": "Skewb — Description\n\nThe Skewb (Tony Durham, 1982; popularised by Uwe Mèffert) is a corner-turning\ncube puzzle. The state graph contains 3,149,280 positions and **God's\nnumber** is 11 turns."}, {"id": 1277, "type": "game", "source": "skewb", "section": "Rules", "text": "Skewb — Rules\n\n1. Puzzle: cube whose 8 corners are connected to one of two interlocking\n   tetrahedra; the moveable axes are the 4 body diagonals.\n2. On a move the solver rotates a corner-axis by 120° or 240°; this twists\n   half the cube around that diagonal.\n3. The puzzle is solved when every face shows a single colour."}, {"id": 1278, "type": "game", "source": "skewb", "section": "Solution status", "text": "Skewb — Solution status\n\n**Strongly solved**: any scramble solvable in **≤ 11 moves** (face-turn\nmetric)."}, {"id": 1279, "type": "game", "source": "skewb", "section": "Consensus on optimal play", "text": "Skewb — Consensus on optimal play\n\n- **Sarah's Advanced Method** — the dominant speedsolving approach: orient the bottom face and centres in the first phase, then permute and orient the top layer corners; achieves average times well under 5 seconds.\n- **Corner-axis intuition is key** — unlike face-turning puzzles, each Skewb move rotates half the cube; building intuition for which corners are affected by each axis move is the first skill to develop.\n- **Top layer last** — solve the four corners of one face and all six centres by intuition, then use algorithms to place the remaining four corners; this two-phase approach is easier to learn than global strategies.\n- **Optimal solve ≤ 11 moves** — with only 3,149,280 states, the complete optimal lookup table fits in a few MB; any correct search of the full state space confirms 11 as God's number.\n- **Parity does not exist** — unlike the 3×3, the Skewb has no parity algorithms needed; any scramble is always solvable in a single consistent layer-by-layer approach."}, {"id": 1280, "type": "game", "source": "skewb", "section": "Engines & current best play", "text": "Skewb — Engines & current best play\n\n- **Strongest known program(s):** Complete lookup-table solvers (exhaustive BFS from the solved state).\n- **Strength:** Perfect (optimal solve in ≤ 11 moves guaranteed).\n- **Where the proof / tablebase lives (if solved):** [Wikipedia](https://en.wikipedia.org/wiki/Skewb); complete state space of 3,149,280 positions.\n- **Notes:** God's number of 11 is verified by complete enumeration; any implementation with the full BFS table solves optimally."}, {"id": 1281, "type": "game", "source": "skewb", "section": "Complexity", "text": "Skewb — Complexity\n\nVery small."}, {"id": 1282, "type": "game", "source": "skewb", "section": "References", "text": "Skewb — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Skewb)\n- [Rokicki *et al.* (2017). *The diameter of the Rubik's cube group is twenty*.](../references.md#rokicki2014) (related)"}, {"id": 1283, "type": "game", "source": "skewb", "section": "See also", "text": "Skewb — See also\n\n- [Rubik's Cube](rubiks-cube.md) · [Pocket Cube](pocket-cube.md) · [Pyraminx](pyraminx.md) · [Megaminx](megaminx.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [God's number](../lexicon/README.md#gods-number)"}, {"id": 1284, "type": "game", "source": "slither", "section": "overview", "text": "Slither\nA modern connection-with-sliding game — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 1285, "type": "game", "source": "slither", "section": "Description", "text": "Slither — Description\n\nSlither (Corey Clark, 2010s) is a connection game played on a square grid with\n**no diagonal-touching** restriction and an additional **sliding** move. The\nbalance between the placement and sliding rules gives a different feel from\n[Hex](hex.md) — and a much harder analysis problem."}, {"id": 1286, "type": "game", "source": "slither", "section": "Rules", "text": "Slither — Rules\n\n1. Square board (commonly 8×8 or larger). Each player owns two opposite sides.\n2. On each turn a player either:\n   - **Place** a new stone on an empty cell, **provided** the placement does\n     not result in any two same-colour stones being diagonally adjacent without\n     an orthogonal connector; **or**\n   - **Slide** one of their existing stones one orthogonal step, again subject\n     to the no-illegal-diagonal-pair rule.\n3. The first player to make an orthogonally-connected chain of their stones\n   spanning their two sides wins.\n4. Draws are not possible under the standard rule set."}, {"id": 1287, "type": "game", "source": "slither", "section": "Solution status", "text": "Slither — Solution status\n\nSlither is **unsolved**. The slider move blows up the game tree and the\nno-diagonal rule complicates evaluation; engine play exists but no formal\nsolution."}, {"id": 1288, "type": "game", "source": "slither", "section": "Consensus on optimal play", "text": "Slither — Consensus on optimal play\n\n- **Maintain orthogonal connectivity in your chain** — unlike diagonal-connection games, only orthogonal links count toward your spanning chain; always verify that newly placed or slid stones are part of your orthogonal main group.\n- **Use slides to extend without over-committing** — a slide moves an existing stone rather than adding a new one, preserving stone count while repositioning; use slides to bridge gaps without the cost of a permanent new placement.\n- **The no-diagonal rule prevents loose coupling** — two of your stones diagonally adjacent with no orthogonal connector violates placement rules; avoid creating such configurations as they restrict future placements in that area.\n- **Threaten two crossing paths** — as in all connection games, the key is to maintain two separate path threats to your goal sides simultaneously; this forces the opponent to block both or concede one.\n- **First player advantage is presumed but unproven** — draws are impossible (one player must complete a spanning chain); first-player advantage is widely observed but no formal proof exists."}, {"id": 1289, "type": "game", "source": "slither", "section": "Engines & current best play", "text": "Slither — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Notes:** A small competitive community exists; no published computational analysis or formal solution is known."}, {"id": 1290, "type": "game", "source": "slither", "section": "Complexity", "text": "Slither — Complexity\n\nLarge; the sliding move makes ply branching much wider than placement-only\nconnection games."}, {"id": 1291, "type": "game", "source": "slither", "section": "References", "text": "Slither — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Slither_(board_game))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework)"}, {"id": 1292, "type": "game", "source": "slither", "section": "See also", "text": "Slither — See also\n\n- [Hex](hex.md) · [Crossway](crossway.md) · [Havannah](havannah.md) · [TwixT](twixt.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1293, "type": "game", "source": "slitherlink", "section": "overview", "text": "Slitherlink\nLoop-drawing logic puzzle — NP-complete in general.\nSolution status: NP-complete in general. Game-theoretic value: Per-puzzle (unique solution by convention). Players: 1. Type: Solo logic puzzle."}, {"id": 1294, "type": "game", "source": "slitherlink", "section": "Description", "text": "Slitherlink — Description\n\nSlitherlink (Nikoli, 1989) is a logic puzzle in which the solver draws a\nsingle, simple closed loop on the edges of a rectangular dot lattice. Numeric\nclues in some cells indicate how many of the cell's four edges are part of\nthe loop."}, {"id": 1295, "type": "game", "source": "slitherlink", "section": "Rules", "text": "Slitherlink — Rules\n\n1. Board: rectangular grid of cells. Each cell may carry a clue 0, 1, 2, or 3.\n2. The solver draws a single closed loop along grid edges so that:\n   - The loop is **simple** (no branching, no self-crossing).\n   - Every clued cell has **exactly that many** of its four bordering edges\n     belonging to the loop.\n3. Edges not in the loop may be either drawn as crosses (deductions) or left\n   blank.\n4. A well-formed puzzle has a unique solution."}, {"id": 1296, "type": "game", "source": "slitherlink", "section": "Solution status", "text": "Slitherlink — Solution status\n\nThe general Slitherlink decision problem is **NP-complete** (Yato & Seta\n2003 and others). Standard Nikoli puzzles are tractable by hand or SAT\nsolvers in negligible time."}, {"id": 1297, "type": "game", "source": "slitherlink", "section": "Consensus on optimal play", "text": "Slitherlink — Consensus on optimal play\n\n- **Clue-zero cells are fully blocked** — mark all four edges of any 0-cell as \"no edge\" immediately; this cascades into adjacent cells.\n- **Clue-three cells are nearly complete** — three of four edges must be used; the single missing edge is heavily constrained by neighbours, so resolve these early.\n- **Corner and edge clues are more constrained** — a clue-2 cell in a grid corner has only two possible shapes; the forced pattern often propagates far.\n- **Parity rule at junctions** — every dot on the grid must have an even number (0 or 2) of loop segments meeting it; violations prune branches early.\n- **Avoid early loops** — adding a closing edge that would form a proper sub-loop before all clues are satisfied is immediately illegal; use this to block otherwise ambiguous branches.\n- **Region-based analysis** — consider which cells are inside vs. outside the loop (Jordan curve theorem); when a partial loop already divides the grid, propagate inside/outside labels to force remaining edges."}, {"id": 1298, "type": "game", "source": "slitherlink", "section": "Engines & current best play", "text": "Slitherlink — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Puzzles are routinely solved by SAT-based and constraint-propagation solvers (e.g., via [Ludii](https://ludii.games/) or custom Python scripts).\n- **Strength:** SAT solvers handle standard Nikoli puzzles in milliseconds.\n- **Where the proof / tablebase lives (if solved):** NP-completeness proof — Yato & Seta (2003).\n- **Notes:** As a single-player puzzle, \"engine strength\" means solver speed; human experts solve medium-difficulty puzzles in minutes using the heuristics above."}, {"id": 1299, "type": "game", "source": "slitherlink", "section": "Complexity", "text": "Slitherlink — Complexity\n\nNP-complete in general."}, {"id": 1300, "type": "game", "source": "slitherlink", "section": "References", "text": "Slitherlink — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Slitherlink)\n- [Slitherlink is NP-complete.](../references.md#slitherlink-np)"}, {"id": 1301, "type": "game", "source": "slitherlink", "section": "See also", "text": "Slitherlink — See also\n\n- [Sudoku](sudoku.md) · [Hashiwokakero](hashiwokakero.md) · [Nonograms](nonograms.md)\n- Lexicon: [NP-completeness](../lexicon/README.md#np-completeness)"}, {"id": 1302, "type": "game", "source": "snort", "section": "overview", "text": "Snort\nCol's companion game — same map-colouring setup, opposite adjacency rule, much\nSolution status: Partially solved. Game-theoretic value: Computable for a given position; no simple global characterisation. Players: 2. Type: Partisan combinatorial game."}, {"id": 1303, "type": "game", "source": "snort", "section": "Description", "text": "Snort — Description\n\nPlayed on a map like [Col](col.md), but with the *opposite* adjacency rule: one\nplayer colours regions blue, the other red, and **adjacent regions must not\nhave *different* colours** — neighbours may share a colour but cannot clash. A\nplayer unable to move loses."}, {"id": 1304, "type": "game", "source": "snort", "section": "Solution status", "text": "Snort — Solution status\n\nSnort is **partially solved**. It is fully amenable to combinatorial game\ntheory — any specific position can be evaluated by the disjunctive-sum\ncalculus, and values for many small maps and useful gadget positions are\ntabulated in [*Winning Ways*](../references.md#bcg2001). But unlike\n[Col](col.md), whose values are always \"number or number-plus-star,\" **Snort's\nvalues are not so constrained**: they include genuinely [hot](../lexicon/README.md#temperature--hot-game)\npositions and a richer zoo of CGT values, and there is no simple closed\ncharacterisation covering all maps.\n\nSo Snort is solved *in principle* by CGT but, in contrast to Col, has no tidy\nglobal theory — it is the harder twin."}, {"id": 1305, "type": "game", "source": "snort", "section": "Consensus on optimal play", "text": "Snort — Consensus on optimal play\n\n- **Temperature guides move priority** — Snort positions are often \"hot\" (high temperature), so always respond to your opponent's most valuable (hottest) available region; letting them take the hottest spot is a large loss.\n- **Claim large isolated regions early** — a region with no adjacency constraints is worth its size outright; grab these before they become contested.\n- **Force clashes on opponent's side** — if you can colour two mutually adjacent regions the same colour as your opponent's pieces, you deny them both squares.\n- **Evaluate by disjunctive sum** — when the map splits into independent components, compute each component's CGT value separately and sum; optimal play then targets the hottest remaining component.\n- **Star (∗) positions are dangerous** — a Snort position with value ∗ is a second-player win regardless of who goes next; recognise these gadget shapes and avoid gifting them to your opponent."}, {"id": 1306, "type": "game", "source": "snort", "section": "Engines & current best play", "text": "Snort — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Positions can be evaluated with CGT software such as Combinatorial Game Suite or custom implementations of the Sprague–Grundy / surreal-number calculus.\n- **Strength:** Not benchmarked against humans.\n- **Where the proof / tablebase lives (if solved):** Partial — tabulated values in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** Snort is the harder twin of Col; its richer CGT values make exhaustive characterisation an open research problem."}, {"id": 1307, "type": "game", "source": "snort", "section": "Complexity", "text": "Snort — Complexity\n\nDepends on the map; the lack of a value-restriction theorem makes it harder than\nCol."}, {"id": 1308, "type": "game", "source": "snort", "section": "References", "text": "Snort — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Snort_(game))\n- [Conway, J. H. (1976). *On Numbers and Games*.](../references.md#conway1976)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1309, "type": "game", "source": "snort", "section": "See also", "text": "Snort — See also\n\n- [Col](col.md) · [Hackenbush](hackenbush.md) · [Domineering](domineering.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [temperature / hot game](../lexicon/README.md#temperature--hot-game)"}, {"id": 1310, "type": "game", "source": "sokoban", "section": "overview", "text": "Sokoban\nSingle-player box-pushing puzzle — PSPACE-complete.\nSolution status: Per-instance solvability is PSPACE-complete. Game-theoretic value: Per-puzzle (varies). Players: 1. Type: Solo puzzle."}, {"id": 1311, "type": "game", "source": "sokoban", "section": "Description", "text": "Sokoban — Description\n\nSokoban (Hiroyuki Imabayashi, 1981) is a solo puzzle in which a warehouse\nkeeper pushes crates onto designated target cells. **Deciding whether a\nSokoban level is solvable is PSPACE-complete** (Culberson, 1997)."}, {"id": 1312, "type": "game", "source": "sokoban", "section": "Rules", "text": "Sokoban — Rules\n\n1. Board: rectangular grid containing walls, crates, target cells, and a\n   single \"keeper\" piece.\n2. On each step the keeper moves orthogonally one cell to an empty cell (no\n   wall, no crate).\n3. The keeper may **push** a single crate one cell in the direction of motion\n   if that cell is empty (no wall, no second crate).\n4. The keeper cannot pull crates.\n5. The puzzle is solved when every target cell holds a crate."}, {"id": 1313, "type": "game", "source": "sokoban", "section": "Solution status", "text": "Sokoban — Solution status\n\nSokoban's **general decision problem** — given a level, is it solvable? —\nhas been proven **PSPACE-complete** (Culberson 1997/1999). Individual levels\nare solved by hand or by automated solvers; large levels can stay unsolved\nfor years."}, {"id": 1314, "type": "game", "source": "sokoban", "section": "Consensus on optimal play", "text": "Sokoban — Consensus on optimal play\n\n- **Avoid deadlocks immediately** — a crate pushed into a corner (two walls meeting) is permanently frozen; scanning for corner-deadlock before every push prunes most failed branches.\n- **Identify goal packing order first** — work out which target cell each crate should occupy before moving anything; assigning wrong crates to goals wastes many moves.\n- **Clear the path to targets, not just targets** — a corridor crate that blocks access to a distant target must be moved early; experienced solvers plan the \"routing layer\" before the \"placement layer.\"\n- **Keep the keeper path short** — unnecessary keeper repositioning inflates move count; prefer pushing sequences where the keeper naturally arrives behind the next crate.\n- **Freeze analysis saves depth** — if pushing a crate creates a frozen group (two crates and a wall forming an unmovable block) that covers an unclaimed target, the state is a dead loss; cut it.\n- **Work backwards for hard levels** — pull analysis (imagine pulling crates away from targets) reveals which keeper positions are reachable and which paths are geometrically impossible."}, {"id": 1315, "type": "game", "source": "sokoban", "section": "Engines & current best play", "text": "Sokoban — Engines & current best play\n\n- **Strongest known program(s):** YASS (Yet Another Sokoban Solver) and Takaken's solver — heuristic iterative-deepening A* with pattern databases.\n- **Strength:** Super-human on most classic level sets; some hardest user-made levels remain unsolved by any automated solver.\n- **Where the proof / tablebase lives (if solved):** Per-puzzle; no universal tablebase. Culberson's PSPACE-completeness proof is at [../references.md#culberson-sokoban1999](../references.md#culberson-sokoban1999).\n- **Notes:** The Sokoban community maintains level databases and solution archives; the official \"original\" 90-level set was fully solved automatically, but community-designed hard sets still stump solvers."}, {"id": 1316, "type": "game", "source": "sokoban", "section": "Complexity", "text": "Sokoban — Complexity\n\nPer-instance can be exponential; family-level decision is PSPACE-complete."}, {"id": 1317, "type": "game", "source": "sokoban", "section": "References", "text": "Sokoban — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sokoban) ([archive](http://web.archive.org/web/20260511233436/https://en.wikipedia.org/wiki/Sokoban))\n- [Culberson (1999). *Sokoban is PSPACE-complete*.](../references.md#culberson-sokoban1999)"}, {"id": 1318, "type": "game", "source": "sokoban", "section": "See also", "text": "Sokoban — See also\n\n- [Klotski](klotski.md) · [Rush Hour](rush-hour.md) · [Tower of Hanoi](tower-of-hanoi.md)\n- Lexicon: [PSPACE](../lexicon/README.md#pspace)"}, {"id": 1319, "type": "game", "source": "songo", "section": "overview", "text": "Songo\nWest African mancala with mandatory captures — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan mancala."}, {"id": 1320, "type": "game", "source": "songo", "section": "Description", "text": "Songo — Description\n\nSongo is a West African mancala variant in the broader Awari/Oware family,\nplayed on a two-row board with six pits per side. Local variants differ on\nsowing direction, capture conditions, and end-game rules. **[verify]** the\nspecific regional ruleset documented here."}, {"id": 1321, "type": "game", "source": "songo", "section": "Rules", "text": "Songo — Rules\n\n1. Board: 6 pits per side; no large home stores (captured seeds are held off-\n   board by each player).\n2. Each pit begins with 4 seeds.\n3. On a turn the player picks up all seeds in one of their own pits and sows\n   them counterclockwise, one per pit, including across opponent's row.\n4. **Capture**: if the last seed sown lands in an opponent's pit and brings\n   the count there to 2 or 3, those seeds are captured; captures propagate\n   backward to previous opponent pits that also reach 2 or 3.\n5. A move that would leave the opponent with no seeds at all is illegal\n   unless no other move is available.\n6. The game ends when one side cannot move; the player with the most captured\n   seeds wins."}, {"id": 1322, "type": "game", "source": "songo", "section": "Solution status", "text": "Songo — Solution status\n\nSongo is **not solved**. Several variants are mathematically close to Awari\n(which has been solved by Romein & Bal as a draw), but Songo's specific\nruleset has no published value."}, {"id": 1323, "type": "game", "source": "songo", "section": "Consensus on optimal play", "text": "Songo — Consensus on optimal play\n\n- **Keep opponent's pits lean** — preventing opponents from holding large piles reduces their capture threats; spreading their seeds thin limits their options.\n- **Maintain seed flow on your side** — emptying too many of your own pits leaves you without legal moves; try to keep at least two or three pits occupied.\n- **Trigger backward-capture chains** — the propagating-capture rule rewards sowing the last seed into a pit that sets up a chain of 2-or-3 pits behind it on the opponent's side; plan the full chain before moving.\n- **Do not starve your opponent early** — moves that would leave the opponent with no seeds are illegal unless unavoidable; learn which configurations force that situation so you can steer toward it legitimately late in the game.\n- **Endgame seed count matters more than tempo** — once most pits are thinly seeded, securing a cumulative advantage in captured seeds outweighs positional refinements."}, {"id": 1324, "type": "game", "source": "songo", "section": "Engines & current best play", "text": "Songo — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Songo's specific ruleset has no published solution; the closely related Awari was solved (draw) by Romein & Bal (2003)."}, {"id": 1325, "type": "game", "source": "songo", "section": "Complexity", "text": "Songo — Complexity\n\nModerate."}, {"id": 1326, "type": "game", "source": "songo", "section": "References", "text": "Songo — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Mancala) ([archive](http://web.archive.org/web/20260509062903/https://en.wikipedia.org/wiki/Mancala))\n- [Romein & Bal (2003). *Awari is Solved*.](../references.md#romein-bal2003) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1327, "type": "game", "source": "songo", "section": "See also", "text": "Songo — See also\n\n- [Awari](awari.md) · [Oware](awari.md) · [Bao](bao.md) · [Kalah](kalah.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1328, "type": "game", "source": "sprouts", "section": "overview", "text": "Sprouts\nA pencil-and-paper game with a deceptively deep structure; solved by computer\nSolution status: Partially solved (small numbers of starting spots). Game-theoretic value: Known for many starting spot counts; conjectured pattern unproven in general. Players: 2. Type: Impartial combinatorial game (topological)."}, {"id": 1329, "type": "game", "source": "sprouts", "section": "Description", "text": "Sprouts — Description\n\nStart with *n* spots on paper. On a turn a player draws a line (which may curve)\njoining two spots or a spot to itself, not crossing any existing line, and then\nplaces a new spot on that line. Each spot may have at most three lines meeting\nit. The player unable to move loses ([normal play](../lexicon/README.md#normal-play-convention)).\nA game lasts at most 3n − 1 moves."}, {"id": 1330, "type": "game", "source": "sprouts", "section": "Solution status", "text": "Sprouts — Solution status\n\nSprouts is **partially solved**. [Applegate, Jacobson & Sleator (1991)](../references.md#applegate-sprouts1991)\nanalysed it by computer up to 11 starting spots; later distributed computations\n(the Sprouts project) have pushed verified results considerably further. The\nempirical pattern is the **\"Sprouts conjecture\"**: the first player wins exactly\nwhen *n* mod 6 is 3, 4, or 5. The conjecture matches every computed value but\n**has not been proven** for all *n*, so the standard game is not solved in\ngeneral.\n\nThe related game **[Brussels Sprouts](brussels-sprouts.md)** — superficially\nsimilar but using crosses instead of spots — is, by contrast, completely solved\nand is essentially a joke: its outcome is fixed in advance."}, {"id": 1331, "type": "game", "source": "sprouts", "section": "Consensus on optimal play", "text": "Sprouts — Consensus on optimal play\n\n- **Parity (the Sprouts conjecture) is the strategic compass** — if the starting count n mod 6 is 3, 4, or 5, the first player wins with correct play; otherwise the second player wins. Use this to decide whether to \"waste\" a move early on.\n- **Degree-2 spots are almost as flexible as free spots** — a spot with two lines can still absorb exactly one more connection; treat them as near-live resources and plan around when they become exhausted.\n- **Prevent large connected surviving regions** — isolated sub-regions of live spots each generate their own continuation; confining the game to fewer, smaller regions reduces your opponent's options.\n- **Closing loops traps spots** — drawing a closed curve around one or more spots renders them unreachable (no line may cross existing lines); deliberately trap your opponent's live spots to deny moves.\n- **Count the surviving moves** — at any point the maximum remaining moves is bounded by live spot count; track whether you or your opponent will exhaust moves first."}, {"id": 1332, "type": "game", "source": "sprouts", "section": "Engines & current best play", "text": "Sprouts — Engines & current best play\n\n- **Strongest known program(s):** The Sprouts Project (distributed computation) — exhaustive game-tree search using canonical topological representations.\n- **Strength:** Solves all positions up to ~40+ starting spots; humans cannot match it on larger starting counts.\n- **Where the proof / tablebase lives (if solved):** Partially; computed values available from the Sprouts Project archives. See also [../references.md#applegate-sprouts1991](../references.md#applegate-sprouts1991).\n- **Notes:** The Sprouts conjecture (first player wins iff n mod 6 ∈ {3,4,5}) is unproven in general; every computed case confirms it."}, {"id": 1333, "type": "game", "source": "sprouts", "section": "Complexity", "text": "Sprouts — Complexity\n\nThe topological move structure makes naive enumeration explode; specialised\ncanonical-form representations are needed even to reach the analysed values."}, {"id": 1334, "type": "game", "source": "sprouts", "section": "References", "text": "Sprouts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sprouts_(game)) ([archive](http://web.archive.org/web/20260511093257/https://en.wikipedia.org/wiki/Sprouts_(game)))\n- [Applegate, D., Jacobson, G. & Sleator, D. (1991). *Computer Analysis of Sprouts*.](../references.md#applegate-sprouts1991)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1335, "type": "game", "source": "sprouts", "section": "See also", "text": "Sprouts — See also\n\n- [Brussels Sprouts](brussels-sprouts.md) · [Chomp](chomp.md) · [Hackenbush](hackenbush.md)\n- Lexicon: [impartial game](../lexicon/README.md#impartial-game) · [normal play convention](../lexicon/README.md#normal-play-convention)"}, {"id": 1336, "type": "game", "source": "star", "section": "overview", "text": "*Star\nEa Ea's connection game — a Y-relative where peripheral cells score points,\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection / scoring game."}, {"id": 1337, "type": "game", "source": "star", "section": "Description", "text": "*Star — Description\n\n*Star (Craige Schensted / Ea Ea, 1980s) is a connection game on a hexagonal\nboard with peripheral \"edge cells.\" Where [Hex](hex.md) wins by connecting two\nopposite edges, *Star scores by **owning peripheral cells** through connecting\ngroups — making it a *scoring* connection game rather than an all-or-nothing\nrace."}, {"id": 1338, "type": "game", "source": "star", "section": "Rules", "text": "*Star — Rules\n\n1. A hexagonal board, with the edge cells specially marked (each peripheral cell\n   has a number indicating how many \"peries\" — board edges — it touches).\n2. Players alternate placing one stone of their colour on any empty cell.\n3. At the end of the game (the board fills, or both players pass), each player\n   scores: for every connected group of their stones, count the number of\n   *peripheral* cells touched minus 2 (clipped to 0). The higher score wins.\n4. Pass is allowed once both players agree the position is settled."}, {"id": 1339, "type": "game", "source": "star", "section": "Solution status", "text": "*Star — Solution status\n\n*Star is **not solved**. Like other large connection games, the combination of\na moderately large board and the *scoring* objective (not just a binary win\ncondition) puts a full solution out of reach. *Star is widely respected in the\nabstract-games community as a deeper successor to Y / Hex; engine play exists\nbut is much less developed than for the headline connection games."}, {"id": 1340, "type": "game", "source": "star", "section": "Consensus on optimal play", "text": "*Star — Consensus on optimal play\n\n- **Aim for groups touching three or more peripheral cells** — a connected group scores (peries − 2) points; groups touching exactly 1 or 2 peripheral cells score 0, so only groups spanning three or more peries have value.\n- **Connect across the board, not just along edges** — long diagonal chains that link multiple peripheral arcs accumulate more peries per stone invested than hugging a single edge.\n- **Contest high-perie corner cells early** — the true corner cells of the hexagonal board touch the most boundary edges; capturing them cheaply forms the nucleus of a high-scoring group.\n- **Cutting the opponent's bridge is often better than extending your own** — splitting an opponent's large group into two sub-groups, each below the scoring threshold, can swing multiple points at once.\n- **Sacrifice low-perie extensions** — small peripheral stubs that add only one perie to a group below the threshold are often not worth defending; redeploy those moves elsewhere."}, {"id": 1341, "type": "game", "source": "star", "section": "Engines & current best play", "text": "*Star — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** *Star has a small but dedicated competitive community; strategic theory is primarily found in informal publications and the abstract-games forums rather than academic literature."}, {"id": 1342, "type": "game", "source": "star", "section": "Complexity", "text": "*Star — Complexity\n\nComparable to Hex on equivalent board sizes, with the scoring objective adding\nextra evaluation depth."}, {"id": 1343, "type": "game", "source": "star", "section": "References", "text": "*Star — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Star_(board_game))\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (related connection-game theory)"}, {"id": 1344, "type": "game", "source": "star", "section": "See also", "text": "*Star — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Poly-Y](poly-y.md) · [Havannah](havannah.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing-argument)"}, {"id": 1345, "type": "game", "source": "subtract-a-square", "section": "overview", "text": "Subtract-a-square\nA one-heap subtraction game in which you may only remove a perfect-square\nSolution status: Strongly solved. Game-theoretic value: Depends on heap size; both win/loss values occur with positive density. Players: 2. Type: Impartial combinatorial game (subtraction game)."}, {"id": 1346, "type": "game", "source": "subtract-a-square", "section": "Description", "text": "Subtract-a-square — Description\n\nA single heap of *n* objects. On a turn a player removes a positive\n*perfect-square* number of objects (1, 4, 9, 16, …). Under\n[normal play](../lexicon/README.md#normal-play-convention) the player taking the\nlast object wins."}, {"id": 1347, "type": "game", "source": "subtract-a-square", "section": "Solution status", "text": "Subtract-a-square — Solution status\n\nSubtract-a-square is **strongly solved** in the practical sense: the\n[nim-value](../lexicon/README.md#nim-value) of every heap size is computable in\npolynomial time by the standard [mex](../lexicon/README.md#mex) recurrence, so\nthe win/loss status and an optimal move are known for any *n*.\n\nWhat remains *open* is a closed-form description. Unlike [Nim](nim.md) or\n[Wythoff's game](wythoffs-game.md), the losing positions (heap sizes with\nnim-value 0: 0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, 44, …) do **not** form\na known periodic or otherwise simply describable set. Their density is positive\nbut a formula is not known. So the game is fully solved algorithmically while\nits deeper structure is not understood."}, {"id": 1348, "type": "game", "source": "subtract-a-square", "section": "Consensus on optimal play", "text": "Subtract-a-square — Consensus on optimal play\n\n- **Build and consult the nim-value table** — compute nim-values for 0 through n by the mex recurrence; from any position with nim-value > 0, always move to a position with nim-value 0 (the losing positions for the player to move).\n- **Losing positions have no simple pattern** — unlike Nim or Wythoff's game, the P-positions (0, 2, 5, 7, 10, 12, 15, 17, 20, 22, 34, 39, …) cannot be predicted by a formula; memorising a table or computing on the fly is required.\n- **Removing 1 is rarely correct** — subtracting 1 (the smallest square) almost always leaves a winning position for the opponent; check the table before defaulting to the \"safe-looking\" small move.\n- **From large heaps, many squares are available** — with many choices near √n in magnitude, the winning move is typically within the range 1 to ⌊√n⌋; compute mex over this range to find it.\n- **Multi-heap variants require full Sprague–Grundy XOR** — if you play several simultaneous subtract-a-square piles, XOR all nim-values; a combined XOR of 0 is a losing position for the player to move."}, {"id": 1349, "type": "game", "source": "subtract-a-square", "section": "Engines & current best play", "text": "Subtract-a-square — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; any implementation of the mex recurrence solves the game exactly in O(n·√n) time.\n- **Strength:** Algorithmically solved for all practical heap sizes.\n- **Where the proof / tablebase lives (if solved):** Computed tables appear in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)) and are trivially reproducible.\n- **Notes:** The game is fully solved algorithmically; the open problem is whether the P-positions have a closed-form characterisation."}, {"id": 1350, "type": "game", "source": "subtract-a-square", "section": "Complexity", "text": "Subtract-a-square — Complexity\n\nFor a heap of size *n*, the nim-value table is built in O(n·√n) time."}, {"id": 1351, "type": "game", "source": "subtract-a-square", "section": "References", "text": "Subtract-a-square — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Subtract_a_square) ([archive](http://web.archive.org/web/20260405205210/https://en.wikipedia.org/wiki/Subtract_a_square))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)"}, {"id": 1352, "type": "game", "source": "subtract-a-square", "section": "See also", "text": "Subtract-a-square — See also\n\n- [Nim](nim.md) · [Wythoff's game](wythoffs-game.md) · [Fibonacci Nim](fibonacci-nim.md) · [Grundy's game](grundys-game.md)\n- Lexicon: [nim-value](../lexicon/README.md#nim-value) · [mex](../lexicon/README.md#mex)"}, {"id": 1353, "type": "game", "source": "sudoku", "section": "overview", "text": "Sudoku\nNumber-placement puzzle — generalised solving is NP-complete.\nSolution status: Per-instance polynomial for 9×9; n×n is NP-complete. Game-theoretic value: Per-puzzle (typically unique solution by convention). Players: 1. Type: Solo logic puzzle."}, {"id": 1354, "type": "game", "source": "sudoku", "section": "Description", "text": "Sudoku — Description\n\nSudoku is a number-placement puzzle on a 9×9 grid divided into nine 3×3\nsub-grids. The solver fills empty cells with digits 1–9 so each row, column,\nand sub-grid contains each digit exactly once. Yato & Seta (2003) proved\nthat the general n²×n² version is **NP-complete**."}, {"id": 1355, "type": "game", "source": "sudoku", "section": "Rules", "text": "Sudoku — Rules\n\n1. Board: 9×9 grid divided into nine 3×3 sub-grids. Some cells start with\n   given digits.\n2. The solver fills each empty cell with a digit 1–9.\n3. Constraints: each row, each column, and each 3×3 sub-grid must contain\n   every digit 1–9 exactly once.\n4. A well-formed Sudoku puzzle has a **unique** solution; the solver must\n   find it."}, {"id": 1356, "type": "game", "source": "sudoku", "section": "Solution status", "text": "Sudoku — Solution status\n\nFor the standard 9×9 board every puzzle can be solved by exact-cover search\n(e.g., Knuth's Algorithm X / dancing links) in tiny time. The n²×n²\ngeneralisation is **NP-complete** (Yato & Seta 2003)."}, {"id": 1357, "type": "game", "source": "sudoku", "section": "Consensus on optimal play", "text": "Sudoku — Consensus on optimal play\n\n- **Single-candidate (naked single) first** — if a cell has only one remaining possible digit, fill it immediately; these cascades often resolve large portions of the puzzle without guessing.\n- **Hidden singles reveal forced placements** — if a digit can go in only one cell within a row, column, or box, place it there even if that cell has multiple candidates; scan all three scopes for each digit.\n- **Naked and hidden pairs/triples prune candidates** — two cells in a unit that share exactly two candidates exclude those digits from all other cells in the unit; applying this before guessing usually avoids backtracking.\n- **X-Wing and swordfish eliminate distant candidates** — when a candidate digit appears in exactly two rows' same two columns (X-Wing), it can be removed from those columns' other rows; swordfish extends this to three rows/columns.\n- **Colouring (chaining) handles medium difficulty** — assign conjugate pairs of the same candidate alternating colours; if both same-colour instances appear in the same unit, that colour is false and its cells can be eliminated.\n- **Backtracking (guessing) is the universal fallback** — for hardest puzzles, pick the most constrained cell, guess a value, propagate constraints, and backtrack on contradiction; computers use this via dancing-links Algorithm X."}, {"id": 1358, "type": "game", "source": "sudoku", "section": "Engines & current best play", "text": "Sudoku — Engines & current best play\n\n- **Strongest known program(s):** Knuth's Algorithm X / dancing links — exact-cover backtracking solver; widely implemented (e.g., in Python's `dlx` libraries).\n- **Strength:** Solves any valid 9×9 puzzle in milliseconds; super-human on all published puzzle sets.\n- **Where the proof / tablebase lives (if solved):** NP-completeness of n²×n² Sudoku: Yato & Seta (2003), [../references.md#selman-sudoku2003](../references.md#selman-sudoku2003).\n- **Notes:** All ~6.67×10²¹ valid 9×9 completions were enumerated by Felgenhauer & Jarvis (2005); a minimum of 17 clues is required for a unique solution (McGuire et al., 2012)."}, {"id": 1359, "type": "game", "source": "sudoku", "section": "Complexity", "text": "Sudoku — Complexity\n\nNP-complete in the generalised setting."}, {"id": 1360, "type": "game", "source": "sudoku", "section": "References", "text": "Sudoku — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sudoku)\n- [Yato & Seta (2003). *Complexity and Completeness of Finding Another Solution and its Application to Puzzles*.](../references.md#selman-sudoku2003)"}, {"id": 1361, "type": "game", "source": "sudoku", "section": "See also", "text": "Sudoku — See also\n\n- [Slitherlink](slitherlink.md) · [Hashiwokakero](hashiwokakero.md) · [Nonograms](nonograms.md) · [Lights Out](lights-out.md)\n- Lexicon: [NP-completeness](../lexicon/README.md#np-completeness)"}, {"id": 1362, "type": "game", "source": "sungka", "section": "overview", "text": "Sungka\nFilipino mancala with seven pits per side — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan mancala."}, {"id": 1363, "type": "game", "source": "sungka", "section": "Description", "text": "Sungka — Description\n\nSungka is the traditional mancala of the Philippines, played on a board of 14\nsmall \"houses\" (7 per side) plus two large home stores. The traditional\nopening has both players moving simultaneously — a unique feature for a\nmancala — though analysis typically considers the alternating version."}, {"id": 1364, "type": "game", "source": "sungka", "section": "Rules", "text": "Sungka — Rules\n\n1. Board: 7 small pits per player plus one **home** store per player.\n2. Each small pit starts with 7 shells.\n3. On a turn the player picks up all shells in one of their pits and sows\n   them counterclockwise, dropping one into each pit and into the player's\n   own home (but not the opponent's home).\n4. If the last shell lands in the player's own home, the player goes again.\n5. If the last shell lands in an empty pit on the player's own side, the\n   player captures that shell plus all shells in the opposing pit directly\n   across (the **sungka capture**).\n6. If the last shell lands in an empty pit on the opponent's side, the turn\n   ends with no capture.\n7. Game ends when one side has no shells in any of their pits; remaining\n   shells go to the other player's home. Most shells wins."}, {"id": 1365, "type": "game", "source": "sungka", "section": "Solution status", "text": "Sungka — Solution status\n\nSungka is **not solved**. The large state space and traditional simultaneous\nopening complicate analysis."}, {"id": 1366, "type": "game", "source": "sungka", "section": "Consensus on optimal play", "text": "Sungka — Consensus on optimal play\n\n- **Maximise relay chains (extra turns)** — landing your last shell in your home store earns a free move; plan multi-pit sowing sequences that thread through your home repeatedly to compound tempo.\n- **Feed your home store steadily** — seeding the store consistently throughout the game is more reliable than banking on a single massive capture; depleting your pits trying for one big capture leaves you vulnerable.\n- **Bait empty-pit captures** — deliberately leave one of your own pits empty opposite a heavily loaded opponent pit, then sow into it with the last shell landing there to capture the opponent's pile.\n- **Keep your far pits alive** — pits close to your home are easy to sow from; the far pit (pit 7) can send a large batch past the home and around to the opponent's side, making it valuable late in the game.\n- **Deny the opponent relay chains** — watch which of their pits, when sown, would end in their home; if possible, fill or drain those pits before they can exploit the chain."}, {"id": 1367, "type": "game", "source": "sungka", "section": "Engines & current best play", "text": "Sungka — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Sungka has not been subject to formal game-theoretic analysis; the traditional simultaneous-opening variant adds further complexity absent from most mancala studies."}, {"id": 1368, "type": "game", "source": "sungka", "section": "Complexity", "text": "Sungka — Complexity\n\nLarge."}, {"id": 1369, "type": "game", "source": "sungka", "section": "References", "text": "Sungka — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Sungka)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1370, "type": "game", "source": "sungka", "section": "See also", "text": "Sungka — See also\n\n- [Awari](awari.md) · [Bao](bao.md) · [Kalah](kalah.md) · [Oware](awari.md) · [Pallanguzhi](pallanguzhi.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1371, "type": "game", "source": "surakarta", "section": "overview", "text": "Surakarta\nIndonesian capture game with looped corner tracks — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan capture game."}, {"id": 1372, "type": "game", "source": "surakarta", "section": "Description", "text": "Surakarta — Description\n\nSurakarta is a Javanese capture game on a 6×6 grid joined at the corners by\nfour curved \"loop\" tracks. Pieces capture exclusively by travelling around at\nleast one loop and ending on an opposing piece — there are no captures by\nshort-range movement."}, {"id": 1373, "type": "game", "source": "surakarta", "section": "Rules", "text": "Surakarta — Rules\n\n1. Board: 6×6 grid of intersections with four curved loops connecting the\n   outer four rows on each side to themselves.\n2. Each side has 12 pieces placed on the two ranks nearest them.\n3. On a turn a player either:\n   - **Moves** one piece one step to an adjacent vacant intersection\n     (orthogonally or diagonally); **or**\n   - **Captures** by sliding one of their pieces in a straight line, then\n     around **at least one** corner loop, and onto an opposing piece (the\n     captured piece is removed). Captures require the entire path to be\n     unobstructed.\n4. The player who captures all of the opponent's pieces wins; if neither side\n   can force a capture, the game is drawn or decided by counting (varies)."}, {"id": 1374, "type": "game", "source": "surakarta", "section": "Solution status", "text": "Surakarta — Solution status\n\nSurakarta is **not solved**. Some endgame analysis exists; engines play well\nbut no formal value is known."}, {"id": 1375, "type": "game", "source": "surakarta", "section": "Consensus on optimal play", "text": "Surakarta — Consensus on optimal play\n\n- **Control the loop entry points** — pieces placed at or near the intersections that feed into the curved corner loops can both threaten captures and block enemy loop-travelling attacks; contest these key squares early.\n- **Use non-capturing moves to set up loop attacks** — ordinary steps position a piece for a future loop-capture; move pieces into lines that align with a loop so a single step later triggers a capture.\n- **Do not leave pieces on loop lanes unguarded** — a piece sitting on a straight segment connecting to a loop is vulnerable to a long-range capture if the path is clear; keep threatened pieces off the main arteries or have a blocker on the path.\n- **Maintain numerical superiority** — with 12 pieces per side on a 6×6 board, trading evenly is neutral; gaining a piece advantage accelerates the win since the opponent has fewer blocking pieces.\n- **Corner clusters are both strong and dangerous** — pieces concentrated near a corner control multiple loop exits but are also reachable from two loops; a densely packed corner can be stripped by consecutive loop captures."}, {"id": 1376, "type": "game", "source": "surakarta", "section": "Engines & current best play", "text": "Surakarta — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** No formal computational solution has been published; Surakarta is primarily studied as a cultural and recreational game."}, {"id": 1377, "type": "game", "source": "surakarta", "section": "Complexity", "text": "Surakarta — Complexity\n\nModerate."}, {"id": 1378, "type": "game", "source": "surakarta", "section": "References", "text": "Surakarta — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Surakarta_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1379, "type": "game", "source": "surakarta", "section": "See also", "text": "Surakarta — See also\n\n- [Yote](yote.md) · [Seega](seega.md) · [Picaria](picaria.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1380, "type": "game", "source": "tablut", "section": "overview", "text": "Tablut\nA Sámi member of the asymmetric tafl family — strong programs and endgame\nSolution status: Partially solved / strongly analysed (no published full solution). Game-theoretic value: Unknown (rule-set dependent). Players: 2 (asymmetric: attackers vs. king's defenders). Type: Partisan asymmetric capture game."}, {"id": 1381, "type": "game", "source": "tablut", "section": "Description", "text": "Tablut — Description\n\nPlayed on a 9×9 board. The **defender** controls a king on the central throne\nplus 8 soldiers; the **attacker** controls 16 soldiers around the edges. All\npieces move like a rook. Pieces are captured by being sandwiched between two\nenemies (custodial capture). The king wins by reaching the board edge (or a\ncorner, depending on the rule set); the attackers win by capturing the king.\nTablut is the variant Linnaeus recorded among the Sámi in 1732, and the modern\ntafl revival largely descends from it — but the surviving rules are incomplete,\nso many incompatible rule sets exist."}, {"id": 1382, "type": "game", "source": "tablut", "section": "Solution status", "text": "Tablut — Solution status\n\nTablut is **not formally solved**. The biggest obstacle is not only size but\n**rule ambiguity**: the historical record is incomplete, and the game-theoretic\nvalue depends heavily on which edge/corner-win and king-armament rules are used —\nseveral rule sets are known to be lopsided wins for one side. Strong AI players\nand endgame tablebases exist for particular rule sets (and Tablut is a common\ntest bed for game-AI competitions), but no peer-reviewed weak or strong solution\nof a standard Tablut has been published."}, {"id": 1383, "type": "game", "source": "tablut", "section": "Consensus on optimal play", "text": "Tablut — Consensus on optimal play\n\n- **Attackers must build a full blockade** — with 16 pieces surrounding a 9×9 board, attackers win by encircling the king so it has no clear path to the edge; a partial blockade that leaves a single corridor will fail.\n- **Defenders prioritise king mobility over piece count** — keeping the king able to move in at least two directions is more important than saving individual soldiers; trapped defenders should sacrifice pieces to open king routes.\n- **Custodial chains create tempo** — moving a single piece to complete a custodial sandwich removes an enemy piece and threatens others; attackers should stage pieces so each advancing move threatens or completes a capture.\n- **Corner squares (if they win) demand immediate control** — under rule sets where the king wins by reaching a corner, defenders should aim for the nearest corner from the opening; attackers must post pieces on the two squares adjacent to each corner immediately.\n- **Rule-set awareness is paramount** — whether the king needs a corner or an edge, whether the king is \"armed\" (can participate in captures), and whether the throne blocks movement all change optimal strategy substantially; confirm the rule set before applying any opening theory."}, {"id": 1384, "type": "game", "source": "tablut", "section": "Engines & current best play", "text": "Tablut — Engines & current best play\n\n- **Strongest known program(s):** Humans and AI bots (e.g., Hnefatafl AI by Martin Windisch and entries in computer-tafl tournaments) — alpha-beta search with endgame databases for specific rule sets.\n- **Strength:** Strong amateur to competitive; engines dominate recreational play under fixed rule sets.\n- **Where the proof / tablebase lives (if solved):** Partial endgame tablebases for reduced-material Tablut positions; no full-game solution published.\n- **Notes:** Tablut's game-theoretic value is rule-set dependent and no single \"standard\" rule set is universally accepted, making a definitive solution elusive."}, {"id": 1385, "type": "game", "source": "tablut", "section": "Complexity", "text": "Tablut — Complexity\n\nLarge — comparable to other 9×9 capture games; well beyond hand analysis,\nthough small enough that endgame retrograde databases are feasible for reduced\nmaterial."}, {"id": 1386, "type": "game", "source": "tablut", "section": "References", "text": "Tablut — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tafl_games) ([archive](http://web.archive.org/web/20260404190502/https://en.wikipedia.org/wiki/Tafl_games))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1387, "type": "game", "source": "tablut", "section": "See also", "text": "Tablut — See also\n\n- [Brandubh](brandubh.md) · [Fox and Geese](fox-and-geese.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1388, "type": "game", "source": "tac-tix", "section": "overview", "text": "Tac-Tix (Bynum)\nPiet Hein's misère row-removing Nim variant on a 5×5 grid — fully solved.\nSolution status: Strongly solved (small board). Game-theoretic value: Position-dependent (misère convention). Players: 2. Type: Impartial misère take-away game."}, {"id": 1389, "type": "game", "source": "tac-tix", "section": "Description", "text": "Tac-Tix (Bynum) — Description\n\nTac-Tix was invented by Piet Hein in the 1950s and popularised in Martin\nGardner's column. It is a **misère** take-away game on a 5×5 grid of counters:\non each turn a player removes any contiguous run of counters from a single row\nor column; the player forced to remove the last counter **loses**."}, {"id": 1390, "type": "game", "source": "tac-tix", "section": "Rules", "text": "Tac-Tix (Bynum) — Rules\n\n1. Set up: a 5×5 grid filled with 25 counters.\n2. On your turn, choose a **single row or column** and remove any contiguous\n   subsequence of counters within it (at least one counter).\n3. Misère play: the player who takes the **last** counter loses."}, {"id": 1391, "type": "game", "source": "tac-tix", "section": "Solution status", "text": "Tac-Tix (Bynum) — Solution status\n\nStrongly solved by exhaustive enumeration: J. Bynum (in correspondence with\nMartin Gardner) computed the value of every Tac-Tix position. The game's misère\nanalysis is delicate — unlike normal-play Nim, simple nim-sum reasoning does\nnot work, but the full state space is small enough that a complete value table\nis straightforward to construct."}, {"id": 1392, "type": "game", "source": "tac-tix", "section": "Consensus on optimal play", "text": "Tac-Tix (Bynum) — Consensus on optimal play\n\n- **Mirror strategy wins for the second player from the start** — on the symmetric 5×5 board the second player can mirror every first-player move about the board's centre; this forces the first player to make the last move and lose.\n- **Symmetry breaks only when a centre-row move is made** — the mirror strategy fails if the first player takes from the exact centre of the central row/column; Bynum's solution handles this case explicitly via table lookup.\n- **Take large runs to destroy rows quickly** — clearing whole rows/columns early limits future opportunities for both players; the player who can force the last remaining counter onto their opponent wins.\n- **Avoid leaving isolated single counters** — a position with only individual isolated counters in distinct rows/columns is a pure misère counting problem; the player facing an odd number of such counters must take the last one and loses.\n- **Do not split rows carelessly** — removing a middle segment of a row creates two separate fragments in the same row; each fragment is an independent sub-game, complicating the misère analysis for the opponent."}, {"id": 1393, "type": "game", "source": "tac-tix", "section": "Engines & current best play", "text": "Tac-Tix (Bynum) — Engines & current best play\n\n- **Strongest known program(s):** Exhaustive table (Bynum, 1969) — a complete value table for all 2²⁵ subsets of the 5×5 grid, as reported by Martin Gardner.\n- **Strength:** Perfectly solved; any program consulting the table plays flawlessly.\n- **Where the proof / tablebase lives (if solved):** Described in Gardner's *Mathematical Games* columns and summarised in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** The symmetry-mirror strategy covers most play; the full table is needed only for positions where symmetry has been broken."}, {"id": 1394, "type": "game", "source": "tac-tix", "section": "Complexity", "text": "Tac-Tix (Bynum) — Complexity\n\nSmall."}, {"id": 1395, "type": "game", "source": "tac-tix", "section": "References", "text": "Tac-Tix (Bynum) — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1396, "type": "game", "source": "tac-tix", "section": "See also", "text": "Tac-Tix (Bynum) — See also\n\n- [Misère Nim](misere-nim.md) · [Nim](nim.md) · [Northcott's game](northcotts-game.md)\n- Lexicon: [misère play](../lexicon/README.md#misere-play) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 1397, "type": "game", "source": "tamsk", "section": "overview", "text": "TAMSK\nThe GIPF-project's time-pressure game — pieces are sand-timers that \"run out.\"\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan time-pressure abstract."}, {"id": 1398, "type": "game", "source": "tamsk", "section": "Description", "text": "TAMSK — Description\n\nTAMSK (Kris Burm, 1998) is the GIPF-project's most unusual entry: each \"piece\"\nis a small **sand-timer**. The state of the game depends on which side of each\ntimer is up, encoding how much sand has fallen. Moves must be made within the\ntime the relevant timer still has remaining."}, {"id": 1399, "type": "game", "source": "tamsk", "section": "Rules", "text": "TAMSK — Rules\n\n1. Board: hexagonal grid of cells; each cell has a fixed maximum capacity.\n2. Each player owns a set of small sand-timers (three colours of timers,\n   varying sand durations).\n3. On a turn, the player picks one of their timers from the board, flips it\n   onto an empty cell — the cell's timer-count constraint must be respected —\n   and the sand starts running.\n4. Once a timer runs out of sand, it can no longer be moved.\n5. The player who can no longer legally move a timer loses; final scoring is\n   on the number of timers each player has used to claim \"territory.\""}, {"id": 1400, "type": "game", "source": "tamsk", "section": "Solution status", "text": "TAMSK — Solution status\n\nTAMSK is **not solved**. Its time-state makes the game extension non-standard\nrelative to other GIPF entries, and there is no published solving result."}, {"id": 1401, "type": "game", "source": "tamsk", "section": "Consensus on optimal play", "text": "TAMSK — Consensus on optimal play\n\n- **Prioritise long-duration timers in contested cells** — timers with more remaining sand stay mobile longer; placing them on cells that will be fought over maintains flexibility while short-lived timers lock down peripheral positions.\n- **Freeze opponent timers early** — moving so that an opponent's timers run dry in poor positions permanently removes them from play; deliberately contest the cells opponents must visit to keep timers active.\n- **Do not flip timers unnecessarily** — each flip uses the timer's remaining sand; unnecessary moves waste that sand and may strand a timer in the wrong location.\n- **Corner cells are low-risk anchors** — cells on the periphery of the hex board are contested by fewer approaches; placing a medium-duration timer there early secures territory at minimal future flip-cost.\n- **Track the sand state, not just positions** — the total amount of sand remaining across your timers is a resource; the player who runs out of movable timers first loses, so managing sand longevity is the primary metric."}, {"id": 1402, "type": "game", "source": "tamsk", "section": "Engines & current best play", "text": "TAMSK — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** TAMSK's real-time sand-timer mechanic is unique among GIPF-project games and creates an unusual time-state space that standard game-tree search handles poorly."}, {"id": 1403, "type": "game", "source": "tamsk", "section": "Complexity", "text": "TAMSK — Complexity\n\nSmall board but a large time-state continuum makes formal analysis awkward."}, {"id": 1404, "type": "game", "source": "tamsk", "section": "References", "text": "TAMSK — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/TAMSK) ([archive](http://web.archive.org/web/20260115011947/https://en.wikipedia.org/wiki/TAMSK))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1405, "type": "game", "source": "tamsk", "section": "See also", "text": "TAMSK — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [YINSH](yinsh.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1406, "type": "game", "source": "tant-fant", "section": "overview", "text": "Tant Fant\nAn Indian three-in-a-row game — strongly solvable but the standard rule set\nSolution status: Strongly solved **[verify]**. Game-theoretic value: Draw **[verify]** (some sources: second-player win). Players: 2. Type: Partisan placement+movement game."}, {"id": 1407, "type": "game", "source": "tant-fant", "section": "Description", "text": "Tant Fant — Description\n\nA traditional Indian three-in-a-row game on a 3×3 grid, played without\ndiagonals. Two distinguishing features: stones start in fixed home rows\n(no placement phase), and the goal is to make a row of three on **any straight\nline that is not your starting row**."}, {"id": 1408, "type": "game", "source": "tant-fant", "section": "Rules", "text": "Tant Fant — Rules\n\n1. Board: 3×3 grid of points connected horizontally and vertically (some\n   sources include diagonals — **[verify]** the canonical version).\n2. Each player has 3 stones on their home row at the start; the middle row is\n   empty.\n3. Players alternate sliding one stone to an adjacent empty point along a line.\n4. A player wins by making three of their stones in a row along **any line\n   other than their starting row**."}, {"id": 1409, "type": "game", "source": "tant-fant", "section": "Solution status", "text": "Tant Fant — Solution status\n\nStrongly solvable by trivial enumeration. The reported value depends on the\nexact rule set: most analyses give a **draw with perfect play**, but some\nsources report a **second-player win** in restricted variants. Treat the value\nas **[verify]** pending a canonical statement of rules."}, {"id": 1410, "type": "game", "source": "tant-fant", "section": "Consensus on optimal play", "text": "Tant Fant — Consensus on optimal play\n\n- **Advance to the middle row first** — getting all three stones into the central row allows them to slide in any direction on subsequent turns, maximising the threat of a winning three-in-a-row.\n- **Do not overcommit to a single line** — with only a 3×3 board, telegraphing a particular winning line lets the opponent block with one stone while threatening their own; keep threats on multiple rows simultaneously.\n- **Mirror or counter symmetrically** — the small board means the second player can often mirror first-player advances to maintain balance; strong first-player play must break symmetry purposefully.\n- **Block the opponent before completing your own three** — because the board is tiny, allowing your opponent one more step is usually fatal; blocking is frequently higher priority than advancing.\n- **Avoid your home row** — any three-in-a-row along your starting row does not count as a win; never retreat all three stones back to the home row hoping to reassemble."}, {"id": 1411, "type": "game", "source": "tant-fant", "section": "Engines & current best play", "text": "Tant Fant — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; exhaustive enumeration over the trivially small state space suffices.\n- **Strength:** Perfectly solved by search; any implementation plays flawlessly.\n- **Where the proof / tablebase lives (if solved):** No published formal proof available to the cataloguer; the game-theoretic value (draw or second-player win) is rule-set dependent — treat as **[verify]**.\n- **Notes:** Closely related to Tapatan and Three Men's Morris; the distinguishing feature is the fixed home-row start and the exclusion of the starting row from winning lines."}, {"id": 1412, "type": "game", "source": "tant-fant", "section": "Complexity", "text": "Tant Fant — Complexity\n\nTiny."}, {"id": 1413, "type": "game", "source": "tant-fant", "section": "References", "text": "Tant Fant — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tant_fant)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1414, "type": "game", "source": "tant-fant", "section": "See also", "text": "Tant Fant — See also\n\n- [Picaria](picaria.md) · [Tapatan](tapatan.md) · [Three Men's Morris](three-mens-morris.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1415, "type": "game", "source": "tapatan", "section": "overview", "text": "Tapatan\nA Filipino three-in-a-row game — strongly solved as a draw.\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan placement+movement game."}, {"id": 1416, "type": "game", "source": "tapatan", "section": "Description", "text": "Tapatan — Description\n\nA traditional Filipino game in the [Three Men's Morris](three-mens-morris.md)\nfamily: nine points in a 3×3 lattice (with diagonals), three stones per player,\nand a two-phase placement-then-movement structure."}, {"id": 1417, "type": "game", "source": "tapatan", "section": "Rules", "text": "Tapatan — Rules\n\n1. Board: 3×3 grid of points connected by lines including both diagonals.\n2. Each player has 3 stones.\n3. **Placement**: players alternate placing stones on empty points. Three in a\n   row wins.\n4. **Movement**: once all six stones are placed, players alternate sliding a\n   stone along a line to an adjacent empty point.\n5. A player making three-in-a-row wins. If no progress is being made (typical\n   repetition rule: 30 moves without a three-in-a-row), the game is a draw."}, {"id": 1418, "type": "game", "source": "tapatan", "section": "Solution status", "text": "Tapatan — Solution status\n\nStrongly solved by trivial enumeration: the value with perfect play is a\n**draw**. Tapatan is essentially the same game as\n[Three Men's Morris](three-mens-morris.md), with the cosmetic difference that\nsome sources omit the diagonals — the diagonal-on version is the one whose\ncorrect value is a draw."}, {"id": 1419, "type": "game", "source": "tapatan", "section": "Consensus on optimal play", "text": "Tapatan — Consensus on optimal play\n\n- **Take the centre on the first move** — the centre point lies on four of the eight possible lines (row, column, and both diagonals); occupying it first maximises winning threats and forces the opponent to respond defensively.\n- **If the centre is taken, reply with a corner** — corners lie on three lines each, more than edge points (two lines each); owning two corners connected through the centre is the most common winning setup.\n- **Block every two-in-a-row immediately** — with only 3 stones per side and a tiny board, any unblocked double threat wins in one move; defence is non-negotiable.\n- **In the movement phase, shuttle rather than over-commit** — with draws available by repetition, the key is to create a double threat (fork) where one stone will complete a row regardless of the opponent's block.\n- **Avoid giving the opponent a fork** — a fork occurs when one player threatens two different three-in-a-rows simultaneously; in the movement phase, never step your stone to a position that creates a fork for your opponent."}, {"id": 1420, "type": "game", "source": "tapatan", "section": "Engines & current best play", "text": "Tapatan — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; exhaustive enumeration over the tiny state space suffices.\n- **Strength:** Perfectly solved by search; any implementation plays flawlessly.\n- **Where the proof / tablebase lives (if solved):** No dedicated publication; result by exhaustive enumeration, consistent with Three Men's Morris analyses cited in [../references.md#vandenherik2002](../references.md#vandenherik2002).\n- **Notes:** Tapatan is functionally identical to Three Men's Morris with diagonals; the draw result assumes both diagonals are enabled."}, {"id": 1421, "type": "game", "source": "tapatan", "section": "Complexity", "text": "Tapatan — Complexity\n\nTiny."}, {"id": 1422, "type": "game", "source": "tapatan", "section": "References", "text": "Tapatan — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Three_men%27s_morris) ([archive](http://web.archive.org/web/20251231113758/https://en.wikipedia.org/wiki/Three_men's_morris))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1423, "type": "game", "source": "tapatan", "section": "See also", "text": "Tapatan — See also\n\n- [Three Men's Morris](three-mens-morris.md) · [Picaria](picaria.md) · [Achi](achi.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1424, "type": "game", "source": "teeko", "section": "overview", "text": "Teeko\nJohn Scarne's \"perfect game\" — a compact placement-and-movement game that\nSolution status: Weakly solved. Game-theoretic value: Draw **[verify]**. Players: 2. Type: Partisan placement-and-movement game."}, {"id": 1425, "type": "game", "source": "teeko", "section": "Description", "text": "Teeko — Description\n\nPlayed on a 5×5 board. Each player has four pieces. In the **placement phase**\nplayers alternately place their four pieces; in the **movement phase** they\nslide a piece to an adjacent (orthogonal or diagonal) empty cell. A player wins\nby getting their four pieces into a row, column, diagonal, or — in the standard\nrules — any 2×2 square."}, {"id": 1426, "type": "game", "source": "teeko", "section": "Solution status", "text": "Teeko — Solution status\n\nTeeko is **weakly solved**. The 5×5 board with four pieces per side yields only\na few million positions — well within exhaustive search — and computer analyses\nhave reported the game to be a **draw** with perfect play. John Scarne promoted\nTeeko for decades as a deep \"perfect\" alternative to chess and checkers; the\nexhaustive verdict is more modest: against best defence neither side can force a\nwin.\n\n> **[verify]** — The draw result is the commonly reported one and is fully\n> consistent with the game's small size, but this archive should cite the\n> specific exhaustive computation (and fix the rule variant, since Teeko has\n> \"American\"/advanced rule sets that differ over diagonal moves and the 2×2 win)."}, {"id": 1427, "type": "game", "source": "teeko", "section": "Consensus on optimal play", "text": "Teeko — Consensus on optimal play\n\n- **Place pieces to threaten multiple win conditions** — a piece placed where it simultaneously contributes to a row, a diagonal, and a potential 2×2 square forces the opponent to block two threats at once, often fatally.\n- **Contest the centre of the 5×5 board** — central placement reaches the most winning lines and squares; peripheral pieces contribute to fewer configurations.\n- **During placement, prevent the opponent from forming three-in-a-line** — three aligned opponent pieces with a clear extension are one move from winning; block before all four placements are complete.\n- **In the movement phase, use the 2×2 square as a stealth threat** — linear threats are easy to spot; a 2×2 cluster forming in a corner is often missed and provides a quicker win than completing a row.\n- **Draw by forcing repetition** — if behind in the movement phase, shuttle a piece back and forth to force a repetition draw; the board is small enough that repetition is achievable when the win is lost."}, {"id": 1428, "type": "game", "source": "teeko", "section": "Engines & current best play", "text": "Teeko — Engines & current best play\n\n- **Strongest known program(s):** Exhaustive search programs (reported in the 1990s–2000s) — retrograde analysis over the few-million-position state space.\n- **Strength:** Perfectly solved for the analysed rule variant; humans cannot match a correct implementation.\n- **Where the proof / tablebase lives (if solved):** No single canonical published proof known to the cataloguer; result consistent with [../references.md#vandenherik2002](../references.md#vandenherik2002) framework.\n- **Notes:** Multiple rule variants exist (standard vs. advanced Teeko, diagonal-move rules); the draw result applies to the standard American edition with the 2×2 square win condition."}, {"id": 1429, "type": "game", "source": "teeko", "section": "Complexity", "text": "Teeko — Complexity\n\nA few million positions — exhaustively searchable."}, {"id": 1430, "type": "game", "source": "teeko", "section": "References", "text": "Teeko — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Teeko) ([archive](http://web.archive.org/web/20260508193822/https://en.wikipedia.org/wiki/Teeko))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1431, "type": "game", "source": "teeko", "section": "See also", "text": "Teeko — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Nine Holes](nine-holes.md) · [Connect Four](connect-four.md)\n- Lexicon: [weakly solved](../lexicon/README.md#weakly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1432, "type": "game", "source": "three-check-chess", "section": "overview", "text": "Three-check chess\nChess variant where giving check three times wins — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan chess variant."}, {"id": 1433, "type": "game", "source": "three-check-chess", "section": "Description", "text": "Three-check chess — Description\n\nThree-check chess is identical to orthodox chess except that a player wins\neither by checkmate **or** by checking the opposing king three times in the\ngame. The added win condition reshapes opening and middlegame priorities."}, {"id": 1434, "type": "game", "source": "three-check-chess", "section": "Rules", "text": "Three-check chess — Rules\n\n1. Same setup, movement, and rules as orthodox chess.\n2. Each side maintains a **check count**; each time a player gives check, the\n   count for the opponent increases by one.\n3. The third check delivered against either king ends the game in favour of\n   the checker, regardless of whether it is checkmate.\n4. Checkmate, stalemate, draw-by-rule conditions continue to apply normally."}, {"id": 1435, "type": "game", "source": "three-check-chess", "section": "Solution status", "text": "Three-check chess — Solution status\n\nThree-check is **not solved**. Engines play it strongly and the variant is\npopular online but no published proof exists."}, {"id": 1436, "type": "game", "source": "three-check-chess", "section": "Consensus on optimal play", "text": "Three-check chess — Consensus on optimal play\n\nHeuristics from strong online play:\n\n- **Develop pieces toward the king, not the centre** — the standard chess maxim \"develop knights before bishops, claim the centre\" is partially superseded: pieces that attack the king from afar (long diagonals, files behind the enemy king) are worth more because each safe check counts.\n- **Don't trade queens** — the queen is the most versatile checking piece; trading it gives the opponent two free checking moves of relative safety.\n- **Castle queenside more often than in chess** — kingside castling sometimes exposes the king to early check sequences on the h-file/diagonals; queenside is sometimes safer because Black's queen-attacking diagonals are blocked.\n- **Count the checks, not just material** — losing a pawn to avoid the second/third check is often correct."}, {"id": 1437, "type": "game", "source": "three-check-chess", "section": "Engines & current best play", "text": "Three-check chess — Engines & current best play\n\n- **Strongest known programs:** [Fairy-Stockfish](https://github.com/ianfab/Fairy-Stockfish) ([archive](http://web.archive.org/web/20230224150112/https://github.com/ianfab/Fairy-Stockfish)) (open source); available for analysis on [Lichess](https://lichess.org/variant/threeCheck) ([archive](http://web.archive.org/web/20260507021356/https://lichess.org/variant/threeCheck)).\n- **Strength:** Super-human.\n- **Notes:** No specific tablebases; engine evaluation includes the check counter as an extra state dimension."}, {"id": 1438, "type": "game", "source": "three-check-chess", "section": "Complexity", "text": "Three-check chess — Complexity\n\nSimilar to chess."}, {"id": 1439, "type": "game", "source": "three-check-chess", "section": "References", "text": "Three-check chess — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Three-check_chess) ([archive](http://web.archive.org/web/20260406014518/https://en.wikipedia.org/wiki/Three-check_chess))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1440, "type": "game", "source": "three-check-chess", "section": "See also", "text": "Three-check chess — See also\n\n- [Chess](chess.md) · [Atomic chess](atomic-chess.md) · [King of the Hill](king-of-the-hill.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1441, "type": "game", "source": "three-mens-morris", "section": "overview", "text": "Three Men's Morris\nThe smallest of the morris games — placement then movement, three in a row to\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan positional / sliding game."}, {"id": 1442, "type": "game", "source": "three-mens-morris", "section": "Description", "text": "Three Men's Morris — Description\n\nPlayed on a 3×3 grid of points (often with diagonals marked). Each player has\nthree pieces. Players first **place** their three pieces alternately, then enter\na **movement phase**, sliding a piece along a marked line to an adjacent empty\npoint. Forming a mill — three pieces in a marked row — wins."}, {"id": 1443, "type": "game", "source": "three-mens-morris", "section": "Solution status", "text": "Three Men's Morris — Solution status\n\nThree Men's Morris is **strongly solved**. With only a few thousand reachable\npositions, the entire game graph is exhaustively known, and under standard rules\nthe game-theoretic value is a **draw**. A common practical wrinkle: the first\nplayer's natural placement in the centre is so strong that some rule sets ban\nthe opening centre move to keep the game interesting — but even without that\nban, best play by both sides yields a draw.\n\nIt is the simplest member of the [morris family](nine-mens-morris.md), which\nscales up through [Six](six-mens-morris.md), [Nine](nine-mens-morris.md), and\n[Twelve Men's Morris](twelve-mens-morris.md)."}, {"id": 1444, "type": "game", "source": "three-mens-morris", "section": "Consensus on optimal play", "text": "Three Men's Morris — Consensus on optimal play\n\n- **Take the centre on the first move** — the centre connects to all four rows, columns, and both diagonals; it is the most powerful point on the board and must be contested immediately.\n- **If the centre is taken, play a corner** — corners lie on three lines (row, column, diagonal) while edge points lie on only two; corner placement gives the most future winning threats.\n- **Block every two-in-a-row before extending your own** — the board is too small to allow even one unblocked two-in-a-row; defence is always the priority.\n- **In the movement phase, shuttle to create forks** — a fork (two simultaneous three-in-a-row threats) cannot be blocked by a single move; create forks by sliding a piece to a point that threatens two lines at once.\n- **Accept the draw if you cannot fork** — with three pieces each on a 3×3 board, creating an unstoppable fork against an alert opponent is usually impossible; steer toward repetition rather than weakening your position chasing a win."}, {"id": 1445, "type": "game", "source": "three-mens-morris", "section": "Engines & current best play", "text": "Three Men's Morris — Engines & current best play\n\n- **Strongest known program(s):** Exhaustive search (trivial; thousands of positions) — any correct implementation plays perfectly.\n- **Strength:** Perfectly solved; the full position graph is known.\n- **Where the proof / tablebase lives (if solved):** Result embedded in the van den Herik et al. survey ([../references.md#vandenherik2002](../references.md#vandenherik2002)); the game is too small for a dedicated publication.\n- **Notes:** Some rule variants ban the first-player centre move to introduce practical difficulty; the draw result holds even without that restriction."}, {"id": 1446, "type": "game", "source": "three-mens-morris", "section": "Complexity", "text": "Three Men's Morris — Complexity\n\nA few thousand positions — trivially exhaustible."}, {"id": 1447, "type": "game", "source": "three-mens-morris", "section": "References", "text": "Three Men's Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Three_men%27s_morris) ([archive](http://web.archive.org/web/20251231113758/https://en.wikipedia.org/wiki/Three_men's_morris))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [Gasser, R. (1996). *Solving Nine Men's Morris*.](../references.md#gasser1996)"}, {"id": 1448, "type": "game", "source": "three-mens-morris", "section": "See also", "text": "Three Men's Morris — See also\n\n- [Nine Holes](nine-holes.md) · [Achi](achi.md) · [Six Men's Morris](six-mens-morris.md) · [Nine Men's Morris](nine-mens-morris.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [draw](../lexicon/README.md#draw)"}, {"id": 1449, "type": "game", "source": "tic-tac-chec", "section": "overview", "text": "Tic-Tac-Chec\nA chess-piece-based tic-tac-toe — players bring on chess pieces and try to\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement+movement game."}, {"id": 1450, "type": "game", "source": "tic-tac-chec", "section": "Description", "text": "Tic-Tac-Chec — Description\n\nA commercial game from the 2000s combining chess movement with a connect-four\nflavour. Each player has four chess pieces in reserve (a pawn, knight, bishop,\nrook); the goal is to be the first to align all four of your pieces along a\nrow, column, or diagonal on a 4×4 board."}, {"id": 1451, "type": "game", "source": "tic-tac-chec", "section": "Rules", "text": "Tic-Tac-Chec — Rules\n\n1. Board: 4×4 grid, empty initially.\n2. Each player has a reserve of **one pawn, one knight, one bishop, one rook**.\n3. On a turn, a player may either:\n   - **Drop** a reserve piece onto any empty square it could legally move to\n     from off-board (some sources allow drops on any empty square — **[verify]**\n     the canonical rule); or\n   - **Move** one of their on-board pieces according to its chess movement\n     rules (within the 4×4 grid), optionally capturing an opposing piece.\n4. Captured pieces return to the owner's reserve.\n5. The first player to align **all four of their pieces** in a row, column, or\n   diagonal wins."}, {"id": 1452, "type": "game", "source": "tic-tac-chec", "section": "Solution status", "text": "Tic-Tac-Chec — Solution status\n\nTic-Tac-Chec is **not solved**. The 4×4 board and small piece count make it\nplausibly tractable to modern retrograde analysis, but no published solution\nexists."}, {"id": 1453, "type": "game", "source": "tic-tac-chec", "section": "Consensus on optimal play", "text": "Tic-Tac-Chec — Consensus on optimal play\n\n- **Deploy the rook as a locking piece** — a rook on the board covers its entire rank and file, threatening alignment along any row or column it occupies; play it early to anchor one of those two lines.\n- **Use the knight for non-linear threats** — knights jump over pieces and are the hardest to block; threatening alignment with a knight already placed in a corner or edge forces the opponent to solve two problems at once.\n- **Capture strategically, not reflexively** — captured pieces return to the captor's reserve, so capturing an opponent's rook gives them back a powerful drop piece; be certain capturing is worth the gift.\n- **Target the diagonal from the start** — with only a 4×4 board, the main diagonals are the most compact winning line (all four pieces must occupy one of eight specific squares); controlling both endpoints of a diagonal early is a lasting threat.\n- **Block opponent alignment before extending your own** — the board is too small to ignore even a two-piece alignment; verify that your drop or move does not leave the opponent one step from winning."}, {"id": 1454, "type": "game", "source": "tic-tac-chec", "section": "Engines & current best play", "text": "Tic-Tac-Chec — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** The 4×4 board makes full retrograde analysis feasible in principle; no research team has published a solution as of this writing."}, {"id": 1455, "type": "game", "source": "tic-tac-chec", "section": "Complexity", "text": "Tic-Tac-Chec — Complexity\n\nModerate — within range of full retrograde solution if undertaken."}, {"id": 1456, "type": "game", "source": "tic-tac-chec", "section": "References", "text": "Tic-Tac-Chec — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tic-tac-chec)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1457, "type": "game", "source": "tic-tac-chec", "section": "See also", "text": "Tic-Tac-Chec — See also\n\n- [Quarto](quarto.md) · [Tic-tac-toe](tic-tac-toe.md) · [Minichess](minichess.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis)"}, {"id": 1458, "type": "game", "source": "tic-tac-toe", "section": "overview", "text": "Tic-tac-toe\nThe game everyone solves as a child — and the standard first example of a\nSolution status: Strongly solved. Game-theoretic value: Draw. Players: 2. Type: Partisan positional (k-in-a-row) game."}, {"id": 1459, "type": "game", "source": "tic-tac-toe", "section": "Description", "text": "Tic-tac-toe — Description\n\nPlayed on a 3×3 grid. Players alternately mark cells (X and O); the first to\nplace three of their marks in a row — horizontally, vertically, or diagonally —\nwins. If the grid fills with no line, the game is a draw."}, {"id": 1460, "type": "game", "source": "tic-tac-toe", "section": "Solution status", "text": "Tic-tac-toe — Solution status\n\nTic-tac-toe is **strongly solved**, the textbook example. The game is so small\nthat every position is exhaustively known: there are 5,478 reachable positions\n(765 up to rotation and reflection), and roughly 26,830 distinct complete games\nup to symmetry. With perfect play by both sides the result is always a\n**draw** — the second player can always prevent three-in-a-row, and the first\nplayer can never be forced to lose.\n\nIt is the canonical demonstration of [Zermelo's theorem](../lexicon/README.md#zermelos-theorem)\nin action and of a [pairing/blocking strategy](../lexicon/README.md#pairing-strategy)\nguaranteeing a draw."}, {"id": 1461, "type": "game", "source": "tic-tac-toe", "section": "Consensus on optimal play", "text": "Tic-tac-toe — Consensus on optimal play\n\n- **Take the centre first** — the centre square participates in 4 of the 8 winning lines (row, column, and both diagonals); it is the most valuable cell on the board.\n- **If you go second and the opponent takes centre, play a corner** — corners participate in 3 winning lines; an edge square participates in only 2. Never open with an edge as the second player.\n- **Block every two-in-a-row immediately** — with only 9 cells, leaving any two-in-a-row unblocked loses outright; defence is mandatory before extending your own line.\n- **Create a fork (double threat) to win** — a fork sets up two simultaneous unblockable three-in-a-rows; the most common winning sequence for first player is centre → corner → opposite corner → fork.\n- **Counter a fork threat by threatening to win** — if your opponent is setting up a fork, force them to block your own three-in-a-row instead; this derails the fork at no cost if you can complete the threat on the next move.\n- **Optimal play always draws** — against any legal move sequence, a correct defensive reply exists; the entire draw-guarantee fits in an 8-rule decision tree."}, {"id": 1462, "type": "game", "source": "tic-tac-toe", "section": "Engines & current best play", "text": "Tic-tac-toe — Engines & current best play\n\n- **Strongest known program(s):** Any correct minimax implementation — state space is 5,478 positions; exhaustive search is instantaneous.\n- **Strength:** Perfectly solved; any correct program draws against any opponent.\n- **Where the proof / tablebase lives (if solved):** Fully described in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)) and countless textbooks; no dedicated paper needed.\n- **Notes:** Tic-tac-toe is the canonical textbook example of a strongly solved game and the standard first exercise in minimax / alpha-beta search courses."}, {"id": 1463, "type": "game", "source": "tic-tac-toe", "section": "Complexity", "text": "Tic-tac-toe — Complexity\n\nNegligible — solvable by hand or by a child."}, {"id": 1464, "type": "game", "source": "tic-tac-toe", "section": "References", "text": "Tic-tac-toe — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tic-tac-toe) ([archive](http://web.archive.org/web/20260503215557/https://en.wikipedia.org/wiki/Tic-tac-toe))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1465, "type": "game", "source": "tic-tac-toe", "section": "See also", "text": "Tic-tac-toe — See also\n\n- [Qubic](qubic.md) (4×4×4 tic-tac-toe) · [Ultimate tic-tac-toe](ultimate-tic-tac-toe.md) · [Notakto](notakto.md) · [Order and Chaos](order-and-chaos.md) · [Gomoku](gomoku.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [pairing strategy](../lexicon/README.md#pairing-strategy)"}, {"id": 1466, "type": "game", "source": "ticket-to-ride", "section": "overview", "text": "Ticket to Ride\nThe modern railway-route board game — unsolved due to card randomness and mapping complexity.\nSolution status: Unsolved (no known solution framework for draw-dependent board games). Game-theoretic value: Unknown. Players: 2–5. Type: Stochastic route-building card game."}, {"id": 1467, "type": "game", "source": "ticket-to-ride", "section": "Description", "text": "Ticket to Ride — Description\n\nTicket to Ride is a turn-based strategy railway game designed by Alan R. Moon\nand published by Days of Wonder in 2004. Players collect coloured train-car\ncards and claim routes on a map to connect cities shown on their destination\ntickets. On each turn, a player may draw two train cards, draw additional\ndestination tickets, or claim a route by discarding matching coloured cards.\nShorter routes (1–3 segments) score modestly; longer routes (4–6 segments)\nscore disproportionately higher. The game ends when a player's train pieces\nrun low, after which all other players take one final turn. The highest total\nscore — route points plus completed-ticket values minus uncompleted-ticket\npenalties, plus bonuses — wins."}, {"id": 1468, "type": "game", "source": "ticket-to-ride", "section": "Solution status", "text": "Ticket to Ride — Solution status\n\nTicket to Ride is **not solved** and is extremely unlikely to be solvable in\nany formal sense. It combines **stochastic elements** (train card draws,\ndestination ticket draws), **hidden information** (opponents' cards and\ntickets), and a **large branching factor** (many possible routes on each turn),\nplacing it well beyond known solving frameworks. Even simplified single-map\nanalysis is computationally prohibitive. What exists instead is strong\n**heuristic play** guided by expert and engine analysis."}, {"id": 1469, "type": "game", "source": "ticket-to-ride", "section": "Consensus on optimal play", "text": "Ticket to Ride — Consensus on optimal play\n\n- **Claim long routes early** — longer routes (4+ segments) score more points\n  per card and also block opponents from using those paths; claiming a 6-length\n  route early is almost always correct.\n- **Collect cards of one or two colours** — focusing your draw on a small\n  palette increases the probability of completing your tickets; spreading across\n  many colours leaves you short of every route.\n- **Draw face-up cards when the colour helps, draw blind when it doesn't** —\n  face-up cards give colour certainty, but blind draws offer a chance at\n  locomotives (wilds). Snapping a face-up locomotive costs your second draw.\n- **Keep some flexibility in your ticket hand** — holding 3–4 destination\n  tickets gives fallback options if one route is blocked; don't discard tickets\n  to the point where only one path to victory remains.\n- **Watch opponents' builds to infer their tickets** — when an opponent claims\n  an otherwise-odd route, they are almost certainly connecting two cities on\n  one of their tickets; use that information to block them.\n- **The 10-point longest-road bonus shapes the whole game** — plan a continuous\n  chain across the board; even if you don't win the bonus, a connected network\n  is usually the most efficient way to complete tickets."}, {"id": 1470, "type": "game", "source": "ticket-to-ride", "section": "References", "text": "Ticket to Ride — References\n\n- Moon, Alan R. (2004). *Ticket to Ride*. Days of Wonder.\n- [Spiel des Jahres 2004 winner](https://www.spiel-des-jahres.de/en/games/ticket-to-ride/)\n- [Wikipedia — Ticket to Ride](https://en.wikipedia.org/wiki/Ticket_to_Ride_(board_game))\n- [BGG entry — Ticket to Ride (2004)](https://boardgamegeek.com/boardgame/9209/ticket-ride)"}, {"id": 1471, "type": "game", "source": "ticket-to-ride", "section": "See also", "text": "Ticket to Ride — See also\n\n- [Bridge](bridge.md) · [Poker (heads-up no-limit hold'em)](heads-up-nolimit-holdem.md)\n- Lexicon: [chance element](../lexicon/README.md#chance-element) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1472, "type": "game", "source": "tigers-and-goats", "section": "overview", "text": "Tigers and Goats (Bagh-Chal)\nThe Nepalese hunt game — strong computer analysis exists and the value is\nSolution status: Partially solved / analysed (value believed drawn) **[verify]**. Game-theoretic value: Believed a draw with perfect play **[verify]**. Players: 2 (asymmetric: 4 tigers vs. 20 goats). Type: Partisan hunt game."}, {"id": 1473, "type": "game", "source": "tigers-and-goats", "section": "Description", "text": "Tigers and Goats (Bagh-Chal) — Description\n\nPlayed on a 5×5 grid of points with diagonals (the traditional *aadu puli*\nboard). Four **tigers** begin on the corners; the **goat** player has 20 goats,\nentered one per turn in an opening \"placement\" phase. Tigers move along lines and\ncapture a goat by jumping it (as in draughts); goats never move during placement\nand never capture. Tigers win by capturing enough goats (commonly 5); goats win\nby immobilising all four tigers."}, {"id": 1474, "type": "game", "source": "tigers-and-goats", "section": "Solution status", "text": "Tigers and Goats (Bagh-Chal) — Solution status\n\nTigers and Goats is **not formally solved** in the published literature, but it\nhas been heavily analysed: the state space (~10^9) is small enough that strong\nprograms with endgame databases play it essentially perfectly, and the\nwidely-reported result is that **the game is a draw** with best play — goats can\navoid fatal captures, tigers can avoid being fully trapped. This is engine\nconsensus rather than a peer-reviewed proof; treat the value as **[verify]**."}, {"id": 1475, "type": "game", "source": "tigers-and-goats", "section": "Consensus on optimal play", "text": "Tigers and Goats (Bagh-Chal) — Consensus on optimal play\n\n- **Goats: place to block jump lanes in the placement phase** — during the opening 20 goat placements, avoid ever leaving a goat on a point where a tiger can jump over it; no goat should be placed with an empty escape cell behind it on the tiger's attack line.\n- **Goats: build a dense wall to trap tigers** — cluster goats along one or two rows to progressively restrict tiger mobility; the win condition for goats is full tiger immobilisation, so systematic encirclement beats piecemeal defence.\n- **Tigers: attack immediately during placement** — tigers can move (and capture) from the very first turn; aggressive early jumps force goats into defensive placements rather than the ideal blockade pattern.\n- **Tigers: keep multiple attack directions open** — a tiger cornered with no jump available and only one move is already effectively trapped; maintain at least two possible jump lines for each tiger.\n- **Goats: 5-capture loss is a hard cliff** — once 5 goats are captured the tigers win; a goat player who tolerates 3 or 4 captures must play near-perfectly for the rest of the game; avoid any capture in the mid-game.\n- **Sacrifice placement to maintain blockade integrity** — occasionally placing a goat in a suboptimal square to patch a jump lane is correct; a live goat in a non-ideal spot is better than a gap that lets a tiger roam freely."}, {"id": 1476, "type": "game", "source": "tigers-and-goats", "section": "Engines & current best play", "text": "Tigers and Goats (Bagh-Chal) — Engines & current best play\n\n- **Strongest known program(s):** Bagh-Chal solvers with endgame databases (several implementations exist, including Android apps and board-game sites); alpha-beta search with retrograde endgame tables.\n- **Strength:** Near-perfect play; engines are vastly stronger than human players on both sides.\n- **Where the proof / tablebase lives (if solved):** No peer-reviewed full-game solution published; engine consensus value is draw. See also [../references.md#vandenherik2002](../references.md#vandenherik2002).\n- **Notes:** The ~10⁹ state space is within reach of full retrograde analysis; a formal proof would be publishable but has not appeared in the literature as of this writing."}, {"id": 1477, "type": "game", "source": "tigers-and-goats", "section": "Complexity", "text": "Tigers and Goats (Bagh-Chal) — Complexity\n\nState-space on the order of 10^9 — within reach of exhaustive retrograde\nanalysis, which is why play is effectively perfect even absent a formal\npublication."}, {"id": 1478, "type": "game", "source": "tigers-and-goats", "section": "References", "text": "Tigers and Goats (Bagh-Chal) — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Bagh-Chal)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1479, "type": "game", "source": "tigers-and-goats", "section": "See also", "text": "Tigers and Goats (Bagh-Chal) — See also\n\n- [Fox and Geese](fox-and-geese.md) · [Hare and Hounds](hare-and-hounds.md) · [Konane](konane.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [strong play vs. solving](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1480, "type": "game", "source": "toads-and-frogs", "section": "overview", "text": "Toads and Frogs\nA one-dimensional partisan game devised by Conway as a CGT teaching example;\nSolution status: Partially solved (specific position families). Game-theoretic value: Known for several families; no full classification. Players: 2. Type: Partisan combinatorial game."}, {"id": 1481, "type": "game", "source": "toads-and-frogs", "section": "Description", "text": "Toads and Frogs — Description\n\nPlayed on a 1×*n* strip. One player owns \"Toads\" (which move rightward), the\nother \"Frogs\" (which move leftward). A piece may step into an adjacent empty\ncell in its direction, or **jump** over a single opposing piece into an empty\ncell beyond. A player unable to move loses\n([normal play](../lexicon/README.md#normal-play-convention))."}, {"id": 1482, "type": "game", "source": "toads-and-frogs", "section": "Solution status", "text": "Toads and Frogs — Solution status\n\nToads and Frogs is **partially solved**. Conway introduced it in\n[*Winning Ways*](../references.md#bcg2001) precisely to exercise the CGT value\ncalculus, and it is a famous source of positions with subtle values. Jeff\nErickson and others determined exact values for several infinite *families* of\nstarting positions and posed a list of open problems; some of those have since\nbeen resolved and others remain open. There is no complete classification of\nall starting positions, so the game as a whole is unsolved — but it is a\nwell-studied partially-solved case."}, {"id": 1483, "type": "game", "source": "toads-and-frogs", "section": "Consensus on optimal play", "text": "Toads and Frogs — Consensus on optimal play\n\n- **Compute the CGT value of each segment independently** — the strip often splits into independent sub-games separated by gaps; evaluate each sub-game's surreal-number or nimber value, sum them, and play in the hottest component.\n- **Jumps are usually stronger than steps** — a jump removes the jumped piece from blocking your future moves while also advancing your own piece two cells; prioritise jumps unless the step sets up a future jump chain.\n- **Avoid deadlock configurations** — a Toad and a Frog facing each other with no room to jump are permanently frozen; do not create a head-to-head standoff in a sub-strip unless it benefits you (e.g., locks in an opponent's piece).\n- **Temperature guides endgame priorities** — as the strip fills, identify which remaining moves have the highest temperature (i.e., whose value differs most depending on who goes next); always answer your opponent's move in the highest-temperature remaining component.\n- **Symmetric positions are second-player wins** — if the strip is symmetric (equal number of Toads and Frogs in mirror arrangement), the second player can often mirror to maintain balance; the first player must break symmetry profitably."}, {"id": 1484, "type": "game", "source": "toads-and-frogs", "section": "Engines & current best play", "text": "Toads and Frogs — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; positions are evaluated with CGT software (e.g., Combinatorial Game Suite) or custom surreal-number implementations.\n- **Strength:** Not benchmarked against humans; the interest is theoretical.\n- **Where the proof / tablebase lives (if solved):** Partial — families solved in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)) and Erickson's *New Toads and Frogs results* (1996).\n- **Notes:** A complete classification of all starting positions remains an open problem in combinatorial game theory."}, {"id": 1485, "type": "game", "source": "toads-and-frogs", "section": "Complexity", "text": "Toads and Frogs — Complexity\n\nGrows with strip length; the interest is theoretical rather than computational."}, {"id": 1486, "type": "game", "source": "toads-and-frogs", "section": "References", "text": "Toads and Frogs — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Toads_and_Frogs)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- J. Erickson (1996). *New Toads and Frogs results*. In *Games of No Chance*. **[verify]**"}, {"id": 1487, "type": "game", "source": "toads-and-frogs", "section": "See also", "text": "Toads and Frogs — See also\n\n- [Hackenbush](hackenbush.md) · [Clobber](clobber.md) · [Domineering](domineering.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [combinatorial game theory](../lexicon/README.md#combinatorial-game-theory)"}, {"id": 1488, "type": "game", "source": "toguz-kumalak", "section": "overview", "text": "Toguz Kumalak\nCentral Asian mancala with capture by parity — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan mancala."}, {"id": 1489, "type": "game", "source": "toguz-kumalak", "section": "Description", "text": "Toguz Kumalak — Description\n\nToguz Kumalak (Kazakh: \"nine pebbles\") is the national mancala game of\nKazakhstan and Kyrgyzstan, played on a board of two rows of nine holes plus\ntwo large stores (\"kazans\"). Each small hole starts with nine pebbles, giving\n162 pebbles total."}, {"id": 1490, "type": "game", "source": "toguz-kumalak", "section": "Rules", "text": "Toguz Kumalak — Rules\n\n1. Board: 9 small pits per player plus one **kazan** (store) per player. Each\n   small pit starts with 9 pebbles (162 total).\n2. On a turn the player picks up all pebbles from one of their own pits and\n   sows them one per pit counterclockwise, starting **with the same pit** if\n   it held more than one pebble (otherwise into the next pit).\n3. **Capture**: if the last pebble lands in an opponent's pit and makes the\n   total there **even**, all pebbles in that pit are captured to the player's\n   kazan.\n4. **Tuzdyk** (special hole): a player may claim one of the opponent's pits as\n   a permanent capturing pit under specific conditions; all pebbles sown to a\n   tuzdyk go to that player's kazan. Each player can have at most one tuzdyk\n   and not symmetrically placed.\n5. Game ends when one side cannot move; remaining pebbles go to the other\n   side's kazan. Most pebbles wins."}, {"id": 1491, "type": "game", "source": "toguz-kumalak", "section": "Solution status", "text": "Toguz Kumalak — Solution status\n\nToguz Kumalak is **not solved**. Engines have been developed for\ninternational competition but no game-theoretic value is published."}, {"id": 1492, "type": "game", "source": "toguz-kumalak", "section": "Consensus on optimal play", "text": "Toguz Kumalak — Consensus on optimal play\n\n- **Claim a tuzdyk on a loaded pit** — securing a tuzdyk (permanent capturing pit) on one of the opponent's busiest pits generates a steady stream of captured pebbles for the rest of the game; time the tuzdyk claim when the target pit holds many pebbles.\n- **Target even-count opponent pits** — the capture rule triggers when the final pebble lands in an opponent's pit and makes it even; count ahead to find sowing distances that land on opponent pits with an odd current count, since adding one pebble makes them even.\n- **Manage your kazan lead, not just the current sow** — Toguz Kumalak is a pebble-majority game; favour moves that increase your kazan count even if they forgo an immediate capture, as small steady gains compound.\n- **Keep large pits for long-range sowing** — a pit with many pebbles lets you sow past the midpoint and into the opponent's half of the board, threatening captures or enabling tuzdyk conditions that a short sow cannot reach.\n- **Prevent the opponent's tuzdyk** — the conditions for claiming a tuzdyk are specific (landing with an odd number ≥ 3 in a pit not directly opposite your own tuzdyk); track when the opponent is one sow away from meeting those conditions and disrupt it."}, {"id": 1493, "type": "game", "source": "toguz-kumalak", "section": "Engines & current best play", "text": "Toguz Kumalak — Engines & current best play\n\n- **Strongest known program(s):** Competition engines developed for Central Asian Toguz Kumalak tournaments (no widely available open-source implementation known to the cataloguer).\n- **Strength:** Competitive with strong human players; used for national-level training in Kazakhstan and Kyrgyzstan.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Toguz Kumalak is an official sport in Kazakhstan and Kyrgyzstan; competitive play has a well-developed opening and endgame theory, though no formal game-theoretic solution has been published."}, {"id": 1494, "type": "game", "source": "toguz-kumalak", "section": "Complexity", "text": "Toguz Kumalak — Complexity\n\nLarge."}, {"id": 1495, "type": "game", "source": "toguz-kumalak", "section": "References", "text": "Toguz Kumalak — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Toguz_korgool) ([archive](http://web.archive.org/web/20260113231550/https://en.wikipedia.org/wiki/Toguz_korgool))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1496, "type": "game", "source": "toguz-kumalak", "section": "See also", "text": "Toguz Kumalak — See also\n\n- [Awari](awari.md) · [Bao](bao.md) · [Kalah](kalah.md) · [Oware](awari.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1497, "type": "game", "source": "toppling-dominoes", "section": "overview", "text": "Toppling Dominoes\nA row-of-dominoes game whose values cover a clean range of switches and\nSolution status: Strongly solved as a theory. Game-theoretic value: Position-dependent (computable closed forms for many rows). Players: 2. Type: Partisan combinatorial game."}, {"id": 1498, "type": "game", "source": "toppling-dominoes", "section": "Description", "text": "Toppling Dominoes — Description\n\nA simple partisan game played on a row of coloured dominoes. The CGT analysis\nyields a value structure of \"switches\" and \"integers\" that demonstrates several\ncore ideas — atomic weight, temperature, and the values of \"hot\" games — in\nminiature."}, {"id": 1499, "type": "game", "source": "toppling-dominoes", "section": "Rules", "text": "Toppling Dominoes — Rules\n\n1. A row of dominoes, each coloured **blue (L)**, **red (R)**, or **green\n   (either)**.\n2. **Left** moves: pick a blue or green domino and topple it **left** or\n   **right** — toppling a domino removes it together with every domino that\n   would fall in the chosen direction (contiguous tiles in that direction).\n3. **Right** moves: pick a red or green domino and topple it left or right.\n4. The player unable to move loses (normal play)."}, {"id": 1500, "type": "game", "source": "toppling-dominoes", "section": "Solution status", "text": "Toppling Dominoes — Solution status\n\nStrongly solved as a theory. For a single row, value formulas in CGT are known\nfor the basic colour patterns; sums of independent rows add by ordinary game\narithmetic. The game is featured prominently in introductory CGT texts because\nits value computations stay tractable while exhibiting nontrivial structure\n(switches, atomic-weight analysis)."}, {"id": 1501, "type": "game", "source": "toppling-dominoes", "section": "Consensus on optimal play", "text": "Toppling Dominoes — Consensus on optimal play\n\n- **Compute each row's CGT value independently** — Toppling Dominoes is a disjunctive sum; evaluate each separate row segment, then sum the values and use standard CGT move selection.\n- **Play the hottest component first** — in a multi-row game, the row with the highest temperature gives the largest advantage to whichever player moves in it; always respond to the opponent's hot move in the same-hot or next-hottest row.\n- **Toppling left vs. right changes which pieces remain** — the direction of topple determines which dominoes are eliminated; choose the direction that leaves a row with the most favourable remaining value for you.\n- **Green (either-player) dominoes are often the key** — green dominoes can be toppled by either side; a green domino sitting between large blue and red blocks can swing the game; contest or use them before pure-colour dominoes.\n- **Switches favour the player who moves in them last** — a row that is a \"switch\" (value {a | b} with a ≠ b) favours the player who gets the last topple there; count the parity of remaining moves in each switch row to decide whether to enter it now or wait."}, {"id": 1502, "type": "game", "source": "toppling-dominoes", "section": "Engines & current best play", "text": "Toppling Dominoes — Engines & current best play\n\n- **Strongest known program(s):** CGT software (e.g., Combinatorial Game Suite) — closed-form value formulas for standard colour patterns.\n- **Strength:** Theoretically solved; any CGT-aware program plays perfectly.\n- **Where the proof / tablebase lives (if solved):** *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)) and *On Numbers and Games* ([../references.md#conway1976](../references.md#conway1976)).\n- **Notes:** Toppling Dominoes is primarily a CGT teaching vehicle; its tractable value structure makes it the standard example of hot games and switch analysis in introductory courses."}, {"id": 1503, "type": "game", "source": "toppling-dominoes", "section": "Complexity", "text": "Toppling Dominoes — Complexity\n\nPolynomial in row length for the standard rules — a major reason it is used as\na teaching game."}, {"id": 1504, "type": "game", "source": "toppling-dominoes", "section": "References", "text": "Toppling Dominoes — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Combinatorial_game_theory) ([archive](http://web.archive.org/web/20260508023449/https://en.wikipedia.org/wiki/Combinatorial_game_theory))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Conway (1976). *On Numbers and Games*.](../references.md#conway1976)"}, {"id": 1505, "type": "game", "source": "toppling-dominoes", "section": "See also", "text": "Toppling Dominoes — See also\n\n- [Push](push.md) · [Shove](shove.md) · [Domineering](domineering.md)\n- Lexicon: [temperature / hot game](../lexicon/README.md#temperature--hot-game) · [surreal number](../lexicon/README.md#surreal-number)"}, {"id": 1506, "type": "game", "source": "tower-of-hanoi", "section": "overview", "text": "Tower of Hanoi\nThe classic recursive disc-stacking puzzle — fully solved.\nSolution status: Strongly solved. Game-theoretic value: Always solvable in 2^n − 1 moves. Players: 1. Type: Solo puzzle."}, {"id": 1507, "type": "game", "source": "tower-of-hanoi", "section": "Description", "text": "Tower of Hanoi — Description\n\nThe Tower of Hanoi (Édouard Lucas, 1883) is the canonical recursion puzzle:\nmove a stack of *n* discs from one peg to another, never placing a larger\ndisc on a smaller. Its minimum solution length is exactly 2^n − 1 moves."}, {"id": 1508, "type": "game", "source": "tower-of-hanoi", "section": "Rules", "text": "Tower of Hanoi — Rules\n\n1. Three pegs labelled A, B, C. *n* discs of distinct sizes start stacked on\n   peg A, largest at the bottom.\n2. On each move the player transfers the top disc of one peg to another peg.\n3. A disc may never be placed on top of a smaller disc.\n4. The puzzle is solved when all *n* discs are stacked, in order, on peg C."}, {"id": 1509, "type": "game", "source": "tower-of-hanoi", "section": "Solution status", "text": "Tower of Hanoi — Solution status\n\n**Strongly solved**: the minimum number of moves is 2^n − 1, and the optimal\nstrategy is well known (recursive: move the top n−1 discs to the spare peg,\nmove disc n to the goal peg, then move the n−1 discs onto it)."}, {"id": 1510, "type": "game", "source": "tower-of-hanoi", "section": "Consensus on optimal play", "text": "Tower of Hanoi — Consensus on optimal play\n\n- **Use the recursive algorithm** — to move n discs from A to C using B as spare: (1) move n−1 discs from A to B, (2) move disc n from A to C, (3) move n−1 discs from B to C. This exactly 2^n − 1-move strategy is provably optimal.\n- **Odd-numbered discs move to the goal peg, even-numbered to the spare** — for three pegs the smallest disc cycles A→C→B→A (or the reverse); knowing the peg-cycle for your disc size eliminates guesswork on each step.\n- **The iterative rule: alternate moving the smallest disc with the unique legal move for all other discs** — on every odd step move the smallest disc one position in its cycle direction; on every even step make the only non-smallest-disc move available. This produces the optimal solution without recursion.\n- **State encodes position** — represent each disc's peg as a ternary digit; the state number from 0 to 3^n − 1 maps exactly to a position, and the move sequence corresponds to reflected Gray-code counting.\n- **Four-peg (Frame–Stewart) variant is harder** — with an extra peg the optimal number of moves is still an open problem in general; the Frame–Stewart algorithm gives the best known solution but its optimality is unproven beyond small n."}, {"id": 1511, "type": "game", "source": "tower-of-hanoi", "section": "Engines & current best play", "text": "Tower of Hanoi — Engines & current best play\n\n- **Strongest known program(s):** Any correct recursive or iterative implementation — the optimal algorithm is closed-form.\n- **Strength:** Perfectly solved; the exact minimum-move solution is always achievable.\n- **Where the proof / tablebase lives (if solved):** Lucas (1883); see also [Wikipedia](https://en.wikipedia.org/wiki/Tower_of_Hanoi) and [../references.md#gardner-tower-of-hanoi](../references.md#gardner-tower-of-hanoi).\n- **Notes:** The standard three-peg puzzle is the textbook recursion example; the four-peg generalisation (Frame–Stewart problem) remains an open problem in combinatorics."}, {"id": 1512, "type": "game", "source": "tower-of-hanoi", "section": "Complexity", "text": "Tower of Hanoi — Complexity\n\nLinear in n for description; exponential in n for total move count."}, {"id": 1513, "type": "game", "source": "tower-of-hanoi", "section": "References", "text": "Tower of Hanoi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tower_of_Hanoi) ([archive](http://web.archive.org/web/20260508043356/https://en.wikipedia.org/wiki/Tower_of_Hanoi))\n- [Gardner. *Mathematical Recreations* (various editions on Hanoi).](../references.md#gardner-tower-of-hanoi)"}, {"id": 1514, "type": "game", "source": "tower-of-hanoi", "section": "See also", "text": "Tower of Hanoi — See also\n\n- [Klotski](klotski.md) · [Sokoban](sokoban.md) · [Pocket Cube](pocket-cube.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 1515, "type": "game", "source": "treblecross", "section": "overview", "text": "Treblecross\nA one-dimensional \"three-in-a-row\" game that is secretly an impartial octal\nSolution status: Strongly solved. Game-theoretic value: Depends on strip length; determined by nim-values. Players: 2. Type: Impartial combinatorial game (octal game 0.007)."}, {"id": 1516, "type": "game", "source": "treblecross", "section": "Description", "text": "Treblecross — Description\n\nPlayed on a 1×*n* strip of cells. **Both** players mark cells with the same\nsymbol (an X). A player who completes three consecutive X's *wins immediately*.\nBecause the players share the marking symbol, the game is\n[impartial](../lexicon/README.md#impartial-game): the moves available depend\nonly on the position, not on whose turn it is."}, {"id": 1517, "type": "game", "source": "treblecross", "section": "Solution status", "text": "Treblecross — Solution status\n\nTreblecross is **strongly solved**. Reformulated as a take-and-break game it is\nthe octal game **0.007**, and [Guy & Smith (1956)](../references.md#guy-smith1956)\nshowed its [nim-value](../lexicon/README.md#nim-value) sequence is eventually\nperiodic. Hence the winner of a strip of any length, and the winning move, is\nknown. Positions composed of several independent segments resolve by\n[nim-sum](../lexicon/README.md#nim-sum).\n\nIt is a tidy example of how a \"make three in a row\" game — which sounds like a\n[k-in-a-row](gomoku.md) Maker game — collapses to standard impartial theory once\nboth players use the same symbol."}, {"id": 1518, "type": "game", "source": "treblecross", "section": "Consensus on optimal play", "text": "Treblecross — Consensus on optimal play\n\n- **Consult the nim-value table** — the nim-value sequence for strip segments is eventually periodic (period 34); look up the nim-value for each independent segment, XOR them all, and move to make the total XOR equal to zero.\n- **Never fill the third cell of three adjacent marked cells yourself** — completing three-in-a-row wins for you, so equally it means you must not place a mark that gives your opponent a winning three-in-a-row next turn.\n- **Leave nim-value-zero segments for the opponent** — a segment of length n with nim-value 0 is a losing position for the player to move; whenever possible, transfer control so your opponent must act in a nim-zero segment.\n- **Multiple segments combine by nim-sum** — when the strip has been split into several independent marked fragments, XOR the nim-values; a non-zero XOR means the player to move wins, and the winning move is the one that makes the XOR zero.\n- **Small strips (n ≤ 4) are trivially losing for the mover** — on strips of length 1, 2, or 4 any mark risks completing or setting up three-in-a-row; the correct response often fills the position that minimises the opponent's threat."}, {"id": 1519, "type": "game", "source": "treblecross", "section": "Engines & current best play", "text": "Treblecross — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; any implementation of the Guy–Smith nim-value table plays perfectly.\n- **Strength:** Perfectly solved via the periodic nim-value sequence.\n- **Where the proof / tablebase lives (if solved):** Guy & Smith (1956) ([../references.md#guy-smith1956](../references.md#guy-smith1956)); see also *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** Treblecross is the standard example of how a \"make three-in-a-row\" game becomes a tractable impartial game once both players use the same symbol."}, {"id": 1520, "type": "game", "source": "treblecross", "section": "Complexity", "text": "Treblecross — Complexity\n\nPer-position analysis is linear in the number of independent segments once the\nperiodic table is known."}, {"id": 1521, "type": "game", "source": "treblecross", "section": "References", "text": "Treblecross — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Treblecross)\n- [Guy, R. K. & Smith, C. A. B. (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1522, "type": "game", "source": "treblecross", "section": "See also", "text": "Treblecross — See also\n\n- [Kayles](kayles.md) · [Dawson's chess](dawsons-chess.md) · [Notakto](notakto.md) · [Gomoku](gomoku.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [impartial game](../lexicon/README.md#impartial-game)"}, {"id": 1523, "type": "game", "source": "tribolo", "section": "overview", "text": "Tribolo\nA three-player area-capture game on a hex grid — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown (multi-player; no unique value). Players: 3. Type: Partisan multi-player placement game."}, {"id": 1524, "type": "game", "source": "tribolo", "section": "Description", "text": "Tribolo — Description\n\nTribolo (Christian Freeling, 1980s/90s) is one of the few well-known three-\nplayer abstract games with Othello-like capture mechanics: stones flip when\nsurrounded, and the player who has the most stones at the end wins. As a\nmulti-player game it lies outside the usual two-player solving framework —\n\"optimal play\" requires non-equilibrium reasoning about alliances and\nthreats."}, {"id": 1525, "type": "game", "source": "tribolo", "section": "Rules", "text": "Tribolo — Rules\n\n1. Board: hexagonal grid of cells.\n2. Three players take turns in fixed order, placing a stone of their colour on\n   an empty cell adjacent to at least one stone of a different colour. (Exact\n   placement and flipping rules vary by sub-variant — **[verify]**.)\n3. When a stone is placed, neighbouring runs of one opposing colour bracketed\n   by the placer and one other colour are flipped to the placer's colour, in\n   an Othello-style sandwich rule generalised to three colours.\n4. When no legal moves remain for any player, the game ends and the player\n   with the most stones on the board wins."}, {"id": 1526, "type": "game", "source": "tribolo", "section": "Solution status", "text": "Tribolo — Solution status\n\nTribolo is **not solved**. Three-player solving lacks a unique game-theoretic\nvalue in the standard sense; the analysis is best framed as a search for Nash\nequilibria, of which there may be many. No published solution exists."}, {"id": 1527, "type": "game", "source": "tribolo", "section": "Consensus on optimal play", "text": "Tribolo — Consensus on optimal play\n\n- **Avoid leaving a clear leader unopposed** — in any three-player game, if one player pulls ahead while the other two compete, the leader usually wins; target the current leader's stones with your placements to maintain a balanced stone count.\n- **Create chains along multiple flanks** — placing a stone that can bracket long runs in more than one direction simultaneously is stronger than a one-direction flip; multi-directional flips maximise your stone gain per move.\n- **Be the kingmaker only under duress** — if you genuinely cannot win, the next goal is deciding which of the remaining players wins; flipping stones toward the weaker opponent is a form of tactical control even in apparent defeat.\n- **Control the centre of the hex grid** — central cells on a hex board adjoin more cells, so centre stones can be brackets for flips across multiple hex directions; peripheral stones can only bracket in fewer directions.\n- **Protect large clusters by caging them** — a group of your stones entirely surrounded by your own stones (so no opponent can bracket them) is safe from flipping; building enclosed territories early secures a floor on your score."}, {"id": 1528, "type": "game", "source": "tribolo", "section": "Engines & current best play", "text": "Tribolo — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Three-player games resist standard game-theoretic solving because \"optimal\" requires specifying a solution concept (e.g., Nash equilibrium or maximin); no published analysis exists for Tribolo."}, {"id": 1529, "type": "game", "source": "tribolo", "section": "Complexity", "text": "Tribolo — Complexity\n\nModerate."}, {"id": 1530, "type": "game", "source": "tribolo", "section": "References", "text": "Tribolo — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tribolo)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1531, "type": "game", "source": "tribolo", "section": "See also", "text": "Tribolo — See also\n\n- [Othello](othello.md) · [Quixo](quixo.md)\n- Lexicon: [Nash equilibrium](../lexicon/README.md#nash-equilibrium) · [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1532, "type": "game", "source": "tribonacci-nim", "section": "overview", "text": "Tribonacci Nim\nA Fibonacci-Nim variant where each move is bounded by *three times* the\nSolution status: Strongly solved. Game-theoretic value: P-positions characterised by tribonacci representations. Players: 2. Type: Impartial subtraction game."}, {"id": 1533, "type": "game", "source": "tribonacci-nim", "section": "Description", "text": "Tribonacci Nim — Description\n\nA generalisation of [Fibonacci Nim](fibonacci-nim.md) in which the take-size\nrestriction is loosened. Where Fibonacci Nim's optimal P-positions sit on\nFibonacci numbers and use Zeckendorf representations, Tribonacci Nim's sit on\n**tribonacci** numbers (the recurrence T(n) = T(n−1) + T(n−2) + T(n−3))."}, {"id": 1534, "type": "game", "source": "tribonacci-nim", "section": "Rules", "text": "Tribonacci Nim — Rules\n\n1. A single heap of n tokens. The first move may remove any positive number\n   strictly less than n.\n2. Each subsequent move may remove at most **three times** the number of tokens\n   just removed by the opponent, but at least one and not more than the heap\n   currently contains.\n3. The player who takes the last token wins (normal play)."}, {"id": 1535, "type": "game", "source": "tribonacci-nim", "section": "Solution status", "text": "Tribonacci Nim — Solution status\n\nStrongly solved by Sprague–Grundy analysis: the **P-positions** are exactly the\ntribonacci numbers, and from any N-position the unique winning move is to\nremove the smallest tribonacci number in the \"tribonacci representation\"\n(analogue of Zeckendorf) of the heap. The proof is the same induction structure\nas for Fibonacci Nim."}, {"id": 1536, "type": "game", "source": "tribonacci-nim", "section": "Consensus on optimal play", "text": "Tribonacci Nim — Consensus on optimal play\n\n- **Identify if the heap is a tribonacci number** — the tribonacci sequence is 1, 1, 2, 4, 7, 13, 24, 44, …; if the current heap equals a tribonacci number you are in a losing (P-) position with best play by your opponent, so choose a move that forces a tribonacci-number heap.\n- **Use the tribonacci (Zeckendorf-like) representation** — write n as a sum of distinct tribonacci numbers using the greedy algorithm; the winning move is to remove the *smallest* summand in that representation.\n- **Respect the \"at most 3×\" constraint** — the winning move from an N-position (heap not a tribonacci number) is always small enough to satisfy the constraint; verify that your chosen removal does not exceed three times your opponent's last move.\n- **Limit the opponent's range by removing small amounts** — removing k tokens lets your opponent remove up to 3k; when you are forced to take from a P-position, take as few as possible (1 token) to limit your opponent's reply range.\n- **First move is unrestricted** — on the very first move any amount from 1 to n−1 is legal; always identify whether n is itself a tribonacci number before making the opening move."}, {"id": 1537, "type": "game", "source": "tribonacci-nim", "section": "Engines & current best play", "text": "Tribonacci Nim — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; the winning strategy is a closed-form computation in O(log n) via the tribonacci representation.\n- **Strength:** Perfectly solved; a correct implementation wins from any N-position in a single computed move.\n- **Where the proof / tablebase lives (if solved):** Follows by induction analogous to Whinihan's Fibonacci Nim proof; referenced in the *Winning Ways* framework ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** Tribonacci Nim is the k=3 case of a general family where the move bound is k times the previous move and P-positions are k-bonacci numbers."}, {"id": 1538, "type": "game", "source": "tribonacci-nim", "section": "Complexity", "text": "Tribonacci Nim — Complexity\n\nPolynomial in log n; trivially solvable on paper for n up to thousands."}, {"id": 1539, "type": "game", "source": "tribonacci-nim", "section": "References", "text": "Tribonacci Nim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001) (general framework)\n- [Sprague (1935). *Über mathematische Kampfspiele*.](../references.md#sprague1935)"}, {"id": 1540, "type": "game", "source": "tribonacci-nim", "section": "See also", "text": "Tribonacci Nim — See also\n\n- [Fibonacci Nim](fibonacci-nim.md) · [Nim](nim.md) · [Wythoff's game](wythoffs-game.md)\n- Lexicon: [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 1541, "type": "game", "source": "triplets", "section": "overview", "text": "Triplets\nAn impartial three-pile subtraction game related to ternary nimbers — solved\nSolution status: Strongly solved. Game-theoretic value: Determined by ternary-based nim-value formula. Players: 2. Type: Impartial subtraction game."}, {"id": 1542, "type": "game", "source": "triplets", "section": "Description", "text": "Triplets — Description\n\nTriplets is one of the take-and-break games catalogued in the\nSprague–Grundy / octal-game literature. Its rules deliberately mix three heaps\nto produce a nim-sequence with ternary structure."}, {"id": 1543, "type": "game", "source": "triplets", "section": "Rules", "text": "Triplets — Rules\n\n1. Three heaps of tokens.\n2. A move chooses one heap and removes one, two, or three tokens (so this is\n   the \"Subtract {1,2,3}\" rule), with the additional constraint inherent to\n   \"Triplets\" that **all three heaps must be touched** by some move under the\n   chosen rule code. (Different sources define Triplets slightly differently —\n   the octal-game tables give the canonical version. **[verify]** the exact\n   variant.)\n3. The player who cannot move loses (normal play)."}, {"id": 1544, "type": "game", "source": "triplets", "section": "Solution status", "text": "Triplets — Solution status\n\nStrongly solved by [Guy & Smith (1956)](../references.md#guy-smith1956): the\nsingle-heap nim-values follow a short period-3 cycle, and the multi-heap\nposition is decided by the [nim-sum](../lexicon/README.md#nim-sum) of pile\nnim-values."}, {"id": 1545, "type": "game", "source": "triplets", "section": "Consensus on optimal play", "text": "Triplets — Consensus on optimal play\n\n- **Look up the nim-value for each pile** — the single-pile nim-values follow a period-3 cycle; determine the cycle phase for each pile's size with a simple modular computation, then read off the nim-value from the short table.\n- **XOR the nim-values (nim-sum)** — combine the three piles' nim-values with bitwise XOR; if the result is non-zero you are in a winning position and a winning move exists.\n- **Make the nim-sum zero** — find a pile whose nim-value you can reduce to bring the total XOR to zero; this is always possible from an N-position and gives the unique (or one of several) optimal moves.\n- **Do not leave a nim-sum of zero** — handing the opponent a position with XOR = 0 is the only error; avoid it on every move.\n- **In period-3 nim, a pile of size divisible by 3 has nim-value 0** — such a pile contributes nothing to the XOR and can be safely ignored when looking for the winning move in the other piles."}, {"id": 1546, "type": "game", "source": "triplets", "section": "Engines & current best play", "text": "Triplets — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; the strategy is a constant-time table lookup plus XOR.\n- **Strength:** Perfectly solved by the Guy–Smith nim-value table.\n- **Where the proof / tablebase lives (if solved):** Guy & Smith (1956) ([../references.md#guy-smith1956](../references.md#guy-smith1956)); *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** The exact rule variant of \"Triplets\" varies across sources; verify the octal-game code before applying these tables."}, {"id": 1547, "type": "game", "source": "triplets", "section": "Complexity", "text": "Triplets — Complexity\n\nTrivial: O(1) per pile to look up the nim-value."}, {"id": 1548, "type": "game", "source": "triplets", "section": "References", "text": "Triplets — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Combinatorial_game_theory) ([archive](http://web.archive.org/web/20260508023449/https://en.wikipedia.org/wiki/Combinatorial_game_theory))\n- [Guy & Smith (1956). *The G-values of various games*.](../references.md#guy-smith1956)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1549, "type": "game", "source": "triplets", "section": "See also", "text": "Triplets — See also\n\n- [Kayles](kayles.md) · [Dawson's chess](dawsons-chess.md) · [Nim](nim.md)\n- Lexicon: [octal game](../lexicon/README.md#octal-game) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)"}, {"id": 1550, "type": "game", "source": "turkish-draughts", "section": "overview", "text": "Turkish draughts\nOrthogonal draughts where pieces move along ranks and files — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan draughts."}, {"id": 1551, "type": "game", "source": "turkish-draughts", "section": "Description", "text": "Turkish draughts — Description\n\nTurkish draughts is played on an 8×8 board where pieces move **orthogonally**\nrather than diagonally. The starting position fills the 2nd and 3rd ranks\nwith 16 men per side, and kings are long-range orthogonal movers."}, {"id": 1552, "type": "game", "source": "turkish-draughts", "section": "Rules", "text": "Turkish draughts — Rules\n\n1. Board: 8×8. Each side has 16 men on the 2nd and 3rd ranks (or 7th and 6th).\n2. Men move one square forward or sideways (never backward, never diagonally).\n3. Men capture by jumping an adjacent enemy piece **forward or sideways** to\n   the next empty square; captures are mandatory.\n4. Multiple captures chain; the player must take the maximum number of pieces.\n5. A man reaching the far rank becomes a **king**, which moves and captures\n   any number of squares along a rank or file in one move (like a rook).\n6. A side with no pieces or no legal moves loses; a single king vs. a single\n   king is drawn."}, {"id": 1553, "type": "game", "source": "turkish-draughts", "section": "Solution status", "text": "Turkish draughts — Solution status\n\nTurkish draughts is **not solved**. The orthogonal geometry makes its\nstate-graph distinct from diagonal draughts variants."}, {"id": 1554, "type": "game", "source": "turkish-draughts", "section": "Consensus on optimal play", "text": "Turkish draughts — Consensus on optimal play\n\n- **Trade men for kings aggressively** — a king in Turkish draughts is vastly more powerful than a man (full rook movement); accepting an unfavourable man trade to promote is usually correct if the resulting king cannot be quickly captured.\n- **Mandatory multi-capture is the dominant tactic** — a chain capture that removes two or three men is almost always better than a positional move; position pieces to enable or threaten long chains, and watch for your opponent setting a trap with an apparent capture bait.\n- **Occupy the centre ranks early** — controlling ranks 4 and 5 with men allows forward-and-sideways threats in multiple directions; edge men are hemmed in and contribute to fewer captures.\n- **Never allow a sideways blockade** — because men can move sideways, a wall of men along a rank can be attacked from both sides; spread your men slightly to avoid a single orthogonal sweep stripping an entire rank.\n- **King vs. king endings often draw** — a lone king against a lone king is explicitly drawn; if losing material, steer for a single-king-vs.-single-king endgame to secure a half-point."}, {"id": 1555, "type": "game", "source": "turkish-draughts", "section": "Engines & current best play", "text": "Turkish draughts — Engines & current best play\n\n- **Strongest known program(s):** No widely-known open-source engine dedicated to Turkish draughts; some general draughts engines and online platforms (e.g., on Turkish gaming sites) include it.\n- **Strength:** Competitive with strong amateur players; no super-human benchmarked program is publicly documented.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Turkish draughts is the primary draughts variant in Turkey and parts of the Middle East; its orthogonal geometry makes it geometrically distinct from diagonal variants like English draughts."}, {"id": 1556, "type": "game", "source": "turkish-draughts", "section": "Complexity", "text": "Turkish draughts — Complexity\n\nSimilar to English draughts."}, {"id": 1557, "type": "game", "source": "turkish-draughts", "section": "References", "text": "Turkish draughts — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Turkish_draughts) ([archive](http://web.archive.org/web/20250927084242/https://en.wikipedia.org/wiki/Turkish_draughts))\n- [Schaeffer et al. (2007). *Checkers is Solved*.](../references.md#schaeffer2007) (related)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1558, "type": "game", "source": "turkish-draughts", "section": "See also", "text": "Turkish draughts — See also\n\n- [English draughts](checkers.md) · [Russian draughts](russian-draughts.md) · [Frisian draughts](frisian-draughts.md) · [Italian draughts](italian-draughts.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1559, "type": "game", "source": "turning-turtles", "section": "overview", "text": "Turning Turtles\nThe simplest coin-turning game — and it turns out to be exactly Nim in\nSolution status: Strongly solved. Game-theoretic value: Equivalent to a Nim position; second-player win iff the nim-sum of head positions is 0. Players: 2. Type: Impartial combinatorial game (coin-turning game)."}, {"id": 1560, "type": "game", "source": "turning-turtles", "section": "Description", "text": "Turning Turtles — Description\n\nA row of coins, each heads or tails, positions numbered 1, 2, 3, …. On a turn a\nplayer turns over **one or two** coins, with the constraint that the\n**rightmost coin turned must go from heads to tails**. Under\n[normal play](../lexicon/README.md#normal-play-convention) the last player to\nmove wins."}, {"id": 1561, "type": "game", "source": "turning-turtles", "section": "Solution status", "text": "Turning Turtles — Solution status\n\nTurning Turtles is **strongly solved** — and the solution is especially clean:\nit is **[Nim](nim.md) in disguise**. A single heads coin at position *n* behaves\nexactly like a Nim heap of size *n*, and a whole row is the\n[nim-sum](../lexicon/README.md#nim-sum) of those heaps. So the position is a\nsecond-player win exactly when the XOR of the positions of all heads coins is 0,\nand otherwise the player to move wins by the ordinary Nim rule.\n\nThis makes Turning Turtles the standard worked example of the coin-turning\nframework: it shows that \"turn coins\" games inherit the entire\n[Sprague–Grundy](../lexicon/README.md#sprague-grundy-theorem) theory."}, {"id": 1562, "type": "game", "source": "turning-turtles", "section": "Consensus on optimal play", "text": "Turning Turtles — Consensus on optimal play\n\n- **Treat every heads coin as a Nim heap** — a heads coin at position n is exactly a Nim heap of size n; the entire row is the disjunctive sum of those heaps.\n- **Compute the nim-sum (XOR) of all heads positions** — XOR together the positions of every heads coin; if the result is 0 you are in a losing position, otherwise you are in a winning position.\n- **Win by turning one or two coins to zero-out the XOR** — find a heads coin at position n and a way to turn it (and optionally one earlier coin) such that the XOR of the remaining heads positions becomes 0; this is the unique (or one of the) winning moves.\n- **Flipping two coins can increase or decrease the effective nim-heap** — when you turn a coin from tails to heads (the leftward coin in a two-coin move) you are adding a new heap; use this to reach the target XOR when a single-coin flip cannot.\n- **Opponent must always flip the rightmost coin from heads to tails** — this constraint is the \"Nim heap removal\" analogue; every legal move reduces the position of at least one heads coin, guaranteeing the game terminates."}, {"id": 1563, "type": "game", "source": "turning-turtles", "section": "Engines & current best play", "text": "Turning Turtles — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed; the Nim XOR strategy is a closed-form O(n) computation.\n- **Strength:** Perfectly solved; any implementation of the XOR strategy wins from all N-positions.\n- **Where the proof / tablebase lives (if solved):** *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)); coin-turning games framework by Berlekamp, Conway & Guy.\n- **Notes:** Turning Turtles is the standard introductory example of the coin-turning game equivalence; its one-to-one correspondence with Nim is the cleanest illustration of the Sprague–Grundy theorem in action."}, {"id": 1564, "type": "game", "source": "turning-turtles", "section": "Complexity", "text": "Turning Turtles — Complexity\n\nLinear in the row length to evaluate a position."}, {"id": 1565, "type": "game", "source": "turning-turtles", "section": "References", "text": "Turning Turtles — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Nim) ([archive](http://web.archive.org/web/20260513001624/https://en.wikipedia.org/wiki/Nim))\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)\n- [Bouton, C. L. (1901). *Nim, A Game with a Complete Mathematical Theory*.](../references.md#bouton1901)"}, {"id": 1566, "type": "game", "source": "turning-turtles", "section": "See also", "text": "Turning Turtles — See also\n\n- [Nim](nim.md) · [Mock Turtles](mock-turtles.md) · [Northcott's game](northcotts-game.md)\n- Lexicon: [nim-sum](../lexicon/README.md#nim-sum) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)"}, {"id": 1567, "type": "game", "source": "twelve-mens-morris", "section": "overview", "text": "Twelve Men's Morris\nThe largest classic morris game — the Nine Men's Morris board plus diagonals —\nSolution status: Partially solved / status not firmly established here. Game-theoretic value: Unknown **[verify]**. Players: 2. Type: Partisan placement-and-movement (\"mill\") game."}, {"id": 1568, "type": "game", "source": "twelve-mens-morris", "section": "Description", "text": "Twelve Men's Morris — Description\n\nPlayed on the 24-point [Nine Men's Morris](nine-mens-morris.md) board **with the\nfour diagonal lines added**, so corner points connect across the diagonals. Each\nplayer has twelve pieces. Rules otherwise follow Nine Men's Morris: a placement\nphase, then a movement phase, with mills removing enemy pieces. With twelve\npieces on 24 points the board is very full after placement, so the game often\nhinges sharply on the placement phase."}, {"id": 1569, "type": "game", "source": "twelve-mens-morris", "section": "Solution status", "text": "Twelve Men's Morris — Solution status\n\nTwelve Men's Morris is **not firmly established as solved** in this archive. The\nmethods that weakly solved [Nine Men's Morris](nine-mens-morris.md)\n([Gasser, 1996](../references.md#gasser1996)) — endgame databases plus search —\nare applicable in principle, and the closely related Southern African game\n*Morabaraba* has received computational study. But this archive has not\nconfirmed a citable primary solution of standard Twelve Men's Morris with a\ndefinite game-theoretic value.\n\n> **[verify]** — If Twelve Men's Morris (or Morabaraba under an equivalent rule\n> set) has been weakly or strongly solved, the result and citation should be\n> added."}, {"id": 1570, "type": "game", "source": "twelve-mens-morris", "section": "Consensus on optimal play", "text": "Twelve Men's Morris — Consensus on optimal play\n\n- **Diagonal points are high-value in Twelve Men's Morris** — adding the four diagonals means corner points now participate in three lines (two sides of the square plus one diagonal); securing corners during placement is more important than in Nine Men's Morris.\n- **Placement phase is decisive** — with 12 pieces filling 24 points the board is fully occupied after placement; forming a mill during the placement phase and removing a key opponent piece can decide the game before movement begins.\n- **Close two mills simultaneously if possible** — any mill lets you remove an opponent piece; threatening to close two mills with one placement forces the opponent to choose which to prevent, and you complete the other.\n- **Protect pieces not in a mill** — pieces that are not part of any current or imminent mill can be removed if the opponent closes a mill; keep non-mill pieces in safe positions or use them to block opponent mill formations.\n- **In the movement phase, open and close mills repeatedly** — sliding a piece one step out of a mill, then back, re-closes the mill and earns another removal each cycle; the opponent must disrupt the pattern or face steady piece loss."}, {"id": 1571, "type": "game", "source": "twelve-mens-morris", "section": "Engines & current best play", "text": "Twelve Men's Morris — Engines & current best play\n\n- **Strongest known program(s):** No widely available dedicated Twelve Men's Morris solver known to the cataloguer; the morris-family methodology (endgame databases + alpha-beta search, as used by Gasser for Nine Men's Morris) is applicable.\n- **Strength:** No benchmarked super-human engine publicly documented for this variant.\n- **Where the proof / tablebase lives (if solved):** No confirmed citable solution; see related Gasser (1996) result for Nine Men's Morris ([../references.md#gasser1996](../references.md#gasser1996)).\n- **Notes:** Morabaraba (the Southern African equivalent) has received some computational study; whether its ruleset exactly matches standard Twelve Men's Morris is **[verify]**."}, {"id": 1572, "type": "game", "source": "twelve-mens-morris", "section": "Complexity", "text": "Twelve Men's Morris — Complexity\n\nLarger than Nine Men's Morris's ~10^10 positions, owing to the extra diagonal\nconnections and twelve pieces per side."}, {"id": 1573, "type": "game", "source": "twelve-mens-morris", "section": "References", "text": "Twelve Men's Morris — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Twelve_men%27s_morris) ([archive](http://web.archive.org/web/20251223020816/https://en.wikipedia.org/wiki/Twelve_men%27s_morris))\n- [Gasser, R. (1996). *Solving Nine Men's Morris*.](../references.md#gasser1996)\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1574, "type": "game", "source": "twelve-mens-morris", "section": "See also", "text": "Twelve Men's Morris — See also\n\n- [Nine Men's Morris](nine-mens-morris.md) · [Six Men's Morris](six-mens-morris.md) · [Three Men's Morris](three-mens-morris.md)\n- Lexicon: [retrograde analysis](../lexicon/README.md#retrograde-analysis) · [weakly solved](../lexicon/README.md#weakly-solved)"}, {"id": 1575, "type": "game", "source": "twixt", "section": "overview", "text": "TwixT\nA connection game of pegs and links; popular, elegant, and unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan connection game."}, {"id": 1576, "type": "game", "source": "twixt", "section": "Description", "text": "TwixT — Description\n\nPlayed on a square grid of holes (commonly 24×24). Players alternately place a\npeg of their colour; whenever two of your pegs are a chess-knight's-move apart\nwith no crossing link in the way, you may connect them with a link. Each player\ntries to build a continuous linked chain between their two opposite edges.\nUnlike [Hex](hex.md), **TwixT can be drawn** — links can block each other so\nthat neither side completes a connection."}, {"id": 1577, "type": "game", "source": "twixt", "section": "Solution status", "text": "TwixT — Solution status\n\nTwixT is **unsolved**. Because draws are possible, the\n[strategy-stealing argument](../lexicon/README.md#strategy-stealing) that settles\nHex and Y does **not** directly give TwixT's value — it would only rule out a\nsecond-player win, not establish a first-player win versus a draw. Small boards\nhave been studied and the game has a long history of strong human and computer\nplay, but the standard board's game-theoretic value is not known."}, {"id": 1578, "type": "game", "source": "twixt", "section": "Consensus on optimal play", "text": "TwixT — Consensus on optimal play\n\n- **Use the swap rule to correct for first-mover advantage** — TwixT is typically played with a swap (pie) rule: if the second player considers the first move too strong, they can swap colours; always open with a move you would be happy to defend from either side.\n- **Build diagonal ladders along the 3/4-column** — chains running at a ~45-degree angle are the most space-efficient routes; experienced players route through the 3rd or 4th column from each edge to leave room for defensive detours.\n- **Block by crossing links, not just placing pegs** — a link between two of your pegs permanently blocks any link that would cross it; strategic link placement can cut off the opponent's entire routing corridor without adding a peg directly in their path.\n- **Avoid isolated pegs far from your chain** — a peg not already connected to your chain offers no immediate benefit and requires future moves to incorporate; keep pegs within knight's-move range of your existing links.\n- **Contest the narrow \"bridging\" points** — the grid has certain bottleneck squares where both sides' optimal paths converge; placing a peg at such a pivot forces the opponent to route around, often gaining a column of space.\n- **Draws arise from deadlocked links** — if the midgame produces a fully cut-off corridor for both players, accept a draw; do not weaken your own formation chasing a win that is geometrically impossible."}, {"id": 1579, "type": "game", "source": "twixt", "section": "Engines & current best play", "text": "TwixT — Engines & current best play\n\n- **Strongest known program(s):** Twixt-playing programs (various, including entries in computer-games competitions); Monte Carlo tree search implementations have been developed.\n- **Strength:** Competitive with strong human players; no super-human benchmarked open-source engine is publicly well-known to the cataloguer.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** TwixT is unique among popular connection games in allowing draws via link-blocking; this makes strategy-stealing inapplicable and the game-theoretic value genuinely open."}, {"id": 1580, "type": "game", "source": "twixt", "section": "Complexity", "text": "TwixT — Complexity\n\nLarge; the 24×24 board and link-blocking rules make exhaustive analysis\ninfeasible with current methods."}, {"id": 1581, "type": "game", "source": "twixt", "section": "References", "text": "TwixT — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/TwixT) ([archive](http://web.archive.org/web/20260324115453/https://en.wikipedia.org/wiki/TwixT))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1582, "type": "game", "source": "twixt", "section": "See also", "text": "TwixT — See also\n\n- [Hex](hex.md) · [Y](y.md) · [Havannah](havannah.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing) · [draw](../lexicon/README.md#draw)"}, {"id": 1583, "type": "game", "source": "tzaar", "section": "overview", "text": "TZAAR\nThe combat-focused GIPF-project game with three piece types — widely held\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan capture game."}, {"id": 1584, "type": "game", "source": "tzaar", "section": "Description", "text": "TZAAR — Description\n\nTZAAR (Kris Burm, 2007) is the sixth and arguably deepest GIPF-project game.\nEach player has three types of pieces (TZAARs, TZARRAs, TOTTs) and must protect\n*all three* types from being captured below a threshold while attacking the\nopponent's pieces."}, {"id": 1585, "type": "game", "source": "tzaar", "section": "Rules", "text": "TZAAR — Rules\n\n1. Board: hexagonal grid of 30 cells, pre-filled at game start with all 30\n   pieces (15 per player: 6 TZAARs, 9 TZARRAs, 15 TOTTs **[verify]** the exact\n   counts).\n2. On each turn (after the first), a player makes **two** actions in order:\n   1. **Capture**: move one of your stacks along a straight line of empty\n      cells to a cell occupied by an opposing stack of **equal or smaller\n      total height**, removing the opposing stack and stacking yours on its\n      cell.\n   2. Either **capture again** (same rule) **or** **stack on a friendly\n      piece** (move a stack onto a friendly stack to consolidate).\n3. The first player's first move is a single capture only.\n4. A player **loses** if at the start of their turn they have **zero pieces\n   left of any one type** (TZAAR, TZARRA, or TOTT) — *or* if they cannot\n   capture."}, {"id": 1586, "type": "game", "source": "tzaar", "section": "Solution status", "text": "TZAAR — Solution status\n\nTZAAR is **not solved**. It is widely regarded by abstract-game enthusiasts as\none of the deepest two-player games of the past quarter-century, with strong\nengines (including neural-network players) but no published solution."}, {"id": 1587, "type": "game", "source": "tzaar", "section": "Consensus on optimal play", "text": "TZAAR — Consensus on optimal play\n\n- **Never let any piece type drop to zero** — losing all TZAARs, TZARRAs, or TOTTs is an immediate loss regardless of total piece count; guarding your minority piece type is always the highest priority.\n- **Target the opponent's rarest piece type** — if the opponent has many TOTTs but few TZAARs, relentlessly capture TZAARs; this exploits the loss condition more directly than capturing by strength.\n- **Use stacking to make pieces invulnerable** — tall stacks can only be captured by equally or taller stacks; stack your smallest or most-threatened piece type to price it out of capture range.\n- **Two actions per turn means you can both attack and consolidate** — a strong pattern is to capture an opponent piece on action 1 and then stack two friendly pieces on action 2 to grow a tall defensive stack; this simultaneously reduces the opponent and strengthens your position.\n- **Control the central hexes** — pieces in the centre of the board can threaten in six straight lines, while edge pieces threaten in fewer; central stacks are both more threatening and harder to isolate.\n- **Count piece-type totals before every turn** — the game can flip from winning to lost in a single turn; tracking each player's count of all three types prevents surprises and reveals the opponent's vulnerabilities."}, {"id": 1588, "type": "game", "source": "tzaar", "section": "Engines & current best play", "text": "TZAAR — Engines & current best play\n\n- **Strongest known program(s):** Neural-network-based and MCTS programs have been developed by the abstract-games community; no single well-known open-source engine dominates.\n- **Strength:** Competitive with strong human club players; top engines likely exceed the best human performance.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** TZAAR succeeded ZERTZ in the GIPF project series and is widely considered the most tactically deep entry; the two-action turn structure creates exceptional combinatorial breadth per ply."}, {"id": 1589, "type": "game", "source": "tzaar", "section": "Complexity", "text": "TZAAR — Complexity\n\nLarge."}, {"id": 1590, "type": "game", "source": "tzaar", "section": "References", "text": "TZAAR — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/TZAAR) ([archive](http://web.archive.org/web/20260113143313/https://en.wikipedia.org/wiki/TZAAR))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1591, "type": "game", "source": "tzaar", "section": "See also", "text": "TZAAR — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [YINSH](yinsh.md) · [LYNGK](lyngk.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1592, "type": "game", "source": "ultimate-tic-tac-toe", "section": "overview", "text": "Ultimate tic-tac-toe\nNine tic-tac-toe boards nested inside one — small-looking, but its standard\nSolution status: Unsolved. Game-theoretic value: Unknown (not established by a verified proof). Players: 2. Type: Partisan positional game."}, {"id": 1593, "type": "game", "source": "ultimate-tic-tac-toe", "section": "Description", "text": "Ultimate tic-tac-toe — Description\n\nPlayed on a 3×3 arrangement of nine small [tic-tac-toe](tic-tac-toe.md) boards.\nA move's cell *within* a small board dictates *which* small board the opponent\nmust play in next. Winning a small board claims it; winning three small boards\nin a row wins the game. (Rules vary on what happens when you are sent to an\nalready-decided board.)"}, {"id": 1594, "type": "game", "source": "ultimate-tic-tac-toe", "section": "Solution status", "text": "Ultimate tic-tac-toe — Solution status\n\nUltimate tic-tac-toe is **unsolved**. The forced-board mechanic makes the game\ntree much larger and far less symmetric than it looks, and although the game is\nsmall enough that a determined solving effort is plausible, this archive is not\naware of a **verified, citable** proof of its game-theoretic value. Claims of a\nsolution circulate informally; until a primary source is confirmed, the standard\ngame should be treated as unsolved.\n\n> **[verify]** — If a rigorous solution (with a fixed rule set for\n> already-won boards) has been published, it should be added here."}, {"id": 1595, "type": "game", "source": "ultimate-tic-tac-toe", "section": "Consensus on optimal play", "text": "Ultimate tic-tac-toe — Consensus on optimal play\n\n- **Winning the centre macro-board is the highest strategic goal** — the centre small board participates in all four winning lines (row, column, and both diagonals) of the meta-board; fight hard for it and send the opponent to weak macro-cells when possible.\n- **Send your opponent to already-decided or unfavourable boards** — a move in cell X of the current small board sends the opponent to small board X; send them to boards you have won (they play in the neutral cell) or to boards where they have few good options.\n- **Control local boards with tic-tac-toe principles** — within each small board, take the centre first, corners second, block two-in-a-rows; strong local play is necessary to claim macro-boards.\n- **Use the \"free choice\" rule wisely** — under most rule variants, when you are sent to an already-won or full board you may play anywhere; this is a powerful tempo advantage, so deliberately fill contested boards to earn free-choice turns.\n- **Balance board wins with strategic sends** — winning a small board is only worthwhile if it does not send the opponent to a macro-pivotal board; sometimes deliberately losing a small board is correct to control where the opponent plays next."}, {"id": 1596, "type": "game", "source": "ultimate-tic-tac-toe", "section": "Engines & current best play", "text": "Ultimate tic-tac-toe — Engines & current best play\n\n- **Strongest known program(s):** MCTS-based programs (various informal implementations); online platforms (e.g., BoardGameArena) host AI opponents.\n- **Strength:** Competitive with strong amateur humans; top implementations are likely stronger than most casual players.\n- **Where the proof / tablebase lives (if solved):** No verified citable solution known to the cataloguer; the game-theoretic value is **[verify]**.\n- **Notes:** Informal claims of a first-player win circulate online, but without a published proof with a fixed rule set for already-won boards, the game should be treated as unsolved."}, {"id": 1597, "type": "game", "source": "ultimate-tic-tac-toe", "section": "Complexity", "text": "Ultimate tic-tac-toe — Complexity\n\nMuch larger than tic-tac-toe; exact standardised figures are not well\nestablished in the literature."}, {"id": 1598, "type": "game", "source": "ultimate-tic-tac-toe", "section": "References", "text": "Ultimate tic-tac-toe — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Ultimate_tic-tac-toe) ([archive](http://web.archive.org/web/20260320194944/https://en.wikipedia.org/wiki/Ultimate_tic-tac-toe))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002) (general framework)"}, {"id": 1599, "type": "game", "source": "ultimate-tic-tac-toe", "section": "See also", "text": "Ultimate tic-tac-toe — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Qubic](qubic.md) · [Order and Chaos](order-and-chaos.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1600, "type": "game", "source": "undirected-vertex-geography", "section": "overview", "text": "Undirected Vertex Geography\nThe polynomial-time twin of generalised Geography — solved by maximum\nSolution status: Strongly solved (polynomial algorithm). Game-theoretic value: First-player wins iff every maximum matching covers the start vertex. Players: 2. Type: Impartial graph game."}, {"id": 1601, "type": "game", "source": "undirected-vertex-geography", "section": "Description", "text": "Undirected Vertex Geography — Description\n\nThe undirected analogue of [Generalised Geography](geography.md). Where the\ndirected problem is PSPACE-complete, the undirected version yields to a\nbeautiful **matching argument**."}, {"id": 1602, "type": "game", "source": "undirected-vertex-geography", "section": "Rules", "text": "Undirected Vertex Geography — Rules\n\n1. An undirected graph G and a starting vertex v are given.\n2. A token starts at v. Players alternate moving the token along an edge to an\n   unvisited vertex.\n3. The player unable to move loses (normal play)."}, {"id": 1603, "type": "game", "source": "undirected-vertex-geography", "section": "Solution status", "text": "Undirected Vertex Geography — Solution status\n\nSolved in polynomial time. The classical result (Fraenkel, Scheinerman &\nUllman, 1993; the matching-based characterisation is sometimes credited\nearlier) is:\n\n> The first player wins Undirected Vertex Geography from v iff **every** maximum\n> matching of G covers v.\n\nEquivalently, the first player wins iff v is *essential* for the maximum\nmatching. A winning strategy is to play along edges of a fixed maximum matching\nthat includes v."}, {"id": 1604, "type": "game", "source": "undirected-vertex-geography", "section": "Consensus on optimal play", "text": "Undirected Vertex Geography — Consensus on optimal play\n\n- **Determine whether the start vertex is essential to a maximum matching** — compute any maximum matching of the graph; if the start vertex v is covered by every maximum matching, the first player wins; if some maximum matching leaves v uncovered, the second player wins.\n- **First player: always move along an edge of a fixed maximum matching** — pick a maximum matching M that covers v; on every turn, move the token along an M-edge to its M-matched partner vertex. This strategy guarantees a win by matching-theoretic argument.\n- **Second player: stay off matching edges if possible** — as the second player (in a position where some maximum matching leaves v uncovered), respond to each first-player move by moving along a matching edge in your chosen maximum matching; this ensures you are never stranded.\n- **The key structural insight** — the value of the game is entirely determined by a single maximum-matching computation; no game-tree search is needed beyond that O(V·E) calculation.\n- **Edge direction is the complexity switch** — in the directed version (Generalised Geography) the same problem is PSPACE-complete; the undirected case is solvable in polynomial time, making UVG the standard textbook example of how undirecting edges can collapse game complexity."}, {"id": 1605, "type": "game", "source": "undirected-vertex-geography", "section": "Engines & current best play", "text": "Undirected Vertex Geography — Engines & current best play\n\n- **Strongest known program(s):** Any implementation of Edmonds' blossom maximum-matching algorithm — O(V·E) to determine the winner and optimal first move.\n- **Strength:** Perfectly solved in polynomial time; no game-tree search needed.\n- **Where the proof / tablebase lives (if solved):** Fraenkel, Scheinerman & Ullman (1993); referenced via [../references.md#schaefer1978](../references.md#schaefer1978) framework.\n- **Notes:** UVG is the canonical example of a game whose complexity drops from PSPACE-complete (directed) to polynomial (undirected) simply by removing edge orientation."}, {"id": 1606, "type": "game", "source": "undirected-vertex-geography", "section": "Complexity", "text": "Undirected Vertex Geography — Complexity\n\nPolynomial — a sharp contrast to the PSPACE-complete directed case. UVG is\noften cited as the canonical example of how the **direction of edges** can\nswing a game's complexity from P to PSPACE-complete."}, {"id": 1607, "type": "game", "source": "undirected-vertex-geography", "section": "References", "text": "Undirected Vertex Geography — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Generalized_geography) ([archive](http://web.archive.org/web/20251116130356/https://en.wikipedia.org/wiki/Generalized_geography))\n- [Schaefer (1978). *On the complexity of some two-person perfect-information games*.](../references.md#schaefer1978)"}, {"id": 1608, "type": "game", "source": "undirected-vertex-geography", "section": "See also", "text": "Undirected Vertex Geography — See also\n\n- [Generalized Geography](geography.md) · [Shannon switching game](shannon-switching-game.md)\n- Lexicon: [PSPACE-complete / EXPTIME-complete](../lexicon/README.md#pspace-complete--exptime-complete)"}, {"id": 1609, "type": "game", "source": "unlur", "section": "overview", "text": "Unlur\nAn asymmetric connection game — the two players win in different ways,\nSolution status: Unsolved. Game-theoretic value: Unknown (balanced by bidding / handicap rule). Players: 2 (asymmetric). Type: Partisan asymmetric connection game."}, {"id": 1610, "type": "game", "source": "unlur", "section": "Description", "text": "Unlur — Description\n\nUnlur (Jorge Gómez Arrausi, 2002) is a hexagonal-board connection game in\nwhich **the two players have different victory conditions**. One player (\"Black\")\nwins by connecting opposite sides; the other (\"White\") wins by *preventing*\nthe connection and forming any closed loop of their own stones. A bidding\nopening rule sets the handicap."}, {"id": 1611, "type": "game", "source": "unlur", "section": "Rules", "text": "Unlur — Rules\n\n1. Hexagonal grid board (commonly side 6 or 7).\n2. **Bidding opening**: one player offers a handicap (a number of free Black\n   moves); the other chooses which side to play. This sets up a near-fair\n   starting position.\n3. Players then alternate placing one stone on an empty cell.\n4. **Black** wins by connecting their two opposite sides with a chain of their\n   own stones.\n5. **White** wins by making a closed loop (\"ring\") of their own stones — any\n   cycle that fully encloses one or more cells."}, {"id": 1612, "type": "game", "source": "unlur", "section": "Solution status", "text": "Unlur — Solution status\n\nUnlur is **unsolved**. The asymmetric win conditions and the bidding handicap\nmake the game harder to analyse than ordinary connection games, and there is no\npublished solution."}, {"id": 1613, "type": "game", "source": "unlur", "section": "Consensus on optimal play", "text": "Unlur — Consensus on optimal play\n\n- **Black (connector) must maintain a spanning threat** — as in Hex, Black needs a connection path from side to side; virtual connections (two half-connections sharing a pivot cell) allow Black to advance efficiently and are harder for White to cut.\n- **White (ring-maker) aims for enclosing loops, not just blocking** — White wins by forming any closed cycle, not by preventing Black's connection per se; White should aim for triangular or small hexagonal loops in the centre-board while still disrupting Black's path.\n- **The bidding handicap sets the tempo for the entire game** — in Unlur's bidding opening, assess whether a large Black handicap gives Black too many pre-placed stones; bid the minimum that keeps the position balanced.\n- **Black should route through the central corridor** — the centre of the hexagonal board offers the shortest path from side to side; detouring to the edge to avoid White's pieces usually costs more moves than the detour saves in safety.\n- **White's loop can form anywhere, so defend globally** — Black cannot simply block one cluster; White can build a loop in any corner, so Black must watch the whole board and cut White's forming cycles before they close."}, {"id": 1614, "type": "game", "source": "unlur", "section": "Engines & current best play", "text": "Unlur — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Unlur's asymmetric win conditions (connection vs. enclosing loop) are unusual among connection games and make standard Hex strategies only partially applicable."}, {"id": 1615, "type": "game", "source": "unlur", "section": "Complexity", "text": "Unlur — Complexity\n\nComparable to Hex on similar boards, with the loop condition adding extra\nevaluation work."}, {"id": 1616, "type": "game", "source": "unlur", "section": "References", "text": "Unlur — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Unlur)\n- [Schensted & Titus (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975) (general framework)"}, {"id": 1617, "type": "game", "source": "unlur", "section": "See also", "text": "Unlur — See also\n\n- [Hex](hex.md) · [Havannah](havannah.md) · [Y](y.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [first-player advantage](../lexicon/README.md#first-player-advantage)"}, {"id": 1618, "type": "game", "source": "volo", "section": "overview", "text": "Volo\nFlock-formation game on a hex grid — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement-and-movement game."}, {"id": 1619, "type": "game", "source": "volo", "section": "Description", "text": "Volo — Description\n\nVolo (Nick Bentley) is a placement-and-movement game on a hex grid in which\neach player tries to form the largest connected \"flock\" of their colour by\ncontrolled placements and movement of birds in formation."}, {"id": 1620, "type": "game", "source": "volo", "section": "Rules", "text": "Volo — Rules\n\n1. Board: hexagonal grid (commonly side 5 or 6).\n2. Players take turns. On a turn the player either:\n   - **Places** two stones of their colour onto empty cells; **or**\n   - **Moves** an entire connected group of their stones one step in a chosen\n     direction, provided all destination cells are empty.\n3. The game ends when neither player can place. The player with the largest\n   single connected group wins; ties are broken by total stones on the board."}, {"id": 1621, "type": "game", "source": "volo", "section": "Solution status", "text": "Volo — Solution status\n\nVolo is **not solved**. The combination of placement and group movement makes\nthe branching factor substantial."}, {"id": 1622, "type": "game", "source": "volo", "section": "Consensus on optimal play", "text": "Volo — Consensus on optimal play\n\n- **Grow one large connected flock rather than many small groups** — only the largest single connected group scores; spreading stones into multiple clusters wastes placements that could extend your dominant group.\n- **Use group movement to consolidate isolated pieces** — when you have a small detached cluster near a large group, move the large group one step toward the cluster or vice versa; merging them can dramatically shift the largest-group count.\n- **Block opponent group merges** — placing stones in the gap between two of the opponent's separate groups prevents them from combining into a dominant flock via movement; a single well-placed stone can keep two large enemy groups permanently apart.\n- **Place pairs of stones adjacent to your largest group** — when placing two stones, always put at least one adjacent to your existing largest group to grow it; orphaned pairs in distant corners are hard to integrate later.\n- **Do not move a group into a corner** — moving a flock to the board edge or corner limits its future movement directions and makes it harder to absorb new placements; keep your main group mobile in the centre."}, {"id": 1623, "type": "game", "source": "volo", "section": "Engines & current best play", "text": "Volo — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Volo was designed by Nick Bentley; the combination of dual-placement and full-group-movement creates a distinctive branching factor that makes it more complex than its small board suggests."}, {"id": 1624, "type": "game", "source": "volo", "section": "Complexity", "text": "Volo — Complexity\n\nModerate."}, {"id": 1625, "type": "game", "source": "volo", "section": "References", "text": "Volo — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Volo_(game))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1626, "type": "game", "source": "volo", "section": "See also", "text": "Volo — See also\n\n- [Catchup](catchup.md) · [Hive](hive.md) · [Hex](hex.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1627, "type": "game", "source": "whim", "section": "overview", "text": "Whim\nConway's whimsical variant of Nim: players may, *once each*, swap to the\nSolution status: Strongly solved. Game-theoretic value: First-player win except on a sparse P-set. Players: 2. Type: Impartial game with a convention-switch."}, {"id": 1628, "type": "game", "source": "whim", "section": "Description", "text": "Whim — Description\n\nWhim is Nim plus a single \"whim\" move available once to each player: at any\nturn, instead of removing tokens, a player may declare that the game's ending\nconvention is *reversed* (last-move-wins becomes last-move-loses, or vice\nversa). Whim is the canonical example of how a slight extension of Nim can be\nanalysed without abandoning the Sprague–Grundy framework."}, {"id": 1629, "type": "game", "source": "whim", "section": "Rules", "text": "Whim — Rules\n\n1. Several heaps of tokens, as in Nim.\n2. On your turn either:\n   - Make an ordinary Nim move (remove any positive number from one heap), **or**\n   - If you have not yet used your \"whim,\" declare the **misère switch**:\n     toggle the ending convention. (Each player may do this at most once.)\n3. Under whatever ending convention is active when the last token is taken, that\n   player wins or loses accordingly."}, {"id": 1630, "type": "game", "source": "whim", "section": "Solution status", "text": "Whim — Solution status\n\nStrongly solved by [Berlekamp, Conway & Guy](../references.md#bcg2001). Whim's\nP-positions are essentially those of Nim, but with a careful endgame correction\nwhen only heaps of size 1 remain — exactly the difference between Nim and\nMisère Nim. The analysis adds two state flags (whether each player still has a\n\"whim\" available) and proceeds by ordinary Grundy bookkeeping."}, {"id": 1631, "type": "game", "source": "whim", "section": "Consensus on optimal play", "text": "Whim — Consensus on optimal play\n\n- **Play ordinary Nim (XOR to zero) while heaps are large** — when all heaps are of size ≥ 2, the whim flags are irrelevant to the immediate move; play standard Nim (XOR all heap sizes to zero) and save your whim for the endgame.\n- **Use your whim when heaps shrink to all-1s** — the critical moment is when the position reduces to heaps of size 1 only; at that point the active ending convention determines the winner, so using the whim switch to flip to the favourable convention (or preventing the opponent from doing so) is the entire endgame.\n- **Whoever uses their whim last in the all-1s endgame wins** — in the final all-1s phase, the last whim used sets the final convention; the player with a whim remaining has the decisive move.\n- **Do not waste your whim prematurely** — using the whim switch while large heaps remain is usually wasted: the opponent can simply play the corrected strategy in whatever convention now applies; preserve the whim for maximum value.\n- **Track both players' whim flags as part of the game state** — there are four possible (my-whim, opponent-whim) flag combinations; know which phase you are in to apply the correct strategy."}, {"id": 1632, "type": "game", "source": "whim", "section": "Engines & current best play", "text": "Whim — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; the strategy is an extension of standard Nim computed in the same O(n log n) time as the Sprague–Grundy XOR table.\n- **Strength:** Perfectly solved; any correct implementation wins from all N-positions.\n- **Where the proof / tablebase lives (if solved):** *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** Whim is included in *Winning Ways* as the canonical example of how a single rule-change token can be incorporated into Sprague–Grundy theory without abandoning the XOR framework."}, {"id": 1633, "type": "game", "source": "whim", "section": "Complexity", "text": "Whim — Complexity\n\nSame as Nim — the whim flags add a factor of 4 to the state space."}, {"id": 1634, "type": "game", "source": "whim", "section": "References", "text": "Whim — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Combinatorial_game_theory) ([archive](http://web.archive.org/web/20260508023449/https://en.wikipedia.org/wiki/Combinatorial_game_theory))\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1635, "type": "game", "source": "whim", "section": "See also", "text": "Whim — See also\n\n- [Nim](nim.md) · [Misère Nim](misere-nim.md) · [Poker Nim](poker-nim.md)\n- Lexicon: [misère play](../lexicon/README.md#misere-play) · [Sprague–Grundy theorem](../lexicon/README.md#sprague-grundy-theorem)"}, {"id": 1636, "type": "game", "source": "wild-tic-tac-toe", "section": "overview", "text": "Wild tic-tac-toe\nTic-tac-toe in which either player may play either symbol — easy to solve,\nSolution status: Strongly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan placement game."}, {"id": 1637, "type": "game", "source": "wild-tic-tac-toe", "section": "Description", "text": "Wild tic-tac-toe — Description\n\nA small variant of [tic-tac-toe](tic-tac-toe.md): on every turn the player on\nthe move chooses whether to place an X **or** an O. The first player to\n*complete* a three-in-a-row of either symbol wins."}, {"id": 1638, "type": "game", "source": "wild-tic-tac-toe", "section": "Rules", "text": "Wild tic-tac-toe — Rules\n\n1. Standard 3×3 board, empty initially.\n2. On each turn the player places **either an X or an O** (their choice) on any\n   empty cell.\n3. The first player to **complete a three-in-a-row** of any symbol — even an\n   opponent's — wins. (Variant: misère version in which the player completing a\n   three-in-a-row loses.)\n4. If the board fills with no three-in-a-row, the game is a draw (rare)."}, {"id": 1639, "type": "game", "source": "wild-tic-tac-toe", "section": "Solution status", "text": "Wild tic-tac-toe — Solution status\n\nStrongly solved by trivial exhaustive search. With both players able to pick\nsymbols, the **first player wins** — they can play the centre, then mirror or\nfork to force completion of a line on the next few moves. Misère Wild\ntic-tac-toe (last-to-complete-loses) is also solved by exhaustive search, and\nthe answer is more nuanced; **[verify]** the precise misère value."}, {"id": 1640, "type": "game", "source": "wild-tic-tac-toe", "section": "Consensus on optimal play", "text": "Wild tic-tac-toe — Consensus on optimal play\n\n- **First player takes the centre on move 1** — the centre participates in 4 of the 8 winning lines; combined with the freedom to place either symbol, the first player immediately threatens a line.\n- **Exploit the \"complete your own or opponent's line\" rule** — you can win by finishing a row of O's even if you have been placing X's; keep track of near-complete lines of both symbols and race to complete one.\n- **Fork by creating two near-complete lines simultaneously** — if you can create a position where two three-in-a-rows are each one cell short, your opponent cannot block both; choose the symbol that contributes to both threatened lines.\n- **The defender cannot easily block both symbols at once** — in ordinary tic-tac-toe the defender knows which symbol to block; in Wild tic-tac-toe you can switch symbols each turn, making defensive reasoning far harder for a naive opponent.\n- **Misère Wild tic-tac-toe is genuinely harder** — in the misère variant (completing a row loses), avoid being the one to place the final piece of any three-in-a-row; the strategy inverts and requires careful symbol selection to force the opponent to complete a row."}, {"id": 1641, "type": "game", "source": "wild-tic-tac-toe", "section": "Engines & current best play", "text": "Wild tic-tac-toe — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine; exhaustive search over the tiny state space gives a complete strategy table.\n- **Strength:** Perfectly solved; the first player wins with correct play in the normal convention.\n- **Where the proof / tablebase lives (if solved):** Follows from trivial exhaustive enumeration; discussed in the *Winning Ways* framework ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** The misère variant's precise first/second-player value should be verified — the normal-play first-player win is unambiguous."}, {"id": 1642, "type": "game", "source": "wild-tic-tac-toe", "section": "Complexity", "text": "Wild tic-tac-toe — Complexity\n\nTiny — the entire game tree fits on a small page."}, {"id": 1643, "type": "game", "source": "wild-tic-tac-toe", "section": "References", "text": "Wild tic-tac-toe — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Tic-tac-toe_variants)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001) (general framework)"}, {"id": 1644, "type": "game", "source": "wild-tic-tac-toe", "section": "See also", "text": "Wild tic-tac-toe — See also\n\n- [Tic-tac-toe](tic-tac-toe.md) · [Order and Chaos](order-and-chaos.md) · [Notakto](notakto.md)\n- Lexicon: [strongly solved](../lexicon/README.md#strongly-solved) · [misère play](../lexicon/README.md#misere-play)"}, {"id": 1645, "type": "game", "source": "wolves-and-sheep", "section": "overview", "text": "Wolves and Sheep\nA draughts-style asymmetric racing game on a small chessboard — solvable by\nSolution status: Strongly solved (small board) **[verify]**. Game-theoretic value: Sheep win with correct play **[verify]**. Players: 2 (asymmetric: 1 wolf vs. 4 sheep). Type: Partisan asymmetric racing game."}, {"id": 1646, "type": "game", "source": "wolves-and-sheep", "section": "Description", "text": "Wolves and Sheep — Description\n\nA traditional asymmetric pursuit game, often shown on an 8×8 chessboard with\ndiagonal moves only. One wolf tries to slip past four sheep advancing in a\nphalanx. It is a classic teaching example of how a coordinated weak group beats\na stronger lone piece — like [Fox and Geese](fox-and-geese.md) in miniature."}, {"id": 1647, "type": "game", "source": "wolves-and-sheep", "section": "Rules", "text": "Wolves and Sheep — Rules\n\n1. On the black squares of an 8×8 board, place **4 sheep** on the black squares\n   of one back rank and **1 wolf** on a black square of the opposite back rank.\n2. Pieces move like draughts men: one square diagonally per turn.\n3. Sheep may only move **forward** (toward the wolf's home rank); the wolf may\n   move forward or backward (any diagonal).\n4. The **wolf** wins by reaching the sheep's home rank. The **sheep** win by\n   trapping the wolf so it cannot move.\n5. No captures."}, {"id": 1648, "type": "game", "source": "wolves-and-sheep", "section": "Solution status", "text": "Wolves and Sheep — Solution status\n\nThe game graph is small enough for full enumeration by hand. Standard analyses\nreport that with correct play the **sheep win** — an unbroken advancing line\nleaves the wolf no diagonal lane forward. The result is **[verify]**: it is\nwidely repeated in popular puzzle books but the archive author has not located\na canonical primary citation."}, {"id": 1649, "type": "game", "source": "wolves-and-sheep", "section": "Consensus on optimal play", "text": "Wolves and Sheep — Consensus on optimal play\n\n- **Sheep: advance as an unbroken diagonal wall** — keep all four sheep on adjacent diagonal cells in a single rank-wide line as they march forward; a complete wall with no gaps leaves the wolf no diagonal lane to slip through.\n- **Sheep: never leave a gap of two or more squares in the wall** — the wolf only needs one open diagonal to slip past; even a single two-square gap lets the wolf dodge through before the sheep can close it.\n- **Sheep: advance the sheep nearest to the wolf's lane first** — when the wolf is threading toward a side, advance the sheep that would block that lane rather than the ones already in its path; this closes the escape route with minimum moves.\n- **Wolf: probe for the gap on one side while threatening the other** — the wolf's only hope is to induce a sheep to advance prematurely on one wing, opening a gap on the opposite wing; threaten one side of the sheep line to draw an overreactive advance.\n- **Wolf: avoid the corners** — being pushed into a corner on your own side means the wolf cannot manoeuvre; keep toward the centre diagonals where two escape lanes remain possible."}, {"id": 1650, "type": "game", "source": "wolves-and-sheep", "section": "Engines & current best play", "text": "Wolves and Sheep — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine; the tiny state space is fully exhaustible by hand or any minimax implementation.\n- **Strength:** Perfectly solved by trivial enumeration.\n- **Where the proof / tablebase lives (if solved):** Widely cited in puzzle books as a sheep win; no canonical primary citation located — treat as **[verify]**.\n- **Notes:** Wolves and Sheep is the standard classroom example of how a coordinated defensive phalanx defeats a more mobile lone attacker."}, {"id": 1651, "type": "game", "source": "wolves-and-sheep", "section": "Complexity", "text": "Wolves and Sheep — Complexity\n\nTiny — the full game graph fits in a small table."}, {"id": 1652, "type": "game", "source": "wolves-and-sheep", "section": "References", "text": "Wolves and Sheep — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Wolf_and_sheep)\n- [Berlekamp, Conway & Guy (2001–2004). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001) (general framework; covers Fox and Geese)"}, {"id": 1653, "type": "game", "source": "wolves-and-sheep", "section": "See also", "text": "Wolves and Sheep — See also\n\n- [Fox and Geese](fox-and-geese.md) · [Hare and Hounds](hare-and-hounds.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game) · [strongly solved](../lexicon/README.md#strongly-solved)"}, {"id": 1654, "type": "game", "source": "wythoffs-game", "section": "overview", "text": "Wythoff's game\nA two-heap Nim variant whose losing positions are governed by the golden\nSolution status: Strongly solved. Game-theoretic value: First-player win unless the start is a \"cold\" position. Players: 2. Type: Impartial combinatorial game."}, {"id": 1655, "type": "game", "source": "wythoffs-game", "section": "Description", "text": "Wythoff's game — Description\n\nTwo heaps of objects. On a turn a player either removes any positive number\nfrom one heap, **or** removes the *same* positive number from both heaps. Under\n[normal play](../lexicon/README.md#normal-play-convention) the player taking the\nlast object(s) wins."}, {"id": 1656, "type": "game", "source": "wythoffs-game", "section": "Solution status", "text": "Wythoff's game — Solution status\n\nWythoff's game is **strongly solved**. [Wythoff (1907)](../references.md#wythoff1907)\nshowed that the *P-positions* (previous-player wins — i.e. losses for the player\nto move) are exactly the pairs\n\n  (⌊nφ⌋, ⌊nφ²⌋) for n = 0, 1, 2, …\n\nwhere φ = (1+√5)/2 is the golden ratio. These pairs form a pair of complementary\n[Beatty sequences](https://en.wikipedia.org/wiki/Beatty_sequence). The first\nplayer loses precisely when the starting heaps form such a pair; otherwise a\nmove to the nearest P-position wins."}, {"id": 1657, "type": "game", "source": "wythoffs-game", "section": "Consensus on optimal play", "text": "Wythoff's game — Consensus on optimal play\n\n- **Check if the start is a P-position using the golden-ratio formula** — compute n = min(a, b) if the heaps are (a, b); if (a, b) = (⌊nφ⌋, ⌊nφ²⌋) for some integer n, you are in a losing position; otherwise you are in a winning position.\n- **Win by moving to the nearest P-position** — from any N-position (a,b) there exists at least one move (either reduce one heap, or reduce both by the same amount) that reaches a P-position; find it using the formula and make that move.\n- **Nim XOR does not work here** — Wythoff's game adds the diagonal move (remove equal amounts from both heaps); unlike ordinary Nim, the losing positions are *not* characterised by XOR = 0 and the golden-ratio Beatty-sequence formula is the only clean characterisation.\n- **Diagonal moves close the gap** — if the two heaps differ by Δ and the pair is not already a P-position, the diagonal move can adjust both heaps simultaneously to hit the P-position; this is often the winning move when the heaps are close in size.\n- **Single-heap moves suffice when heaps are very unequal** — when one heap is much larger, reducing only that heap (as in standard Nim) can land directly on the nearest P-position; check single-heap options before computing diagonal moves."}, {"id": 1658, "type": "game", "source": "wythoffs-game", "section": "Engines & current best play", "text": "Wythoff's game — Engines & current best play\n\n- **Strongest known program(s):** No game-specific engine needed; the golden-ratio formula gives the optimal move in O(1) time.\n- **Strength:** Perfectly solved with a closed-form expression.\n- **Where the proof / tablebase lives (if solved):** Wythoff (1907) ([../references.md#wythoff1907](../references.md#wythoff1907)); also in *Winning Ways* ([../references.md#bcg2001](../references.md#bcg2001)).\n- **Notes:** The connection between Wythoff's game and the golden ratio / Beatty sequences makes it one of the most elegant results in combinatorial game theory."}, {"id": 1659, "type": "game", "source": "wythoffs-game", "section": "Complexity", "text": "Wythoff's game — Complexity\n\nA family of positions; optimal play for a given start is computable directly\nfrom the formula."}, {"id": 1660, "type": "game", "source": "wythoffs-game", "section": "References", "text": "Wythoff's game — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Wythoff%27s_game) ([archive](http://web.archive.org/web/20251206112849/https://en.wikipedia.org/wiki/Wythoff%27s_game))\n- [Wythoff, W. A. (1907). *A modification of the game of Nim*.](../references.md#wythoff1907)\n- [Berlekamp, Conway & Guy (2001). *Winning Ways for Your Mathematical Plays*.](../references.md#bcg2001)"}, {"id": 1661, "type": "game", "source": "wythoffs-game", "section": "See also", "text": "Wythoff's game — See also\n\n- [Nim](nim.md) · [Fibonacci Nim](fibonacci-nim.md) · [Subtract-a-square](subtract-a-square.md)\n- Lexicon: [impartial game](../lexicon/README.md#impartial-game) · [nim-value](../lexicon/README.md#nim-value)"}, {"id": 1662, "type": "game", "source": "xiangqi", "section": "overview", "text": "Xiangqi\nChinese chess — comparable in complexity to Western chess, and likewise\nSolution status: Unsolved (some endgame tables computed). Game-theoretic value: Unknown. Players: 2. Type: Partisan board game."}, {"id": 1663, "type": "game", "source": "xiangqi", "section": "Description", "text": "Xiangqi — Description\n\nPlayed on a 9×10 board, with pieces placed on line intersections. Distinctive\nelements include the **river** dividing the board, the **palace** that confines\neach general, and pieces such as the Cannon (which captures by jumping). The\ngenerals may not face each other directly along an open file."}, {"id": 1664, "type": "game", "source": "xiangqi", "section": "Solution status", "text": "Xiangqi — Solution status\n\nXiangqi is **unsolved**. Its state-space (~10^40) and game-tree (~10^150)\ncomplexity are broadly comparable to [chess](chess.md) — far beyond exhaustive\nsearch. As with chess, **endgame tablebases** have been built for material\nconfigurations with few pieces, giving strong (exact) solutions to those\nsub-games, and xiangqi engines play at a superhuman level. But the\ngame-theoretic value of the standard opening position is not known."}, {"id": 1665, "type": "game", "source": "xiangqi", "section": "Consensus on optimal play", "text": "Xiangqi — Consensus on optimal play\n\n- **Cannon forks before crossing the river** — a Cannon on the back rank can pivot to attack along ranks and files using friendly or enemy screens; establishing early cannon pressure (especially targeting the palace) constrains the opponent before they can develop.\n- **Control the river-crossing with Horses** — Horses (which move like a knight but can be blocked) are most effective once they cross the river; the two central river-crossing points are natural staging areas; occupy them to threaten the opponent's back ranks.\n- **Protect the General from \"facing\" (Flying General)** — two generals may not stand on the same open file; always verify that advancing a piece does not open a check via the Flying General rule, and use it offensively to threaten the opponent's General on an open file.\n- **Restrict the opponent's Elephants early** — Elephants (which move exactly two points diagonally and cannot cross the river) are purely defensive; attacking the squares that would block their paths limits the opponent's palace defence.\n- **Palace control wins the endgame** — the 3×3 palace confines each General to only 9 squares; in endgames a Rook supported by a Cannon or Horse in or near the palace is usually decisive; aim to penetrate the palace with a supported piece.\n- **Rooks belong on open files and the palace approach** — as in chess, doubled Rooks on an open central file or aimed at the palace are dominant; connect Rooks as early as possible."}, {"id": 1666, "type": "game", "source": "xiangqi", "section": "Engines & current best play", "text": "Xiangqi — Engines & current best play\n\n- **Strongest known program(s):** ElephantEye, Cyclone, and various Chinese commercial engines — alpha-beta search with neural-network evaluation; top engines are significantly stronger than top human players.\n- **Strength:** Super-human; top engines defeat world champions consistently.\n- **Where the proof / tablebase lives (if solved):** Endgame tablebases for small piece counts are built and used by competitive engines; no full-game solution. See [../lexicon/README.md#endgame-tablebase](../lexicon/README.md#endgame-tablebase).\n- **Notes:** Xiangqi's game-tree complexity (~10¹⁵⁰) is comparable to chess; it is practically unsolvable by exhaustive means for the foreseeable future."}, {"id": 1667, "type": "game", "source": "xiangqi", "section": "Complexity", "text": "Xiangqi — Complexity\n\nState-space ~10^40; game-tree ~10^150\n([van den Herik et al., 2002](../references.md#vandenherik2002)). Generalised\nxiangqi is EXPTIME-complete."}, {"id": 1668, "type": "game", "source": "xiangqi", "section": "References", "text": "Xiangqi — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Xiangqi) ([archive](http://web.archive.org/web/20260429225338/https://en.wikipedia.org/wiki/Xiangqi))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1669, "type": "game", "source": "xiangqi", "section": "See also", "text": "Xiangqi — See also\n\n- [Chess](chess.md) · [Janggi](janggi.md) · [Shogi](shogi.md) · [Makruk](makruk.md)\n- Lexicon: [endgame tablebase](../lexicon/README.md#endgame-tablebase) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1670, "type": "game", "source": "y", "section": "overview", "text": "Y\nA connection game even purer than Hex — every game has a winner, so the first\nSolution status: Ultra-weakly solved (standard board); small boards weakly solved. Game-theoretic value: First-player win. Players: 2. Type: Partisan connection game."}, {"id": 1671, "type": "game", "source": "y", "section": "Description", "text": "Y — Description\n\nPlayed on a triangular board tiled with hexagons, with three sides. Players\nalternately place stones of their colour; a player wins by forming a single\nconnected chain that touches **all three sides** of the triangle. Hex is in fact\na special case of Y played on a corner of the board."}, {"id": 1672, "type": "game", "source": "y", "section": "Solution status", "text": "Y — Solution status\n\nY is **ultra-weakly solved**, by the same logic as [Hex](hex.md):\n\n- **Y cannot be drawn.** A completely filled Y board always contains exactly one\n  winning three-side connection — a fact provable by a neat reduction in which\n  the board \"shrinks\" cell by cell to a single deciding cell.\n- Since there are no draws and the game is symmetric, the\n  [strategy-stealing argument](../lexicon/README.md#strategy-stealing) proves\n  the **first player has a winning strategy** — without exhibiting one.\n\nSo, like Hex, Y's *value* is settled while its *strategy* is not, on full-size\nboards. Small boards are weakly solved by exhaustive search. Y is described in\n[Schensted & Titus's *Mudcrack Y and Poly-Y*](../references.md#schensted-titus1975)."}, {"id": 1673, "type": "game", "source": "y", "section": "Consensus on optimal play", "text": "Y — Consensus on optimal play\n\n- **Aim for the board's centroid, not its centre cell** — unlike Hex, where the exact centre point is on the shortest path between two sides, Y's three-way connection requirement means the ideal \"hub\" is roughly equidistant from all three sides; pieces near the board's centre of mass anchor a spanning structure efficiently.\n- **Virtual connections reduce the number of required moves** — a virtual connection between two stones (a bridge using two pivots that the opponent cannot simultaneously block) effectively extends your chain safely; maintain virtual connections toward all three sides.\n- **All three sides must be reached, so balance your expansion** — focusing on a two-side connection early is wasteful if the third side is unaddressed; ensure your extending stones stay roughly equidistant from all three sides.\n- **Cutting the opponent's bridge is often the best move** — taking the single pivot of an opponent's virtual connection destroys their only clean route to a side; identify these pivots and contest them before the opponent solidifies.\n- **The swap (pie) rule addresses the first-mover advantage** — Y is typically played with a swap rule; open with a stone that you would be content to defend from either side, usually near the centroid."}, {"id": 1674, "type": "game", "source": "y", "section": "Engines & current best play", "text": "Y — Engines & current best play\n\n- **Strongest known program(s):** MoHex (adapted for Y) and other MCTS/neural-network programs for hex-family games; no single dominant open-source engine dedicated specifically to Y is publicly well-known to the cataloguer.\n- **Strength:** Strong amateur to competitive; engines are stronger than most human players.\n- **Where the proof / tablebase lives (if solved):** Ultra-weak solution via strategy-stealing; small boards weakly solved by exhaustive search. See [Schensted & Titus (1975)](../references.md#schensted-titus1975).\n- **Notes:** Y is the \"purest\" connection game — the three-side condition means no pairing arguments based on two-side symmetry apply, yet the no-draw property plus strategy-stealing fully settles the game's value."}, {"id": 1675, "type": "game", "source": "y", "section": "Complexity", "text": "Y — Complexity\n\nComparable to Hex of similar board size; generalised Y is PSPACE-hard."}, {"id": 1676, "type": "game", "source": "y", "section": "References", "text": "Y — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Y_(game))\n- [Schensted, C. & Titus, C. (1975). *Mudcrack Y and Poly-Y*.](../references.md#schensted-titus1975)\n- [Gale, D. (1979). *The Game of Hex and the Brouwer Fixed-Point Theorem*.](../references.md#gale1979)"}, {"id": 1677, "type": "game", "source": "y", "section": "See also", "text": "Y — See also\n\n- [Hex](hex.md) · [Havannah](havannah.md) · [TwixT](twixt.md)\n- Lexicon: [strategy-stealing argument](../lexicon/README.md#strategy-stealing) · [ultra-weakly solved](../lexicon/README.md#ultra-weakly-solved)"}, {"id": 1678, "type": "game", "source": "yahtzee", "section": "overview", "text": "Yahtzee\nThe dice game — solved in the solitaire sense: the strategy maximising\nSolution status: Solved (optimal solitaire strategy computed). Game-theoretic value: N/A (solitaire) — optimal expected score ≈ 254.59. Players: 1+ (the optimal-play result is for the solitaire scoring problem). Type: Stochastic dice game."}, {"id": 1679, "type": "game", "source": "yahtzee", "section": "Description", "text": "Yahtzee — Description\n\nOn a turn a player rolls five dice, may re-roll any subset up to two more times,\nthen must enter a score in one of 13 categories on their scorecard (each used\nexactly once). Categories include the upper section (count of each face value,\nwith a bonus for reaching 63) and the lower section (three/four of a kind, full\nhouse, straights, the 50-point \"Yahtzee,\" and chance). The game involves chance\nbut **no hidden information** — every relevant state is fully observable."}, {"id": 1680, "type": "game", "source": "yahtzee", "section": "Solution status", "text": "Yahtzee — Solution status\n\nYahtzee is **solved** in the relevant sense: as a single-player\nexpected-value-maximisation problem it is a finite **Markov decision process**,\nand the optimal policy can be computed exactly by dynamic programming over the\nscorecard-and-dice state space. [Glenn (2006)](../references.md#glenn-yahtzee2006)\ncomputed the strategy maximising expected total score — about **254.59 points**\nper game — and the policy maximising the probability of beating a fixed target\ncan be computed the same way. The \"solution\" is a large lookup table of optimal\ndecisions, not a game-theoretic value, because solo Yahtzee is an optimisation\nproblem, not an adversarial game."}, {"id": 1681, "type": "game", "source": "yahtzee", "section": "Consensus on optimal play", "text": "Yahtzee — Consensus on optimal play\n\n- **Always aim for the upper-section bonus first** — the 35-point bonus (for scoring ≥ 63 in the upper section) is worth pursuing aggressively; an optimal strategy keeps the upper section on track by accepting slightly suboptimal lower-section plays.\n- **Yahtzee bonus plays are high-leverage** — the 100-point bonus for each additional Yahtzee (after the first) has extremely high expected value; when you have four of a kind on the first roll, keep all five dice on both rerolls to chase it.\n- **Sacrifice chance early** — the \"Chance\" category is a flexible dump for any five dice; preserving it until the last few turns lets you use it as a free safety net when all better categories are used.\n- **Take an upper-section category over zeros when forced** — entering a 1 (e.g., scoring 1 in \"Ones\" to preserve a slot) is usually better than taking zero in a lower-section category; zeros in the upper section forfeit bonus progress permanently.\n- **In multiplayer, shift strategy to target your opponent's score** — the optimal solitaire policy maximises expected score; if you need to beat a specific opponent total, switch to the probability-of-exceeding-target policy, which often differs significantly (e.g., taking higher-variance plays when behind)."}, {"id": 1682, "type": "game", "source": "yahtzee", "section": "Engines & current best play", "text": "Yahtzee — Engines & current best play\n\n- **Strongest known program(s):** Exact dynamic-programming solvers — Glenn (2006) and Verhoeff computed the full optimal-policy lookup table by backward induction over the (scorecard, dice-state, rerolls-remaining) state space.\n- **Strength:** Exactly optimal; expected score ≈ 254.59 points per game.\n- **Where the proof / tablebase lives (if solved):** Glenn (2006) ([../references.md#glenn-yahtzee2006](../references.md#glenn-yahtzee2006)); the full policy table is available from several researchers' websites.\n- **Notes:** Yahtzee is solved as a Markov decision process, not as a two-player adversarial game; the \"solution\" is a decision policy, not a game-theoretic value."}, {"id": 1683, "type": "game", "source": "yahtzee", "section": "Complexity", "text": "Yahtzee — Complexity\n\nThe state space (current scorecard × current dice × rerolls remaining) is large\nbut very much within reach of exact dynamic programming, which is why the optimal\npolicy is known precisely rather than approximated."}, {"id": 1684, "type": "game", "source": "yahtzee", "section": "References", "text": "Yahtzee — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Yahtzee) ([archive](http://web.archive.org/web/20260507114418/https://en.wikipedia.org/wiki/Yahtzee))\n- [Glenn (2006). *An Optimal Strategy for Yahtzee*.](../references.md#glenn-yahtzee2006)"}, {"id": 1685, "type": "game", "source": "yahtzee", "section": "See also", "text": "Yahtzee — See also\n\n- [Klondike solitaire](klondike-solitaire.md) · [Backgammon](backgammon.md) · [Liar's dice](liars-dice.md)\n- Lexicon: [chance element](../lexicon/README.md#chance-element) · [solving vs. strong play](../lexicon/README.md#solving-vs-strong-play)"}, {"id": 1686, "type": "game", "source": "yinsh", "section": "overview", "text": "YINSH\nA flipping-discs connection-and-capture game on a hexagonal board —\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan placement+movement game."}, {"id": 1687, "type": "game", "source": "yinsh", "section": "Description", "text": "YINSH — Description\n\nYINSH (Kris Burm, 2003) is the fourth GIPF-project game. Each player has five\n**rings**; markers placed inside rings flip when jumped over, and rows of five\nsame-coloured markers are removed (along with a ring) for points."}, {"id": 1688, "type": "game", "source": "yinsh", "section": "Rules", "text": "YINSH — Rules\n\n1. Board: 85-cell hexagonal grid.\n2. Each player has 5 rings of their colour and a shared pool of\n   two-sided markers (black on one side, white on the other).\n3. **Setup**: players alternate placing all 10 rings on empty cells.\n4. **Each turn**:\n   1. Place one marker of your colour inside one of your rings.\n   2. Move that ring along a straight line of empty cells, then jumping over\n      any contiguous run of markers, landing on the first empty cell beyond.\n      All markers jumped over are **flipped** to the opposite colour.\n   3. If this move creates **any** five-in-a-row of one colour, the entire row\n      is removed from the board, and the player whose colour matches removes\n      one of their rings (and scores a point).\n5. The first player to remove **3 of their own rings** wins."}, {"id": 1689, "type": "game", "source": "yinsh", "section": "Solution status", "text": "YINSH — Solution status\n\nYINSH is **not solved**. The flipping mechanic gives the game a deep,\nnon-monotone evaluation surface; engines exist but no published solution."}, {"id": 1690, "type": "game", "source": "yinsh", "section": "Consensus on optimal play", "text": "YINSH — Consensus on optimal play\n\n- **Place rings to threaten rows in multiple directions** — a ring on the hex board can project along six lines; position rings so that a single move can create a five-in-a-row along more than one axis, forcing the opponent to decide which threat to block.\n- **Flipping opponent markers is as good as placing your own** — every ring move flips all markers it jumps over; moving a ring through a cluster of opponent markers can instantly convert them to your colour and open or close rows simultaneously.\n- **Sacrifice a ring strategically, not defensively** — removing a ring when you score shrinks your mobility; score with a ring you planned to remove anyway (e.g., a ring that has become hemmed in), preserving your most flexible rings for longer.\n- **Deny opponent five-in-a-rows by disrupting their colour chains** — move a ring through a near-complete opponent row to flip one or more markers and break the sequence; a single flip can invalidate two or three near-wins at once.\n- **The endgame (final ring race) is decisive** — when both players need only one more ring removal to win, every move must either score or prevent the opponent from scoring; tactical calculation in this phase overrides all positional considerations."}, {"id": 1691, "type": "game", "source": "yinsh", "section": "Engines & current best play", "text": "YINSH — Engines & current best play\n\n- **Strongest known program(s):** MCTS and neural-network-based programs (developed within the abstract-games community); no single well-known open-source engine dedicated to YINSH is publicly dominant.\n- **Strength:** Competitive with strong human players; top engines likely exceed most club-level humans.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** YINSH's non-monotone evaluation (flipping reverses piece ownership) makes it particularly challenging for traditional alpha-beta search; neural-network approaches are better suited to its evaluation landscape."}, {"id": 1692, "type": "game", "source": "yinsh", "section": "Complexity", "text": "YINSH — Complexity\n\nLarge."}, {"id": 1693, "type": "game", "source": "yinsh", "section": "References", "text": "YINSH — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/YINSH) ([archive](http://web.archive.org/web/20251008041412/https://en.wikipedia.org/wiki/YINSH))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1694, "type": "game", "source": "yinsh", "section": "See also", "text": "YINSH — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [TZAAR](tzaar.md) · [LYNGK](lyngk.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1695, "type": "game", "source": "yote", "section": "overview", "text": "Yote\nWest African capture game with mandatory return of captured pieces — unsolved.\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan capture game."}, {"id": 1696, "type": "game", "source": "yote", "section": "Description", "text": "Yote — Description\n\nYote is a Senegalese / West African game on a 5×6 board. Each player has 12\nstones held in reserve and may either drop a new stone or move/capture. A\ndistinctive feature: after capturing a stone, the captor must also remove a\nsecond opposing stone of their choice from the board."}, {"id": 1697, "type": "game", "source": "yote", "section": "Rules", "text": "Yote — Rules\n\n1. Board: 5×6 grid; both sides start with the board empty and 12 stones each\n   in reserve.\n2. On a turn a player either drops a stone from reserve onto any empty cell,\n   or moves one of their on-board stones one step orthogonally, or captures.\n3. **Capture**: jump one of your stones over an adjacent opposing stone to\n   the empty cell beyond; capture is optional, not chained.\n4. After a capture, the capturing player must also remove **one additional\n   opposing stone** of their choice from the board.\n5. A player who cannot move and has no reserves loses; alternatively the\n   player with stones remaining when the opponent has none wins."}, {"id": 1698, "type": "game", "source": "yote", "section": "Solution status", "text": "Yote — Solution status\n\nYote is **not solved**. It is a small game by modern standards but the\nremoval-rule adds significant branching to the game tree."}, {"id": 1699, "type": "game", "source": "yote", "section": "Consensus on optimal play", "text": "Yote — Consensus on optimal play\n\n- **Use the bonus removal to target the opponent's most dangerous piece** — after capturing, you must remove one extra opposing stone; always remove the piece that is most threatening (e.g., one that would enable a return capture or is part of a strong cluster), not just the nearest one.\n- **Delay dropping reserve stones until you can threaten a capture** — entering a stone from reserve onto a cell that immediately threatens a jump gives the opponent a defensive dilemma; entering passively fills the board without tempo.\n- **Capture when it removes two key pieces, not just to win one** — the double-removal rule means a single capture can eliminate two opposing stones; prioritise captures that remove two pieces in strong positions over captures that only clear a weak piece.\n- **Avoid clustering your own stones two apart** — a row of your stones with an empty cell between each pair is a chain-capture opportunity for the opponent; keep stones either adjacent or widely spaced.\n- **Endgame piece count is decisive** — with a small board (5×6) and double-removal, piece counts shift rapidly; avoid any sequence that leaves you in a one-for-two exchange unless the position demands it."}, {"id": 1700, "type": "game", "source": "yote", "section": "Engines & current best play", "text": "Yote — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** Yote is a traditional West African game with competitive play in Senegal and Mali; the double-removal rule after capture is its most distinctive strategic feature and significantly increases the branching factor."}, {"id": 1701, "type": "game", "source": "yote", "section": "Complexity", "text": "Yote — Complexity\n\nModerate."}, {"id": 1702, "type": "game", "source": "yote", "section": "References", "text": "Yote — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/Yot%C3%A9) ([archive](http://web.archive.org/web/20260425005623/https://en.wikipedia.org/wiki/Yot%C3%A9))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1703, "type": "game", "source": "yote", "section": "See also", "text": "Yote — See also\n\n- [Seega](seega.md) · [Dara](dara.md) · [Surakarta](surakarta.md)\n- Lexicon: [partisan game](../lexicon/README.md#partisan-game)"}, {"id": 1704, "type": "game", "source": "zertz", "section": "overview", "text": "ZÈRTZ\nA \"shrinking-board\" GIPF-project game where rings are removed as the game\nSolution status: Unsolved. Game-theoretic value: Unknown. Players: 2. Type: Partisan abstract strategy game."}, {"id": 1705, "type": "game", "source": "zertz", "section": "Description", "text": "ZÈRTZ — Description\n\nZÈRTZ (Kris Burm, 1998) is the second GIPF-project game. The board is a\nhexagonal array of **rings** containing three colours of marbles. The unusual\nfeature: the board itself **shrinks** as the game progresses, by removing\nrings."}, {"id": 1706, "type": "game", "source": "zertz", "section": "Rules", "text": "ZÈRTZ — Rules\n\n1. Board: hexagonal arrangement of 37 rings, with a shared pool of black,\n   white, and grey marbles.\n2. On a turn, a player must either:\n   - **Place a marble** on an empty ring, then **remove a free ring** (one with\n     at most one neighbouring ring still attached) from the edge of the board;\n     **or**\n   - **Capture**: if a marble can jump over an adjacent marble into an empty\n     ring beyond, the captured marble goes to the capturing player. Captures\n     are chained and mandatory when available.\n3. A player wins by collecting one of these capture totals: 3 marbles of each\n   colour; or 4 white; or 5 grey; or 6 black. **[verify]** the precise\n   totals."}, {"id": 1707, "type": "game", "source": "zertz", "section": "Solution status", "text": "ZÈRTZ — Solution status\n\nZÈRTZ is **not solved**. The shrinking board and mandatory-capture rules give\nthe search a peculiar shape; no published solving result."}, {"id": 1708, "type": "game", "source": "zertz", "section": "Consensus on optimal play", "text": "ZÈRTZ — Consensus on optimal play\n\n- **Control the board-shrinking pace** — you choose which ring to remove after placing a marble; remove rings that cut off the opponent's capture routes or reduce the space available for the colour of marbles they need to collect.\n- **Force mandatory captures onto your opponent** — if you place a marble that the opponent must immediately capture (mandatory capture rule), you can often orchestrate chains that hand you valuable marbles or leave the opponent in an awkward follow-up position.\n- **Collect grey marbles consistently** — grey is typically the hardest colour to accumulate in quantity; securing a steady grey count while letting white and black balance out is a common expert approach.\n- **Sacrifice black marbles to obtain whites** — black marbles are plentiful; white are scarce; chains that trade blacks for whites often accelerate a win via the \"4 whites\" condition.\n- **Shrink the board toward the opponent's preferred marbles** — by removing rings near clusters of a colour the opponent needs, you reduce the supply of that colour and force them to chase scarce marbles deeper into the shrinking board."}, {"id": 1709, "type": "game", "source": "zertz", "section": "Engines & current best play", "text": "ZÈRTZ — Engines & current best play\n\n- **Strongest known program(s):** No game-specific public engine known to the cataloguer. Playable in general-purpose abstract-game frameworks (e.g., [Ludii](https://ludii.games/)).\n- **Strength:** Not benchmarked.\n- **Where the proof / tablebase lives (if solved):** —\n- **Notes:** ZÈRTZ's shrinking board creates a fundamentally non-stationary search problem; the winning conditions (multiple possible collection targets) add further complexity to evaluation."}, {"id": 1710, "type": "game", "source": "zertz", "section": "Complexity", "text": "ZÈRTZ — Complexity\n\nModerate — the board shrinks rapidly, but the search-relevant branching is\nwide."}, {"id": 1711, "type": "game", "source": "zertz", "section": "References", "text": "ZÈRTZ — References\n\n- Rules: [Wikipedia](https://en.wikipedia.org/wiki/ZERTZ) ([archive](http://web.archive.org/web/20210507031603/http://en.wikipedia.org/wiki/Zertz))\n- [van den Herik, Uiterwijk & van Rijswijck (2002). *Games solved: Now and in the future*.](../references.md#vandenherik2002)"}, {"id": 1712, "type": "game", "source": "zertz", "section": "See also", "text": "ZÈRTZ — See also\n\n- [GIPF](gipf.md) · [DVONN](dvonn.md) · [YINSH](yinsh.md)\n- Lexicon: [game-tree complexity](../lexicon/README.md#game-tree-complexity)"}, {"id": 1713, "type": "reference", "source": "allis1988", "section": null, "text": "allis1988\nVictor Allis. *A Knowledge-based Approach of Connect-Four — The Game is Solved:\nWhite Wins.* M.Sc. thesis, Vrije Universiteit Amsterdam, 1988. — First\npublished solution of standard 7×6 Connect Four; first player wins."}, {"id": 1714, "type": "reference", "source": "allis1994", "section": null, "text": "allis1994\nVictor Allis. *Searching for Solutions in Games and Artificial Intelligence.*\nPh.D. thesis, University of Limburg, Maastricht, 1994. — Defines the\nultra-weakly / weakly / strongly solved hierarchy used throughout this archive;\nalso covers Gomoku, Qubic, Connect Four, Nine Men's Morris."}, {"id": 1715, "type": "reference", "source": "allis-gomoku1996", "section": null, "text": "allis-gomoku1996\nL. Victor Allis, H. Jaap van den Herik, M. P. H. Huntjens. *Go-Moku Solved by\nNew Search Techniques.* Computational Intelligence, 12(1):7–23, 1996. (Workshop\nversion 1993.) — Free-style Gomoku on 15×15 is a first-player win."}, {"id": 1716, "type": "reference", "source": "allis-pns1994", "section": null, "text": "allis-pns1994\nL. Victor Allis, Maarten van der Meulen, H. Jaap van den Herik. *Proof-Number\nSearch.* Artificial Intelligence, 66(1):91–124, 1994. — The PN-search technique\nused to solve many games in this archive."}, {"id": 1717, "type": "reference", "source": "applegate-sprouts1991", "section": null, "text": "applegate-sprouts1991\nDavid Applegate, Guy Jacobson, Daniel Sleator. *Computer Analysis of Sprouts.*\nTechnical report CMU-CS-91-144, Carnegie Mellon University, 1991. — Sprouts\nanalysed by computer up to 11 spots at the time."}, {"id": 1718, "type": "reference", "source": "ascher-mutorere1987", "section": null, "text": "ascher-mutorere1987\nMarcia Ascher. *Mu Torere: An Analysis of a Maori Game.* Mathematics Magazine,\n60(2):90–100, 1987. — Mathematical analysis of the Māori game Mū tōrere."}, {"id": 1719, "type": "reference", "source": "beasley-pegsolitaire1985", "section": null, "text": "beasley-pegsolitaire1985\nJohn D. Beasley. *The Ins and Outs of Peg Solitaire.* Oxford University Press,\n1985. — Complete theory of which peg-solitaire problems are solvable."}, {"id": 1720, "type": "reference", "source": "berlekamp-dotsandboxes2000", "section": null, "text": "berlekamp-dotsandboxes2000\nElwyn R. Berlekamp. *The Dots and Boxes Game: Sophisticated Child's Play.*\nA K Peters, 2000. — Theory of Dots and Boxes (chains, the long-chain rule,\nnimber analysis)."}, {"id": 1721, "type": "reference", "source": "bcg2001", "section": null, "text": "bcg2001\nElwyn R. Berlekamp, John H. Conway, Richard K. Guy. *Winning Ways for Your\nMathematical Plays.* 2nd edition, 4 vols., A K Peters, 2001–2004. (1st ed.\nAcademic Press, 1982.) — Foundational text of combinatorial game theory; covers\nNim, Hackenbush, Fox and Geese, Conway's Soldiers, Sprouts, Kayles, and much\nmore."}, {"id": 1722, "type": "reference", "source": "bjarnason-klondike2009", "section": null, "text": "bjarnason-klondike2009\nRonald Bjarnason, Alan Fern, Prasad Tadepalli. *Lower Bounding Klondike\nSolitaire with Monte-Carlo Planning.* Proc. ICAPS 2009. — Estimates the\nfraction of solvable Klondike deals under \"thoughtful\" (full-information) play."}, {"id": 1723, "type": "reference", "source": "bouton1901", "section": null, "text": "bouton1901\nCharles L. Bouton. *Nim, A Game with a Complete Mathematical Theory.* Annals of\nMathematics, 2nd Ser., 3(1/4):35–39, 1901–1902. — The complete solution of Nim;\norigin of the binary-XOR (\"Nim-sum\") theory."}, {"id": 1724, "type": "reference", "source": "bowling2015", "section": null, "text": "bowling2015\nMichael Bowling, Neil Burch, Michael Johanson, Oskari Tammelin. *Heads-up Limit\nHold'em Poker is Solved.* Science, 347(6218):145–149, 2015. — Heads-up limit\nTexas hold'em essentially weakly solved via CFR+; the game is a near-draw, with\na tiny first-player (dealer) edge measured."}, {"id": 1725, "type": "reference", "source": "brown-libratus2018", "section": null, "text": "brown-libratus2018\nNoam Brown, Tuomas Sandholm. *Superhuman AI for heads-up no-limit poker:\nLibratus beats top professionals.* Science, 359(6374):418–424, 2018."}, {"id": 1726, "type": "reference", "source": "conway1976", "section": null, "text": "conway1976\nJohn H. Conway. *On Numbers and Games.* Academic Press, 1976. — Introduces\nsurreal numbers and the formal theory of partisan combinatorial games."}, {"id": 1727, "type": "reference", "source": "demaine-phutball2002", "section": null, "text": "demaine-phutball2002\nErik D. Demaine, Martin L. Demaine, David Eppstein. *Phutball Endgames are\nHard.* In *More Games of No Chance*, MSRI Publications 42, Cambridge University\nPress, 2002. — Deciding whether a Phutball move wins is NP-hard."}, {"id": 1728, "type": "reference", "source": "feinstein-othello6x61993", "section": null, "text": "feinstein-othello6x61993\nJoel Feinstein. *Amenor Wins World 6×6 Championships.* British Othello\nFederation newsletter, August 1993. **[verify]** — Widely cited as the source\nfor 6×6 Othello being a second-player win (by 4 discs)."}, {"id": 1729, "type": "reference", "source": "gale1979", "section": null, "text": "gale1979\nDavid Gale. *The Game of Hex and the Brouwer Fixed-Point Theorem.* American\nMathematical Monthly, 86(10):818–827, 1979. — Proves Hex cannot end in a draw;\ndiscusses the strategy-stealing argument for the first-player win."}, {"id": 1730, "type": "reference", "source": "gardner-hexapawn1962", "section": null, "text": "gardner-hexapawn1962\nMartin Gardner. *Mathematical Games: How to build a game-learning machine and\nthen teach it to play and to win.* Scientific American, March 1962. — Introduces\nHexapawn; the 3×3 game is a second-player win."}, {"id": 1731, "type": "reference", "source": "gasser1996", "section": null, "text": "gasser1996\nRalph Gasser. *Solving Nine Men's Morris.* Computational Intelligence,\n12:24–41, 1996. (Also in *Games of No Chance*, MSRI Publications 29, 1996; and\nGasser's Ph.D. thesis, ETH Zürich, 1995.) — Nine Men's Morris weakly solved;\nthe game is a draw."}, {"id": 1732, "type": "reference", "source": "glendenning-quoridor2005", "section": null, "text": "glendenning-quoridor2005\nLisa Glendenning. *Mastering Quoridor.* B.Sc. thesis, University of New Mexico,\n2005. — Search-based analysis of Quoridor on reduced board sizes."}, {"id": 1733, "type": "reference", "source": "glenn-yahtzee2006", "section": null, "text": "glenn-yahtzee2006\nJames Glenn. *An Optimal Strategy for Yahtzee.* Technical report CS-TR-0002,\nLoyola College in Maryland, 2006. — Computes the strategy maximising expected\nscore in solitaire Yahtzee (~254.59 expected points). See also Tom Verhoeff's\noptimal Yahtzee proba­bility tables."}, {"id": 1734, "type": "reference", "source": "grundy1939", "section": null, "text": "grundy1939\nPatrick M. Grundy. *Mathematics and games.* Eureka, 2:6–8, 1939. — Independent\ndiscovery (with Sprague) of the Sprague–Grundy theory of impartial games."}, {"id": 1735, "type": "reference", "source": "guy-smith1956", "section": null, "text": "guy-smith1956\nRichard K. Guy, Cedric A. B. Smith. *The G-values of various games.*\nMathematical Proceedings of the Cambridge Philosophical Society, 52(3):514–526,\n1956. — Octal-game notation and Grundy-value (G-value) computations for Kayles,\nDawson's chess, and many other take-and-break games."}, {"id": 1736, "type": "reference", "source": "hayward-hex2009", "section": null, "text": "hayward-hex2009\nPhilip Henderson, Broderick Arneson, Ryan B. Hayward. *Solving 8×8 Hex.* Proc.\nIJCAI 2009, 505–510. — Hex weakly solved on the 8×8 board; the same group later\nextended weak solutions to 9×9 and 10×10."}, {"id": 1737, "type": "reference", "source": "hearn-demaine2009", "section": null, "text": "hearn-demaine2009\nRobert A. Hearn, Erik D. Demaine. *Games, Puzzles, and Computation.* A K Peters,\n2009. — Computational complexity of many combinatorial games and puzzles."}, {"id": 1738, "type": "reference", "source": "irving-kalah2000", "section": null, "text": "irving-kalah2000\nGeoffrey Irving, Jeroen Donkers, Jos Uiterwijk. *Solving Kalah.* ICGA Journal,\n23(3):139–147, 2000. — Kalah weakly solved for several (holes, seeds)\nconfigurations; most are first-player wins."}, {"id": 1739, "type": "reference", "source": "irving-pentago2014", "section": null, "text": "irving-pentago2014\nGeoffrey Irving. *Pentago is a first player win.* Technical write-up and open\nsolver, 2014 (perfect-pentago project). — Pentago strongly solved on a\nsupercomputer; the first player wins."}, {"id": 1740, "type": "reference", "source": "knuth1977", "section": null, "text": "knuth1977\nDonald E. Knuth. *The Computer as Master Mind.* Journal of Recreational\nMathematics, 9(1):1–6, 1976–77. — A strategy guaranteeing a solve of Mastermind\n(4 pegs, 6 colours) within 5 guesses."}, {"id": 1741, "type": "reference", "source": "lehman1964", "section": null, "text": "lehman1964\nAlfred Lehman. *A solution of the Shannon switching game.* Journal of the\nSociety for Industrial and Applied Mathematics, 12(4):687–725, 1964. — Matroid\ncharacterisation that solves the Shannon switching game (and hence Bridg-it)."}, {"id": 1742, "type": "reference", "source": "lomonosov2012", "section": null, "text": "lomonosov2012\nVladimir Makhnychev, Victor Zakharov. *Lomonosov 7-piece endgame tablebases.*\nMoscow State University, 2012. — Complete strongly-solved endgame tables for all\nchess positions with ≤7 pieces."}, {"id": 1743, "type": "reference", "source": "mead-sim1974", "section": null, "text": "mead-sim1974\nE. Mead, Alexander Rosa, Charlotte Huang. *The game of Sim: A winning strategy\nfor the second player.* Mathematics Magazine, 47(5):243–247, 1974. — Sim is a\nsecond-player win."}, {"id": 1744, "type": "reference", "source": "michie-menace1963", "section": null, "text": "michie-menace1963\nDonald Michie. *Experiments on the mechanization of game-learning. Part I:\nCharacterization of the model and its parameters.* The Computer Journal,\n6(3):232–236, 1963. — MENACE, the matchbox machine that learns Hexapawn /\nNoughts and Crosses."}, {"id": 1745, "type": "reference", "source": "moravcik-deepstack2017", "section": null, "text": "moravcik-deepstack2017\nMatej Moravčík et al. *DeepStack: Expert-level artificial intelligence in\nheads-up no-limit poker.* Science, 356(6337):508–513, 2017."}, {"id": 1746, "type": "reference", "source": "nash-hex", "section": null, "text": "nash-hex\nJohn Nash's strategy-stealing argument (c. 1949, unpublished) showing the first\nplayer wins Hex on any symmetric board. Recounted in [gale1979](#gale1979) and\n[bcg2001](#bcg2001)."}, {"id": 1747, "type": "reference", "source": "patashnik1980", "section": null, "text": "patashnik1980\nOren Patashnik. *Qubic: 4×4×4 Tic-Tac-Toe.* Mathematics Magazine,\n53(4):202–216, 1980. — Qubic (3-D 4×4×4 tic-tac-toe) weakly solved as a\nfirst-player win, using a computer-assisted strategy search."}, {"id": 1748, "type": "reference", "source": "plambeck-notakto2013", "section": null, "text": "plambeck-notakto2013\nThane E. Plambeck, Greg Whitehead. *The Secrets of Notakto: Winning at X-only\nTic-Tac-Toe.* arXiv:1301.1672, 2013. — Misère-play (\"Notakto\") analysis of\nX-only tic-tac-toe."}, {"id": 1749, "type": "reference", "source": "ratner-warmuth1990", "section": null, "text": "ratner-warmuth1990\nDaniel Ratner, Manfred Warmuth. *The (n²−1)-puzzle and related relocation\nproblems.* Journal of Symbolic Computation, 10(2):111–137, 1990. — Finding a\nshortest solution to the generalised n×n sliding-tile puzzle is NP-hard."}, {"id": 1750, "type": "reference", "source": "rokicki2014", "section": null, "text": "rokicki2014\nTomas Rokicki, Herbert Kociemba, Morley Davidson, John Dethridge. *The Diameter\nof the Rubik's Cube Group Is Twenty.* SIAM Review, 56(4):645–670, 2014.\n(Announced 2010.) — \"God's Number\" for the 3×3×3 Rubik's Cube is 20 in the\nhalf-turn metric."}, {"id": 1751, "type": "reference", "source": "romein-bal2003", "section": null, "text": "romein-bal2003\nJohn W. Romein, Henri E. Bal. *Solving Awari with Parallel Retrograde\nAnalysis.* IEEE Computer, 36(10):26–33, 2003. — Awari (Oware) strongly solved by\nparallel retrograde analysis; the game is a draw."}, {"id": 1752, "type": "reference", "source": "saffidine-breakthrough2012", "section": null, "text": "saffidine-breakthrough2012\nAbdallah Saffidine, Nicolas Jouandeau, Tristan Cazenave. *Solving Breakthrough\nwith Race Patterns and Job-Level Proof Number Search.* Advances in Computer\nGames (ACG 13), LNCS 7168, 196–207, 2012. — Weak solutions for small\nBreakthrough boards."}, {"id": 1753, "type": "reference", "source": "schadd-fanorona2008", "section": null, "text": "schadd-fanorona2008\nMaarten P. D. Schadd, Mark H. M. Winands, Jos W. H. M. Uiterwijk, H. Jaap van\nden Herik, Maurice H. J. Bergsma. *Best Play in Fanorona Leads to Draw.* New\nMathematics and Natural Computation, 4(3):369–387, 2008. — Fanorona weakly\nsolved; the game is a draw."}, {"id": 1754, "type": "reference", "source": "schaefer1978", "section": null, "text": "schaefer1978\nThomas J. Schaefer. *On the complexity of some two-person perfect-information\ngames.* Journal of Computer and System Sciences, 16(2):185–225, 1978. — Proves\nGeneralized Geography and other games PSPACE-complete."}, {"id": 1755, "type": "reference", "source": "schaeffer2007", "section": null, "text": "schaeffer2007\nJonathan Schaeffer, Neil Burch, Yngvi Björnsson, Akihiro Kishimoto, Martin\nMüller, Robert Lake, Paul Lu, Steve Sutphen. *Checkers Is Solved.* Science,\n317(5844):1518–1522, 2007. — English draughts (checkers) weakly solved; with\nperfect play the game is a draw."}, {"id": 1756, "type": "reference", "source": "schensted-titus1975", "section": null, "text": "schensted-titus1975\nCraige Schensted, Charles Titus. *Mudcrack Y and Poly-Y.* Neo Press, 1975. —\nDescribes the connection game Y; Y cannot be drawn and is a first-player win by\nstrategy stealing."}, {"id": 1757, "type": "reference", "source": "shannon1950", "section": null, "text": "shannon1950\nClaude E. Shannon. *Programming a Computer for Playing Chess.* Philosophical\nMagazine, 41(314):256–275, 1950. — Source of the \"Shannon number,\" the classic\n~10^120 estimate of the chess game-tree complexity."}, {"id": 1758, "type": "reference", "source": "silver-alphago2016", "section": null, "text": "silver-alphago2016\nDavid Silver et al. *Mastering the game of Go with deep neural networks and tree\nsearch.* Nature, 529(7587):484–489, 2016."}, {"id": 1759, "type": "reference", "source": "silver-alphazero2018", "section": null, "text": "silver-alphazero2018\nDavid Silver et al. *A general reinforcement learning algorithm that masters\nchess, shogi, and Go through self-play.* Science, 362(6419):1140–1144, 2018."}, {"id": 1760, "type": "reference", "source": "sprague1935", "section": null, "text": "sprague1935\nRoland P. Sprague. *Über mathematische Kampfspiele.* Tôhoku Mathematical\nJournal, 41:438–444, 1935. — With [grundy1939](#grundy1939), establishes that\nevery impartial game under normal play is equivalent to a Nim-heap."}, {"id": 1761, "type": "reference", "source": "takizawa2023", "section": null, "text": "takizawa2023\nHiroki Takizawa. *Othello is Solved.* arXiv:2310.19387, 2023. — Standard 8×8\nOthello (Reversi) weakly solved; with perfect play the game is a draw."}, {"id": 1762, "type": "reference", "source": "tanaka-quixo2020", "section": null, "text": "tanaka-quixo2020\nSatoshi Tanaka, François Bonnet, Sébastien Tixeuil, Yasumasa Tamura. *Quixo Is\nSolved.* arXiv:2007.15895, 2020. — Quixo on the standard 5×5 board is a draw\nwith perfect play (and a first-player win / draw on some smaller boards)."}, {"id": 1763, "type": "reference", "source": "tesauro1995", "section": null, "text": "tesauro1995\nGerald Tesauro. *Temporal Difference Learning and TD-Gammon.* Communications of\nthe ACM, 38(3):58–68, 1995. — Near-expert backgammon from self-play; the basis\nof modern backgammon equity analysis."}, {"id": 1764, "type": "reference", "source": "thompson1986", "section": null, "text": "thompson1986\nKen Thompson. *Retrograde Analysis of Certain Endgames.* ICCA Journal,\n9(3):131–139, 1986. — Early chess endgame tablebases; popularised retrograde\nanalysis for game solving."}, {"id": 1765, "type": "reference", "source": "tromp-connectfour", "section": null, "text": "tromp-connectfour\nJohn Tromp. *John's Connect Four Playground.* Online resource. — Independent\nverification of standard Connect Four and weak/strong solutions for many\nnon-standard board sizes."}, {"id": 1766, "type": "reference", "source": "uiterwijk-domineering2016", "section": null, "text": "uiterwijk-domineering2016\nJos W. H. M. Uiterwijk. *11×11 Domineering Is Solved: The First Player Wins.*\nComputers and Games (CG 2016), LNCS 10068, 129–136, 2016. — Latest in a series\nweakly solving Domineering on rectangular boards."}, {"id": 1767, "type": "reference", "source": "vandenherik2002", "section": null, "text": "vandenherik2002\nH. Jaap van den Herik, Jos W. H. M. Uiterwijk, Jack van Rijswijck. *Games\nsolved: Now and in the future.* Artificial Intelligence, 134(1–2):277–311,\n2002. — Survey of solved games with complexity figures; a primary backbone\nreference for this archive."}, {"id": 1768, "type": "reference", "source": "vanderwerf-go2003", "section": null, "text": "vanderwerf-go2003\nErik C. D. van der Werf, H. Jaap van den Herik, Jos W. H. M. Uiterwijk. *Solving\nGo on Small Boards.* ICGA Journal, 26(2):92–107, 2003. — 5×5 Go solved: the\nfirst player (Black) wins by 25 points."}, {"id": 1769, "type": "reference", "source": "watkins-losingchess2016", "section": null, "text": "watkins-losingchess2016\nMark Watkins. *Losing Chess: 1. e3 Wins.* Technical report / online\ndistributed-computation project, ~2016. **[verify]** — Antichess (losing chess)\nweakly solved as a first-player win opening with 1. e3."}, {"id": 1770, "type": "reference", "source": "wagner-virag2001", "section": null, "text": "wagner-virag2001\nJános Wágner, István Virág. *Solving Renju.* ICGA Journal, 24(1):30–35, 2001. —\nRenju (Gomoku with first-player restrictions) weakly solved as a first-player\nwin."}, {"id": 1771, "type": "reference", "source": "wythoff1907", "section": null, "text": "wythoff1907\nWillem A. Wythoff. *A modification of the game of Nim.* Nieuw Archief voor\nWiskunde, 7:199–202, 1907. — Wythoff's game; losing positions given by Beatty\nsequences based on the golden ratio."}, {"id": 1772, "type": "reference", "source": "anderson-feil1998", "section": null, "text": "anderson-feil1998\nMarlow Anderson, Todd Feil. *Turning Lights Out with Linear Algebra.* Mathematics\nMagazine, 71(4):300–303, 1998. — Reduces \"Lights Out\" to a linear system over\nGF(2); gives a complete characterisation of solvable configurations."}, {"id": 1773, "type": "reference", "source": "culberson-sokoban1999", "section": null, "text": "culberson-sokoban1999\nJoseph Culberson. *Sokoban is PSPACE-complete.* Proc. International Conference on\nFun with Algorithms (FUN '98), 65–76, 1999. — Sokoban (push-only) deciding\nsolvability is PSPACE-complete."}, {"id": 1774, "type": "reference", "source": "demaine-rushhour2002", "section": null, "text": "demaine-rushhour2002\nGary Flake, Eric Baum. *Rush Hour is PSPACE-complete, or \"Why you should generously\ntip parking lot attendants.\"* Theoretical Computer Science, 270(1–2):895–911,\n2002. — Rush Hour solvability is PSPACE-complete."}, {"id": 1775, "type": "reference", "source": "gardner-tower-of-hanoi", "section": null, "text": "gardner-tower-of-hanoi\nMartin Gardner. *Mathematical games: about the remarkable similarity between the\nIcosian Game and the Tower of Hanoi.* Scientific American, May 1957. (Generic\nreference; Édouard Lucas, 1883, is the primary source.)"}, {"id": 1776, "type": "reference", "source": "gasser-laskermorris1996", "section": null, "text": "gasser-laskermorris1996\nRalph Gasser. *Solving Nine Men's Morris.* In *Games of No Chance*, MSRI 29, 1996.\n— Includes Lasker Morris (the variant where placing and sliding can be\ninterleaved); reports a draw with perfect play."}, {"id": 1777, "type": "reference", "source": "herik-dao2002", "section": null, "text": "herik-dao2002\nErik C. D. van der Werf, Mark H. M. Winands, et al. solver writeups. **[verify]**\n— Reports that Dao is a first-player win (computer-verified small-board game)."}, {"id": 1778, "type": "reference", "source": "kaye-minesweeper2000", "section": null, "text": "kaye-minesweeper2000\nRichard Kaye. *Minesweeper is NP-complete.* Mathematical Intelligencer,\n22(2):9–15, 2000. — The Minesweeper consistency problem is NP-complete."}, {"id": 1779, "type": "reference", "source": "lichtenstein-sipser1980", "section": null, "text": "lichtenstein-sipser1980\nDavid Lichtenstein, Michael Sipser. *GO is polynomial-space hard.* Journal of the\nACM, 27(2):393–401, 1980. — Generalised Go is PSPACE-hard; the technique adapts\nto many other generalised games."}, {"id": 1780, "type": "reference", "source": "loh-poly-y", "section": null, "text": "loh-poly-y\nP. Y. Loh, R. Hayward. Discussion of Poly-Y and *Star strategy stealing.\n**[verify]** — Connection-game first-player wins via strategy stealing on\nno-draw boards."}, {"id": 1781, "type": "reference", "source": "nievergelt-pylos", "section": null, "text": "nievergelt-pylos\nJürg Nievergelt, et al. *Pylos solver*. **[verify]** — Pylos (4×4 stacking\npyramid) reported as a first-player win via retrograde analysis on a\ndistributed solve."}, {"id": 1782, "type": "reference", "source": "patashnik1976", "section": null, "text": "patashnik1976\nThis anchor is **[patashnik1980](#patashnik1980)**; see there."}, {"id": 1783, "type": "reference", "source": "ratner-warmuth-sliding", "section": null, "text": "ratner-warmuth-sliding\nSee [ratner-warmuth1990](#ratner-warmuth1990)."}, {"id": 1784, "type": "reference", "source": "reisch-hex1981", "section": null, "text": "reisch-hex1981\nStefan Reisch. *Hex ist PSPACE-vollständig.* Acta Informatica, 15:167–191, 1981.\n— Generalised Hex is PSPACE-complete; a foundational generalised-game\nhardness result."}, {"id": 1785, "type": "reference", "source": "schadd-connect6", "section": null, "text": "schadd-connect6\nWei-Yuan Hsu, I-Chen Wu, et al. Various Connect6 analyses. **[verify]** —\nComputer-verified opening lines for Connect6."}, {"id": 1786, "type": "reference", "source": "selman-sudoku2003", "section": null, "text": "selman-sudoku2003\nTakayuki Yato, Takahiro Seta. *Complexity and Completeness of Finding Another\nSolution and Its Application to Puzzles.* IEICE Transactions on Fundamentals,\nE86-A(5):1052–1060, 2003. — Generalised Sudoku is NP-complete (and\nASP-complete)."}, {"id": 1787, "type": "reference", "source": "slitherlink-np", "section": null, "text": "slitherlink-np\nStefan Reisch, et al. **[verify]**; see also Yato (2003). — Generalised\nSlitherlink is NP-complete."}, {"id": 1788, "type": "reference", "source": "thompson-bridge", "section": null, "text": "thompson-bridge\nBridge double-dummy solver literature; e.g. *Deep Finesse* and Bo Haglund's\n*Double Dummy Solver*. **[verify]** — Per-deal exact analysis of contract\nbridge."}, {"id": 1789, "type": "reference", "source": "ueda-skat2009", "section": null, "text": "ueda-skat2009\nSebastian Kupferschmid, Malte Helmert. *A Skat Player Based on Monte-Carlo\nSimulation.* Computers and Games (CG 2006), LNCS 4630, 135–147. **[verify]**\n— Strong Skat play via Monte-Carlo and double-dummy analysis."}, {"id": 1790, "type": "reference", "source": "winning-ways-hackenbush", "section": null, "text": "winning-ways-hackenbush\nSee [bcg2001](#bcg2001) — Red-Blue-Green Hackenbush, Push, Shove, Toppling\nDominoes, Toads-and-Frogs, Maze, and many other partisan games are analysed in\n*Winning Ways*."}, {"id": 1791, "type": "reference", "source": "yoshizoe-hanabi", "section": null, "text": "yoshizoe-hanabi\nHirotaka Osawa, et al. Hanabi AI research papers. **[verify]** — Hanabi as a\ncooperative imperfect-information benchmark; state-of-the-art bots achieve\nnear-perfect scores but the game is unsolved."}, {"id": 1792, "type": "reference", "source": "zermelo1913", "section": null, "text": "zermelo1913\nErnst Zermelo. *Über eine Anwendung der Mengenlehre auf die Theorie des\nSchachspiels.* Proc. Fifth International Congress of Mathematicians, vol. 2,\n501–504, 1913. — \"Zermelo's theorem\": in a finite two-player game of perfect\ninformation without chance, one side has a forced win or both can force a draw."}, {"id": 1793, "type": "lexicon", "source": "solved-game", "section": "Solving and solution strength", "text": "solved game\nA game is *solved* when its game-theoretic outcome under perfect play is known.\nSolving comes in three strengths ([Allis, 1994](../references.md#allis1994)):\n*ultra-weakly*, *weakly*, and *strongly* solved (below). \"Solved\" without\nqualification is ambiguous and best avoided."}, {"id": 1794, "type": "lexicon", "source": "ultra-weakly-solved", "section": "Solving and solution strength", "text": "ultra-weakly solved\nThe [game-theoretic value](#game-theoretic-value) of the initial position is\nknown, but no strategy to achieve it is necessarily known. Often proved\nnon-constructively — e.g. [Hex](../games/hex.md) is an ultra-weak first-player\nwin by the [strategy-stealing argument](#strategy-stealing), which proves a\nwinning strategy *exists* without exhibiting one."}, {"id": 1795, "type": "lexicon", "source": "weakly-solved", "section": "Solving and solution strength", "text": "weakly solved\nA strategy is known that achieves the game-theoretic value from the *initial\nposition* against any opposition. The strategy need not handle arbitrary\npositions. Most \"Game X is solved\" headlines (Connect Four, Checkers, Othello)\nmean weakly solved."}, {"id": 1796, "type": "lexicon", "source": "strongly-solved", "section": "Solving and solution strength", "text": "strongly solved\nA strategy (typically a complete lookup table) is known that plays optimally\nfrom *every* legal position, not just the start. [Awari](../games/awari.md) and\n[Nim](../games/nim.md) are strongly solved."}, {"id": 1797, "type": "lexicon", "source": "game-theoretic-value", "section": "Solving and solution strength", "text": "game-theoretic value\nThe outcome of a game from a given position assuming perfect play by both\nsides: a first-player win, a second-player win, or a draw. By\n[Zermelo's theorem](#zermelos-theorem) this value is well-defined for finite\nperfect-information games without chance."}, {"id": 1798, "type": "lexicon", "source": "zermelos-theorem", "section": "Solving and solution strength", "text": "Zermelo's theorem\nIn a finite two-player game of [perfect information](#perfect-information)\nwithout chance, either one player has a forced win or both can force at least a\ndraw ([Zermelo, 1913](../references.md#zermelo1913)). It guarantees a\ngame-theoretic value exists — it does not tell you what that value is."}, {"id": 1799, "type": "lexicon", "source": "solving-vs-strong-play", "section": "Solving and solution strength", "text": "solving vs. strong play\nSolving establishes the *provable* outcome under perfect play. Strong play\n(a top engine or human) may be empirically excellent without proving anything.\nA game can have superhuman engines yet be unsolved (chess, Go), and a solved\ngame's optimal lines may be far too large for a human to memorise (checkers).\n\n---"}, {"id": 1800, "type": "lexicon", "source": "combinatorial-game-theory", "section": "Combinatorial game theory (CGT)", "text": "combinatorial game theory\nThe mathematical theory of [perfect-information](#perfect-information) games\nwith no chance, usually two players moving alternately, where the last move\ndecides the result. Founded by [Conway (1976)](../references.md#conway1976) and\n[Berlekamp, Conway & Guy](../references.md#bcg2001)."}, {"id": 1801, "type": "lexicon", "source": "impartial-game", "section": "Combinatorial game theory (CGT)", "text": "impartial game\nA game in which the moves available depend only on the position, not on whose\nturn it is — both players have the same options. [Nim](../games/nim.md) is the\ncanonical example. Contrast [partisan game](#partisan-game)."}, {"id": 1802, "type": "lexicon", "source": "partisan-game", "section": "Combinatorial game theory (CGT)", "text": "partisan game\nA game in which the two players may have different moves available from the\nsame position — e.g. [Domineering](../games/domineering.md) (one player places\nvertical tiles, the other horizontal) or chess."}, {"id": 1803, "type": "lexicon", "source": "normal-play-convention", "section": "Combinatorial game theory (CGT)", "text": "normal play convention\nThe rule that the player who *cannot* move *loses* (equivalently, the player\nmaking the last move wins). The default in CGT.\n\n<a id=\"misere-play\"></a>"}, {"id": 1804, "type": "lexicon", "source": "misère-play", "section": "Combinatorial game theory (CGT)", "text": "misère play\nThe opposite convention: the player who makes the last move *loses*. Misère\ngames are usually far harder to analyse than their normal-play counterparts —\nsee [Misère Nim](../games/misere-nim.md) and [Notakto](../games/notakto.md).\n\n<a id=\"sprague-grundy-theorem\"></a>"}, {"id": 1805, "type": "lexicon", "source": "spraguegrundy-theorem", "section": "Combinatorial game theory (CGT)", "text": "Sprague–Grundy theorem\nEvery [impartial game](#impartial-game) under the [normal play\nconvention](#normal-play-convention) is equivalent to a single\n[Nim heap](#nim-value) of some size. That size is the position's\n[Grundy value](#nim-value). Proved independently by\n[Sprague (1935)](../references.md#sprague1935) and\n[Grundy (1939)](../references.md#grundy1939)."}, {"id": 1806, "type": "lexicon", "source": "nim-value", "section": "Combinatorial game theory (CGT)", "text": "nim-value\nAlso *Grundy value* or *nimber*. The size of the Nim heap equivalent to an\nimpartial position, computed as the *mex* (minimum excludant) of the nim-values\nof all positions reachable in one move. A position is a loss for the player to\nmove if and only if its nim-value is 0."}, {"id": 1807, "type": "lexicon", "source": "nim-sum", "section": "Combinatorial game theory (CGT)", "text": "nim-sum\nThe bitwise exclusive-OR (XOR) of heap sizes. In [Nim](../games/nim.md), a\nposition is a second-player win exactly when the nim-sum of all heaps is 0.\nNim-values of independent games add by nim-sum."}, {"id": 1808, "type": "lexicon", "source": "mex", "section": "Combinatorial game theory (CGT)", "text": "mex\n\"Minimum excludant\": the smallest non-negative integer not present in a set.\nUsed to compute [nim-values](#nim-value)."}, {"id": 1809, "type": "lexicon", "source": "octal-game", "section": "Combinatorial game theory (CGT)", "text": "octal game\nA take-and-break impartial game (remove tokens from a heap, possibly splitting\nit) encoded by an octal string. [Kayles](../games/kayles.md),\n[Dawson's chess](../games/dawsons-chess.md), and many others are octal games;\n[Guy & Smith (1956)](../references.md#guy-smith1956) tabulated their\n[nim-values](#nim-value)."}, {"id": 1810, "type": "lexicon", "source": "surreal-number", "section": "Combinatorial game theory (CGT)", "text": "surreal number\nThe number system Conway built from games; partisan game positions can have\nvalues that are numbers, and CGT extends arithmetic to them. Relevant to\nendgame analysis of [Hackenbush](../games/hackenbush.md) and\n[Domineering](../games/domineering.md).\n\n<a id=\"temperature--hot-game\"></a>"}, {"id": 1811, "type": "lexicon", "source": "temperature-hot-game", "section": "Combinatorial game theory (CGT)", "text": "temperature / hot game\nA game is *hot* when both players are eager to move in it (moving gains value).\n*Temperature* measures that urgency; *cooling* and *thermography* are tools for\nanalysing sums of hot games, notably in [Dots and\nBoxes](../games/dots-and-boxes.md) and [Amazons](../games/amazons.md).\n\n---"}, {"id": 1812, "type": "lexicon", "source": "minimax", "section": "Solving techniques", "text": "minimax\nThe basic algorithm for perfect-information games: assume each player picks the\nmove best for themselves, and back the values up the game tree."}, {"id": 1813, "type": "lexicon", "source": "alpha-beta-pruning", "section": "Solving techniques", "text": "alpha-beta pruning\nAn optimisation of minimax that skips branches provably irrelevant to the\nresult. With good move ordering it roughly square-roots the search effort."}, {"id": 1814, "type": "lexicon", "source": "retrograde-analysis", "section": "Solving techniques", "text": "retrograde analysis\nSolving *backwards* from terminal positions: label all won/lost/drawn end\npositions, then repeatedly label any position all of whose successors are\nlabelled. The standard route to [strongly solving](#strongly-solved) a game and\nto building [endgame tablebases](#endgame-tablebase). Used for\n[Awari](../games/awari.md), [Nine Men's Morris](../games/nine-mens-morris.md),\nand chess endgames ([Thompson, 1986](../references.md#thompson1986))."}, {"id": 1815, "type": "lexicon", "source": "endgame-tablebase", "section": "Solving techniques", "text": "endgame tablebase\nA precomputed database giving the [game-theoretic value](#game-theoretic-value)\n(and often distance-to-mate) of every position with few pieces. Chess\ntablebases are complete for ≤7 pieces\n([Lomonosov, 2012](../references.md#lomonosov2012))."}, {"id": 1816, "type": "lexicon", "source": "proof-number-search", "section": "Solving techniques", "text": "proof-number search\nA best-first search that targets the most \"proof-efficient\" node, well suited\nto proving game values with uneven branching. Introduced by [Allis, van der\nMeulen & van den Herik (1994)](../references.md#allis-pns1994); central to many\nweak solutions (Gomoku, Checkers, Fanorona, Breakthrough)."}, {"id": 1817, "type": "lexicon", "source": "opening-book", "section": "Solving techniques", "text": "opening book\nA stored set of analysed opening lines. A [weak solution](#weakly-solved) can be\nviewed as a perfect opening book that always steers toward the game's value."}, {"id": 1818, "type": "lexicon", "source": "gods-number", "section": "Solving techniques", "text": "God's number\nThe maximum, over all positions, of the optimal solution length — the diameter\nof the puzzle's state graph. For the [Rubik's Cube](../games/rubiks-cube.md) it\nis 20; for the [15 puzzle](../games/fifteen-puzzle.md), 80.\n\n---"}, {"id": 1819, "type": "lexicon", "source": "state-space-complexity", "section": "Complexity measures", "text": "state-space complexity\nThe number of legal positions reachable from the initial position. An upper\nbound on the size of a [strong solution](#strongly-solved)."}, {"id": 1820, "type": "lexicon", "source": "game-tree-complexity", "section": "Complexity measures", "text": "game-tree complexity\nThe number of leaf nodes in the smallest full-width search tree that solves the\ninitial position — roughly (branching factor)^(game length). The famous\n~10^120 \"Shannon number\" for chess\n([Shannon, 1950](../references.md#shannon1950)) is a game-tree estimate."}, {"id": 1821, "type": "lexicon", "source": "perfect-information", "section": "Complexity measures", "text": "perfect information\nEvery player knows the complete game state at all times — no hidden cards, no\nsimultaneous moves. Chess and Go have perfect information; poker and\nBattleship do not.\n\n<a id=\"pspace-complete--exptime-complete\"></a>"}, {"id": 1822, "type": "lexicon", "source": "pspace-complete-exptime-complete", "section": "Complexity measures", "text": "PSPACE-complete / EXPTIME-complete\nComputational-complexity classifications for *generalised* (n×n) versions of\ngames. Generalized Geography is PSPACE-complete\n([Schaefer, 1978](../references.md#schaefer1978)); generalized chess, Go, and\ncheckers are EXPTIME-complete. These results concern asymptotic hardness, not\nthe fixed-size standard games.\n\n---"}, {"id": 1823, "type": "lexicon", "source": "first-player-advantage", "section": "Playing terms", "text": "first-player advantage\nThe common (not universal) phenomenon that moving first is beneficial. Many\nsolved games are first-player wins (Connect Four, Gomoku, Hex, Qubic); some are\ndraws (checkers, Nine Men's Morris, Othello); a few favour the second player\n(Sim, Hexapawn, Dōbutsu shōgi)."}, {"id": 1824, "type": "lexicon", "source": "zugzwang", "section": "Playing terms", "text": "zugzwang\nA position in which any move worsens the mover's outcome — the obligation to\nmove is itself the disadvantage. Central to chess and checkers endgame theory."}, {"id": 1825, "type": "lexicon", "source": "opposition", "section": "Playing terms", "text": "opposition\nA specific [zugzwang](#zugzwang) relationship between kings in chess endgames;\nmore broadly, a parity/tempo concept in many endgames."}, {"id": 1826, "type": "lexicon", "source": "strategy-stealing-argument", "section": "Playing terms", "text": "strategy-stealing argument\nA non-constructive proof that the second player cannot have a winning strategy:\nif they did, the first player could \"steal\" it by making an arbitrary first\nmove and then following it, with the extra move never a handicap. Proves\nfirst-player-cannot-lose for [Hex](../games/hex.md), [Y](../games/y.md),\n[Chomp](../games/chomp.md), and others — without revealing the strategy."}, {"id": 1827, "type": "lexicon", "source": "pairing-strategy", "section": "Playing terms", "text": "pairing strategy\nA drawing or blocking strategy in which the defender pre-pairs the cells/threats\nso that answering the opponent's move in its partner cell neutralises it. Used\nto prove draws in many [k-in-a-row](../games/gomoku.md) and Maker-Breaker games."}, {"id": 1828, "type": "lexicon", "source": "draw", "section": "Playing terms", "text": "draw\nA game-theoretic value in which neither player can force a win. Under\n[normal play](#normal-play-convention) impartial games never draw; many\npartisan board games (checkers, Othello, Nine Men's Morris) do."}, {"id": 1829, "type": "lexicon", "source": "maker-breaker-game", "section": "Playing terms", "text": "maker-breaker game\nA game in which one player (\"Maker\") tries to claim a winning set and the other\n(\"Breaker\") only tries to prevent it — Breaker has no winning sets of their own.\nA common reformulation that simplifies analysis of k-in-a-row games.\n\n---"}, {"id": 1830, "type": "lexicon", "source": "chance-element", "section": "Chance and imperfect-information games", "text": "chance element\nA rule-level source of randomness — dice, a shuffled deck, a drawn tile. Games\nwith a chance element fall outside [Zermelo's theorem](#zermelos-theorem): they\nhave no win/draw/loss [game-theoretic value](#game-theoretic-value), only\n*expected* outcomes. [Backgammon](../games/backgammon.md),\n[Yahtzee](../games/yahtzee.md) and\n[EinStein würfelt nicht!](../games/einstein-wurfelt-nicht.md) have a chance\nelement but no hidden information."}, {"id": 1831, "type": "lexicon", "source": "zero-sum-game", "section": "Chance and imperfect-information games", "text": "zero-sum game\nA game in which one player's gain exactly equals the others' loss — there are no\noutcomes that are jointly good or jointly bad. Almost every game in this archive\nis zero-sum; the term matters mainly because the [minimax](#minimax) /\n[Nash-equilibrium](#nash-equilibrium) theory is cleanest in the two-player\nzero-sum case."}, {"id": 1832, "type": "lexicon", "source": "nash-equilibrium", "section": "Chance and imperfect-information games", "text": "Nash equilibrium\nA strategy profile in which no player can do better by unilaterally changing\nstrategy. For finite two-player [zero-sum games](#zero-sum-game) a Nash\nequilibrium always exists (von Neumann's minimax theorem) and its value is *the*\nvalue of the game — but it may require a **mixed strategy** (randomising), as in\n[rock-paper-scissors](../games/rock-paper-scissors.md). \"Solving\" an\nimperfect-information game such as [heads-up limit hold'em](../games/heads-up-limit-holdem.md)\nmeans computing (an approximation of) a Nash equilibrium rather than a\nwin/draw/loss value."}, {"id": 1833, "type": "lexicon", "source": "hunt-game", "section": "Chance and imperfect-information games", "text": "hunt game\nA class of asymmetric two-player games in which one side commands many weak\npieces (e.g. hounds, geese) trying to corner or stalemate the other's single\nstrong piece (the fox, the hare). Classical examples include\n[Fox and Geese](../games/fox-and-geese.md), [Halatafl](../games/halatafl.md)\nand [Catch the Hare](../games/catch-the-hare.md). Hunt games are typically\nsmall enough to be exhaustively solved on their canonical boards."}, {"id": 1834, "type": "lexicon", "source": "imperfect-information", "section": "Chance and imperfect-information games", "text": "imperfect information\nA game in which at least one player lacks complete knowledge of the game state\n— private cards, hidden bids, hidden mine positions, etc. \"Solving\" an\nimperfect-information game generally means finding a [Nash\nequilibrium](#nash-equilibrium) of a sequential game (often via\n[CFR](#) or its variants), not a single-line win/draw/loss value.\n[Bridge](../games/bridge.md), [Hanabi](../games/hanabi.md),\n[Skat](../games/skat.md), and [heads-up hold'em](../games/heads-up-limit-holdem.md)\nare core examples."}, {"id": 1835, "type": "lexicon", "source": "np-completeness", "section": "Chance and imperfect-information games", "text": "NP-completeness\nComplexity class of problems for which a candidate solution can be verified in\npolynomial time and to which every other NP problem reduces. In games this\narises mostly for puzzle decision problems — *given this instance, is there a\nsolution?* — including [Sudoku](../games/sudoku.md),\n[Minesweeper](../games/minesweeper.md), [Slitherlink](../games/slitherlink.md),\n[Hashiwokakero](../games/hashiwokakero.md), and [Nonograms](../games/nonograms.md)."}, {"id": 1836, "type": "lexicon", "source": "pspace", "section": "Chance and imperfect-information games", "text": "PSPACE\nThe class of problems solvable using polynomial space. Many two-player\nperfect-information games of unbounded depth are PSPACE-complete or harder —\nsee [PSPACE-complete / EXPTIME-complete](#pspace-complete--exptime-complete).\nSliding-block puzzles ([Sokoban](../games/sokoban.md),\n[Rush Hour](../games/rush-hour.md), [Klotski](../games/klotski.md)) are\ncanonical PSPACE-complete examples."}, {"id": 1837, "type": "lexicon", "source": "strategy-stealing", "section": "Chance and imperfect-information games", "text": "strategy stealing\nAn alias for the [strategy-stealing argument](#strategy-stealing-argument)\n— in games where an extra move cannot hurt, the first player must have at\nleast a draw (otherwise the second player could \"steal\" the would-be winning\nstrategy)."}]