Solved game

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

Ultra-weak

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

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

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 an example of a strong solution, the game of tic-tac-toe is 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.[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.

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 2 players both play perfectly, the game will go on indefinitely.

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).

Fanorona

Weakly solved by Maarten Schadd. The game is a draw.[5]

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.[6]

Hexapawn

3×3 variant solved as a win for black, several other larger variants also solved.[7]

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.[8][9] 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.[10]

Maharajah and the Sepoys

This asymmetrical game is a win for the sepoys player with correct play.

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. (Dakon is, however, not identical to Ohvalhu, the game which actually had been observed by de Voogt)

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.

Pentago

Strongly solved by Geoffrey Irving with use of a supercomputer at NERSC. The first player wins.

Pentominoes

Weakly solved by H. K. Orman.[16] It is a win for the first player.

Quarto

Solved by Luc Goossens (1998). Two perfect players will always draw.[17][18][19]

Qubic

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.

Sim

Weakly solved: win for the second player.

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.

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

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.

Tigers and Goats

Weakly solved by Yew Jin Lim (2007). The game is a draw.[24]

Wythoff's game

Strongly solved by W. A. Wythoff in 1907.[25]

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.[26] Checkers is the largest game that has been solved to date, with a search space of 5×10²⁰.[27] 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.[28]

Tigers and Goats

Weakly solved by Yew Jin Lim (2007). The game is a draw.[24]

Pentominoes

Weakly solved by H. K. Orman. It is a win for the first player.[16]

Chess

Main article: Solving chess

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.

Go

The 5×5 board was weakly solved for all opening moves in 2002.[29] The 7×7 board was weakly solved in 2015.[30] Humans usually play on a 19×19 board which is over 145 orders of magnitude more complex than 7×7.[31]

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). 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.[32] 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.[33]

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.

Reversi (Othello)

Weakly solved on a 4×4 and 6×6 board as a second player win in July 1993 by Joel Feinstein.[34] On an 8×8 board (the standard one) it is mathematically unsolved, though computer analysis shows a likely draw. No strongly supposed estimates other than increased chances for the starting player (Black) on 10×10 and greater boards exist.

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

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

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