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 and/or computer assistance.
- Ultra-weak solution
- Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides. This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any moves of the perfect play.
- Weak solution
- Provide an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game.
- Strong solution
- Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.
Despite their name, many game theorists believe that "ultra-weak" proofs are the deepest, most interesting and valuable. "Ultra-weak" proofs require a scholar to reason about the abstract properties of the game, and show how these properties lead to certain outcomes if perfect play is realized.
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.
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. 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.
- 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.
- Strongly solved. If 2 players both play perfectly, the game will go on indefinitely.
- Connect Four
- Solved first by James D. Allen on October 1, 1988, and independently by Victor Allis on October 16, 1988. The first player can force a win. Strongly solved by John Tromp's 8-ply database (Feb 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (as well as 8×8 in late 2015) (Feb 18, 2006).
- Weakly solved by Maarten Schadd. The game is a draw.
- Free gomoku
- Solved by Victor Allis (1993). The first player can force a win without opening rules.
- Solved by Alan Frank using the Official Scrabble Players Dictionary in 1987.
- 3×3 variant solved as a win for black, several other larger variants also solved.
- 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. Mark Rawlings, of Gaithersburg, MD, has quantified the magnitude of the first player win in the (6/6) variant (2015). After creation of 39 GB of endgame databases, searches totaling 106 days of CPU time and over 55 trillion nodes, it was proven that, with perfect play, the first player wins by 2. Note that all these results refer to the Empty-pit Capture variant and therefore are of very limited interest for the standard game. Analysis of the standard rule game has now been posted for Kalah(6,4), which is a win by 8 for the first player, and Kalah(6,5), which is a win by 10 for the first player. Analysis of Kalah(6,6) with the standard rules is on-going, however, it has been proven that it is a win by at least 4 for the first player.
- L game
- Easily solvable. Either player can force the game into a draw.
- Losing chess
- Weakly solved as a win for white beginning with 1. e3.
- Maharajah and the Sepoys
- This asymmetrical game is a win for the sepoys player with correct play.
- Strongly solved.
- Nine men's morris
- Solved by Ralph Gasser (1993). Either player can force the game into a draw.
- Order and Chaos
- Order (First player) wins.
- Weakly solved by humans, but proven by computers. (Dakon is, however, not identical to Ohvalhu, the game which actually had been observed by de Voogt)
- Strongly solved by Jason Doucette (2001). 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.
- Strongly solved by Geoffrey Irving with use of a supercomputer at NERSC. The first player wins.
- Weakly solved by H. K. Orman. It is a win for the first player.
- Solved by Luc Goossens (1998). Two perfect players will always draw.
- Weakly solved by Oren Patashnik (1980) and Victor Allis. The first player wins.
- Renju-like game without opening rules involved
- Claimed to be solved by János Wagner and István Virág (2001). A first-player win.
- Weakly solved: win for the second player.
- Solved by Guy Steele (1998). Depending on the variant either a first-player win or a draw.
- Three men's morris
- Trivially solvable. Either player can force the game into a draw.
- Three Musketeers
- Strongly solved by Johannes Laire in 2009, and weakly solved by Ali Elabridi in 2017. It is a win for the blue pieces (Cardinal Richelieu's men, or, the enemy).
- Trivially strongly solvable because of the small game tree. The game is a draw if no mistakes are made, with no mistake possible on the opening move.
- Tigers and Goats
- Weakly solved by Yew Jin Lim (2007). The game is a draw.
- Wythoff's game
- Strongly solved by W. A. Wythoff in 1907.
- 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. Checkers has a search space of 5×1020 possible game positions. 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.
- Othello (Reversi)
- Weakly solved in 2023 by Hiroki Takizawa, a researcher at Preferred Networks. From the standard starting position on an 8x8 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.
Partially solved games
- Fully solving chess remains elusive, and it is speculated that the complexity of the game may preclude its ever being solved. Through retrograde computer analysis, 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. The 7×7 board was weakly solved in 2015. Humans usually play on a 19×19 board which is over 145 orders of magnitude more complex than 7×7.
- 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). On particular board sizes, more is known: it is strongly solved by several computers for board sizes up to 6×6. Weak solutions are known for board sizes 7×7 (using a swapping strategy), 8×8, and 9×9; in the 8×8 case, a weak solution is known for all opening moves. Strongly solving Hex on an N×N board is unlikely as the problem has been shown to be PSPACE-complete. 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.
- International draughts
- All endgame positions with two through seven pieces were solved, as well as positions with 4×4 and 5×3 pieces where each side had one king or fewer, positions with five men versus four men, positions with five men versus three men and one king, and positions with four men and one king versus four men. The endgame positions were solved in 2007 by Ed Gilbert of the United States. Computer analysis showed that it was highly likely to end in a draw if both players played perfectly.[better source needed]
- 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.
- Computer chess
- Computer Go
- Computer Othello
- Game complexity
- God's algorithm
- Zermelo's theorem (game theory)
- Allis, Louis Victor (1994-09-23). Searching for Solutions in Games and Artificial Intelligence (PhD thesis). Maastricht: Rijksuniversiteit Limburg. ISBN 90-9007488-0.
- H. Jaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck, Games solved: Now and in the future, Artificial Intelligence 134 (2002) 277–311.
- "John's Connect Four Playground". tromp.github.io.
- "UCI Machine Learning Repository: Connect-4 Data Set". archive.ics.uci.edu.
- 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" (PDF). New Mathematics and Natural Computation. 4 (3): 369–387. doi:10.1142/S1793005708001124. Archived from the original (PDF) on 2016-03-04. Retrieved 2015-04-08.
- Frank, Alan (1987-08-01). "Ghostbusters". Word Ways. 20 (4).
- Price, Robert. "Hexapawn". www.chessvariants.com.
- Solving Kalah by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk.
- Solving (6,6)-Kalaha by Anders Carstensen.
- Watkins, Mark. "Losing Chess: 1. e3 wins for White" (PDF). Retrieved 17 January 2017.
- Bouton, C. L. (1901–1902), "Nim, a game with a complete mathematical theory", Annals of Mathematics, 3 (14): 35–39, doi:10.2307/1967631, JSTOR 1967631
- Gasser, Ralph (1996). "Solving Nine Men's Morris". In Nowakowski, Richard (ed.). Games of No Chance (PDF). Vol. 29. Cambridge: Cambridge University Press. pp. 101–113. ISBN 9780521574112. Archived from the original (PDF) on 2015-07-24. Retrieved 2022-01-03.
- Nine Men's Morris is a Draw by Ralph Gasser
- "solved: Order wins - Order and Chaos".
- Pangki is strongly solved as a Draw by Jason Doucette
- Hilarie K. Orman: Pentominoes: A First Player Win in Games of no chance, MSRI Publications – Volume 29, 1996, pages 339-344. Online: pdf.
- http://wouterkoolen.info/Talks/quarto.pdf[bare URL PDF]
- "414298141056 Quarto Draws Suffice!".
- "Quarto". Archived from the original on 2004-10-12.
- Teeko, by E. Weisstein
- Elabridi, Ali. "Weakly Solving the Three Musketeers Game Using Artificial Intelligence and Game Theory" (PDF).
- Three Musketeers, by J. Lemaire
- Tic-Tac-Toe, by R. Munroe
- Yew Jin Lim. On Forward Pruning in Game-Tree Search Archived 2009-03-25 at the Wayback Machine. Ph.D. Thesis, National University of Singapore, 2007.
- Wythoff, W. A. (1907), "A modification of the game of nim", Nieuw Archief voor Wiskunde, 7 (2): 199–202
- Schaeffer, Jonathan (2007-07-19). "Checkers Is Solved". Science. 317 (5844): 1518–22. Bibcode:2007Sci...317.1518S. doi:10.1126/science.1144079. PMID 17641166. S2CID 10274228.
- "Project - Chinook - World Man-Machine Checkers Champion". Retrieved 2007-07-19.
- Mullins, Justin (2007-07-19). "Checkers 'solved' after years of number crunching". NewScientist.com news service. Retrieved 2020-12-06.
- Takizawa, Hiroki (2023-10-30). "Othello is Solved". arXiv:2310.19387 [cs.AI].
- 5×5 Go is solved by Erik van der Werf
- "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". blog.sina.com.cn. (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)
- Counting legal positions in Go Archived 2007-09-30 at the Wayback Machine, Tromp and Farnebäck, accessed 2007-08-24.
- 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.
- Some of the nine-piece endgame tablebase by Ed Gilbert
- Allis, Beating the World Champion? The state-of-the-art in computer game playing. in New Approaches to Board Games Research.