Solved game

Short description: Game whose outcome can be correctly predicted

A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance; solving such a game may use combinatorial game theory or computer assistance.

## Contents

* 1 Overview
* 1.1 Ultra-weak solution
* 1.2 Weak solution
* 1.3 Strong solution
* 2 Perfect play
* 3 Solved games
* 4 Weak-solves
* 5 Partially solved games
* 6 See also
* 7 References
* 8 Further reading
* 9 External links

## Overview

A two-player game can be solved on several levels:[1][2]

### Ultra-weak solution

Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides (see § Perfect play, below). This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any details of the perfect play.

### Weak solution

Provide one algorithm for each of the two players, such that the player using it can achieve at least the optimal outcome, regardless of the opponent's moves, from the start of the game, using reasonable computational resources.

### Strong solution

Provide an algorithm that uses reasonable computational resources and finds optimal plays for both players from all legal positions.

By contrast, "strong" proofs often proceed by brute force—using a computer to exhaustively search a game tree to figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board. However, these proofs are not as helpful in understanding deeper reasons why some games are solvable as a draw, and other, seemingly very similar games are solvable as a win.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree. However, since for many non-trivial games such an algorithm would require an infeasible amount of time to generate a move in a given position, a game is not considered to be solved weakly or strongly unless the algorithm can be run by existing hardware in a reasonable time. Many algorithms rely on a huge pre-generated database and are effectively nothing more.

As a simple example of a strong solution, the game of tic-tac-toe is easily solvable as a draw for both players with perfect play (a result manually determinable). Games like nim also admit a rigorous analysis using combinatorial game theory.

Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Even a strongly solved game can still be interesting if its solution is too complex to be memorized; conversely, a weakly solved game may lose its attraction if the winning strategy is simple enough to remember (e.g., Maharajah and the Sepoys). An ultra-weak solution (e.g., Chomp or Hex on a sufficiently large board) generally does not affect playability.

## Perfect play

In game theory, perfect play is the behavior or strategy of a player that leads to the best possible outcome for that player regardless of the response by the opponent. Perfect play for a game is known when the game is solved.[1] Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). By backward reasoning, one can recursively evaluate a non-final position as identical to the position that is one move away and best valued for the player whose move it is. Thus a transition between positions can never result in a better evaluation for the moving player, and a perfect move in a position would be a transition between positions that are equally evaluated. As an example, a perfect player in a drawn position would always get a draw or win, never a loss. If there are multiple options with the same outcome, perfect play is sometimes considered the fastest method leading to a good result, or the slowest method leading to a bad result.

Perfect play can be generalized to non-perfect information games, as the strategy that would guarantee the highest minimal expected outcome regardless of the strategy of the opponent. As an example, the perfect strategy for rock paper scissors would be to randomly choose each of the options with equal (1/3) probability. The disadvantage in this example is that this strategy will never exploit non-optimal strategies of the opponent, so the expected outcome of this strategy versus any strategy will always be equal to the minimal expected outcome.

Although the optimal strategy of a game may not (yet) be known, a game-playing computer might still benefit from solutions of the game from certain endgame positions (in the form of endgame tablebases), which will allow it to play perfectly after some point in the game. Computer chess programs are well known for doing this.

## Solved games

Awari (a game of the Mancala family)
The variant of Oware allowing game ending "grand slams" was strongly solved by Henri Bal and John Romein at the Vrije Universiteit in Amsterdam, Netherlands (2002). Either player can force the game into a draw.
Chopsticks
Strongly solved. If two players both play perfectly, the game will go on indefinitely.[citation needed]
Connect Four
The game of Connect Four has been solved. Solved first by James D. Allen on October 1, 1988, and independently by Victor Allis on October 16, 1988.[3] The first player can force a win. Strongly solved by John Tromp's 8-ply database[4] (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (as well as 8×8 in late 2015)[3] (Feb 18, 2006). Solved for all boardsizes where width+height equals 16 on May 22, 2024.[5] In 2025, the classic 7x6 board was strongly solved in terms of a win-draw-loss look-up table.[6]
Free gomoku
Solved by Victor Allis (1993). The first player can force a win without opening rules.[1]
Ghost
Solved by Alan Frank using the Official Scrabble Players Dictionary in 1987.[7]
Hexapawn
3×3 variant solved as a win for black, several other larger variants also solved.[8]
Kalah
Most variants solved by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk (2000) except Kalah (6/6). The (6/6) variant was solved by Anders Carstensen (2011). Strong first-player advantage was proven in most cases.[9][10]
L game
Easily solvable. Either player can force the game into a draw.
Maharajah and the Sepoys
This asymmetrical game is a win for the sepoys player with correct play.[citation needed]
Nim
Strongly solved.[11]
Nine men's morris
Solved by Ralph Gasser (1993). Either player can force the game into a draw.[12][13]
Order and Chaos
Order (First player) wins.[14]
Ohvalhu
Weakly solved by humans, but proven by computers.[citation needed] (Dakon is, however, not identical to Ohvalhu, the game which actually had been observed by de Voogt)[citation needed]
Pangki
Strongly solved by Jason Doucette (2001).[15] The game is a draw. There are only two unique first moves if you discard mirrored positions. One forces the draw, and the other gives the opponent a forced win in 15 moves.
Pentago
Strongly solved by Geoffrey Irving with use of a supercomputer at NERSC. The first player wins.
Quarto
Solved by Luc Goossens (1998). Two perfect players will always draw.[16][17][18]
Renju-like game without opening rules involved
Claimed to be solved by János Wagner and István Virág (2001).[19] A first-player win.
Teeko
Solved by Guy Steele (1998). Depending on the variant either a first-player win or a draw.[20]
Three men's morris
Trivially solvable. Either player can force the game into a draw.[citation needed]
Three musketeers
Strongly solved by Johannes Laire in 2009, and weakly solved by Ali Elabridi in 2017.[21] It is a win for the blue pieces (Cardinal Richelieu's men, or, the enemy).[22]
Tic-tac-toe
Extremely trivially strongly solvable because of the small game tree.[23] The game is a draw if no mistakes are made, with no mistake possible on the opening move.
Wythoff's game
Strongly solved by W. A. Wythoff in 1907.[24]

## Weak-solves

English draughts (checkers)
This 8×8 variant of draughts was weakly solved on April 29, 2007, by the team of Jonathan Schaeffer. From the standard starting position, both players can guarantee a draw with perfect play.[25] Checkers has a search space of 5×1020 possible game positions.[26] The number of calculations involved was 1014, which were done over a period of 18 years. The process involved from 200 desktop computers at its peak down to around 50.[27]

Fanorona
Weakly solved by Maarten Schadd. The game is a draw.[28]

Losing chess
Weakly solved in 2016 as a win for White beginning with 1. e3.[29]

Othello (Reversi)
Weakly solved in 2023 by Hiroki Takizawa, a researcher at Preferred Networks.[30] However, the paper's conclusions are contested.[31] From the standard starting position on an 8×8 board, a perfect play by both players will result in a draw. Othello is the largest game solved to date, with a search space of 1028 possible game positions.

Pentominoes
Weakly solved by H. K. Orman.[32] It is a win for the first player.

Qubic
Weakly solved by Oren Patashnik (1980) and Victor Allis. The first player wins.

Sim
Weakly solved: win for the second player.[citation needed]

Lambs and tigers
Weakly solved by Yew Jin Lim (2007). The game is a draw.[33]

## Partially solved games

Chess

Main article: Solving chess

Fully solving chess remains elusive, and it is speculated that the complexity of the game may preclude it ever being solved. Through retrograde computer analysis and endgame tablebases, strong solutions have been found for all three- to seven-piece endgames, counting the two kings as pieces.
Some variants of chess on a smaller board with reduced numbers of pieces have been solved. Some other popular variants have also been solved; for example, a weak solution to Maharajah and the Sepoys is an easily memorable series of moves that guarantees victory to the "sepoys" player.

The 5×5 board was weakly solved for all opening moves in 2002.[34] The 7×7 board was weakly solved in 2015.[35] Humans usually play on a 19×19 board, which is over 145 orders of magnitude more complex than 7×7.[36]
Hex
A strategy-stealing argument (as used by John Nash) shows that all square board sizes cannot be lost by the first player. Combined with a proof of the impossibility of a draw, this shows that the game is a first player win (so it is ultra-weak solved).[citation needed] On particular board sizes, more is known: it is strongly solved by several computers for board sizes up to 6×6.[citation needed] Weak solutions are known for board sizes 7×7 (using a swapping strategy), 8×8, and 9×9;[citation needed] in the 8×8 case, a weak solution is known for all opening moves.[37] Strongly solving Hex on an N×N board is unlikely as the problem has been shown to be PSPACE-complete.[citation needed] If Hex is played on an N×(N \+ 1) board then the player who has the shorter distance to connect can always win by a simple pairing strategy, even with the disadvantage of playing second.[citation needed]

International draughts

Morabaraba
Strongly solved by Gábor E. Gévay (2015). The first player wins in optimal play.[38]

m,n,k-game
It is trivial to show that the second player can never win; see strategy-stealing argument. Almost all cases have been solved weakly for k ≤ 4. Some results are known for k = 5. The games are drawn for k ≥ 8.[citation needed]

## See also

* Computer chess
* Computer Go
* Computer Othello
* Game complexity
* God's algorithm
* Zermelo's theorem (game theory)

## References

1. ↑ 1.0 1.1 1.2 Allis, L.V. (1994). Searching for solutions in games and artificial intelligence (Thesis). Maastricht University. doi:10.26481/dis.19940923la. ISBN 90-90-07488-0.
2. ↑ van den Herik, H.Jaap; Uiterwijk, Jos W.H.M.; van Rijswijck, Jack (2002). "Games solved: Now and in the future" (in en). Artificial Intelligence 134 (1–2): 277–311. doi:10.1016/S0004-3702(01)00152-7. https://linkinghub.elsevier.com/retrieve/pii/S0004370201001527.
3. ↑ 3.0 3.1 "John's Connect Four Playground". https://tromp.github.io/c4/c4.html.
4. ↑ "UCI Machine Learning Repository: Connect-4 Data Set". https://archive.ics.uci.edu/ml/datasets/Connect-4.
5. ↑ "ChristopheSteininger/c4". https://github.com/ChristopheSteininger/c4.
6. ↑ Böck, Markus (1 Jul 2025). "Strongly Solving 7×6 Connect-Four on Consumer Grade Hardware". arXiv:2507.05267 [cs.AI].
7. ↑ Frank, Alan (1987-08-01). "Ghostbusters". Word Ways 20 (4). https://digitalcommons.butler.edu/wordways/vol20/iss4/4.
8. ↑ Price, Robert. "Hexapawn". http://www.chessvariants.com/small.dir/hexapawn.html.
9. ↑ Solving Kalah by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk.
10. ↑ Solving (6,6)-Kalaha by Anders Carstensen.
11. ↑ Bouton, C. L. (1901–1902), "Nim, a game with a complete mathematical theory", Annals of Mathematics 3 (14): 35–39, doi:10.2307/1967631
12. ↑ Gasser, Ralph (1996). "Solving Nine Men’s Morris". in Nowakowski, Richard (in en). Games of No Chance. 29. Cambridge: Cambridge University Press. pp. 101–113. ISBN 9780521574112. http://library.msri.org/books/Book29/files/gasser.pdf. Retrieved 2022-01-03.
13. ↑ Nine Men's Morris is a Draw by Ralph Gasser
14. ↑ "solved: Order wins - Order and Chaos". https://boardgamegeek.com/thread/1579397/solved-order-wins.
15. ↑ Pangki is strongly solved as a draw by Jason Doucette
16. ↑ "Quarto". https://wouterkoolen.info/Talks/quarto.pdf.
17. ↑ "414298141056 Quarto Draws Suffice!". https://www.mathpages.com/home/kmath352.htm.
18. ↑ "Quarto". http://ssel.vub.ac.be/Members/LucGoossens/quarto/quartotext.htm.
19. ↑ Wágner, János; Virág, István (March 2001). "Solving Renju" (in en). p. 30. http://www.sze.hu/~gtakacs/download/wagnervirag_2001.pdf.
20. ↑ Teeko, by E. Weisstein
21. ↑ Elabridi, Ali. "Weakly Solving the Three Musketeers Game Using Artificial Intelligence and Game Theory". http://www.aui.ma/sse-capstone-repository/pdf/fall2017/WEAKLY SOLVING THE THREE MUSKETEERS GAME USING ARTIFICIAL INTELLIGENCE AND GAME THEORY Ali Elabridi.pdf.
22. ↑ Three Musketeers, by J. Lemaire
23. ↑ Tic-Tac-Toe, by R. Munroe
24. ↑ Wythoff, W. A. (1907), "A modification of the game of nim", Nieuw Archief voor Wiskunde 7 (2): 199–202, https://archive.org/details/nieuwarchiefvoo02genogoog/page/n219/mode/2up
25. ↑ Schaeffer, Jonathan (2007-07-19). "Checkers Is Solved". Science 317 (5844): 1518–22. doi:10.1126/science.1144079. PMID 17641166. Bibcode: 2007Sci...317.1518S.
26. ↑ "Project - Chinook - World Man-Machine Checkers Champion". http://www.cs.ualberta.ca/~chinook/project/.
27. ↑ Mullins, Justin (2007-07-19). "Checkers 'solved' after years of number crunching". NewScientist.com news service. https://www.newscientist.com/article/dn12296-checkers-solved-after-years-of-number-crunching/.
28. ↑ M.P.D. Schadd; M.H.M. Winands; J.W.H.M. Uiterwijk; H.J. van den Herik; M.H.J. Bergsma (2008). "Best Play in Fanorona leads to Draw". New Mathematics and Natural Computation 4 (3): 369–387. doi:10.1142/S1793005708001124. http://schadd.com/Papers/2008FanoronaJNMNC.pdf. Retrieved 2015-04-08.
29. ↑ Watkins, Mark. "Losing Chess: 1. e3 wins for White". http://magma.maths.usyd.edu.au/~watkins/LOSING_CHESS/ICGA2016.pdf.
30. ↑ Takizawa, Hiroki (2023-10-30). "Othello is Solved". arXiv:2310.19387 [cs.AI].
31. ↑ "HN discussion". 2024-11-03. https://news.ycombinator.com/item?id=38141366.
32. ↑ Hilarie K. Orman: Pentominoes: A First Player Win in Games of no chance, MSRI Publications - Volume 29, 1996, pages 339-344. Online: pdf.
33. ↑ Yew Jin Lim. On Forward Pruning in Game-Tree Search . Ph.D. Thesis, National University of Singapore, 2007.
34. ↑ 5×5 Go is solved by Erik van der Werf
35. ↑ "首期喆理围棋沙龙举行 李喆7路盘最优解具有里程碑意义_下棋想赢怕输_新浪博客". http://blog.sina.com.cn/s/blog_53a2e03d0102vyt5.html. (which says the 7x7 solution is only weakly solved and it's still under research, 1. the correct komi is 9 (4.5 stone); 2. there are multiple optimal trees - the first 3 moves are unique - but within the first 7 moves there are 5 optimal trees; 3. There are many ways to play that don't affect the result)
36. ↑ Counting legal positions in Go , Tromp and Farnebäck, accessed 2007-08-24.
37. ↑ P. Henderson, B. Arneson, and R. Hayward, [webdocs.cs.ualberta.ca/~hayward/papers/solve8.pdf Solving 8×8 Hex ], Proc. IJCAI-09 505-510 (2009) Retrieved 29 June 2010.
38. ↑ Gevay, Gabor E.; Danner, Gabor (September 2016). "Calculating Ultrastrong and Extended Solutions for Nine Men's Morris, Morabaraba, and Lasker Morris". IEEE Transactions on Computational Intelligence and AI in Games 8 (3): 256–267. doi:10.1109/TCIAIG.2015.2420191. ISSN 1943-068X. Bibcode: 2016ITCIA...8..256G.

## Further reading

* Allis, Beating the World Champion? The state-of-the-art in computer game playing. in New Approaches to Board Games Research.

## External links

* Computational Complexity of Games and Puzzles by David Eppstein.
* GamesCrafters solving two-person games with perfect information and no chance

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Topics in game theory

Definitions|

* Cooperative game
* Determinacy
* Escalation of commitment
* Extensive-form game
* First-player and second-player win
* Game complexity
* Graphical game
* Hierarchy of beliefs
* Information set
* Normal-form game
* Preference
* Sequential game
* Simultaneous game
* Simultaneous action selection
* Solved game
* Succinct game

Equilibrium
concepts|

* Nash equilibrium
* Subgame perfection
* Mertens-stable equilibrium
* Bayesian Nash equilibrium
* Perfect Bayesian equilibrium
* Trembling hand
* Proper equilibrium
* Epsilon-equilibrium
* Correlated equilibrium
* Sequential equilibrium
* Quasi-perfect equilibrium
* Evolutionarily stable strategy
* Risk dominance
* Core
* Shapley value
* Pareto efficiency
* Gibbs equilibrium
* Quantal response equilibrium
* Self-confirming equilibrium
* Strong Nash equilibrium
* Markov perfect equilibrium

Strategies|

* Dominant strategies
* Pure strategy
* Mixed strategy
* Strategy-stealing argument
* Tit for tat
* Grim trigger
* Collusion
* Backward induction
* Forward induction
* Markov strategy

Classes
of games|

* Symmetric game
* Perfect information
* Repeated game
* Signaling game
* Screening game
* Cheap talk
* Zero-sum game
* Mechanism design
* Bargaining problem
* Stochastic game
* n-player game
* Large Poisson game
* Nontransitive game
* Global game
* Strictly determined game
* Potential game

Games|

* Go
* Chess
* Infinite chess
* Checkers
* Tic-tac-toe
* Prisoner's dilemma
* Optional prisoner's dilemma
* Traveler's dilemma
* Coordination game
* Chicken
* Centipede game
* Volunteer's dilemma
* Dollar auction
* Battle of the sexes
* Stag hunt
* Matching pennies
* Ultimatum game
* Rock paper scissors
* Pirate game
* Dictator game
* Public goods game
* Blotto game
* War of attrition
* El Farol Bar problem
* Fair division
* Fair cake-cutting
* Cournot game
* Deadlock
* Diner's dilemma
* Guess 2/3 of the average
* Kuhn poker
* Nash bargaining game
* Prisoners and hats puzzle
* Trust game
* Princess and Monster game
* Rendezvous problem

Theorems|

* Arrow's impossibility theorem
* Aumann's agreement theorem
* Folk theorem
* Minimax theorem
* Nash's theorem
* Purification theorem
* Revelation principle
* Zermelo's theorem

Key
figures|

* Albert W. Tucker
* Amos Tversky
* Ariel Rubinstein
* Claude Shannon
* Daniel Kahneman
* David K. Levine
* David M. Kreps
* Donald B. Gillies
* Drew Fudenberg
* Eric Maskin
* Harold W. Kuhn
* Herbert Simon
* Hervé Moulin
* Jean Tirole
* Jean-François Mertens
* Jennifer Tour Chayes
* John Harsanyi
* John Maynard Smith
* Antoine Augustin Cournot
* John Nash
* John von Neumann
* Kenneth Arrow
* Kenneth Binmore
* Leonid Hurwicz
* Lloyd Shapley
* Melvin Dresher
* Merrill M. Flood
* Olga Bondareva
* Oskar Morgenstern
* Paul Milgrom
* Peyton Young
* Reinhard Selten
* Robert Axelrod
* Robert Aumann
* Robert B. Wilson
* Roger Myerson
* Samuel Bowles
* Suzanne Scotchmer
* Thomas Schelling
* William Vickrey

See also|

* All-pay auction
* Alpha–beta pruning
* Bertrand paradox
* Bounded rationality
* Combinatorial game theory
* Confrontation analysis
* Coopetition
* First-move advantage in chess
* Game mechanics
* Glossary of game theory
* List of game theorists
* List of games in game theory
* No-win situation
* Solving chess
* Topological game
* Tragedy of the commons
* Tyranny of small decisions

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