Othello is Solved

TL;DR

The paper addresses solving Othello, a game with an enormous state space estimated at around $10^{28}$ positions and $10^{58}$ possible game records, by achieving a weak solution on the $8\times 8$ board in which perfect play by both players leads to a draw. The approach combines modified Edax alpha-beta search with targeted subproblems: generating a subset of positions with $50$ empty squares and estimating values for positions with $36$ empty squares, aided by a series of algorithms to traverse the game-graph and verify hypotheses. The key contributions include the first weak solution for Othello, a detailed procedure for proving the initial position's draw via a finite set of verifiable hypotheses, and publicly released software and data to enable reproducibility. The work demonstrates a path toward solving complex perfect-information games and discusses future directions toward semi-strong solving and potential extensions to chess, reinforcing the practical significance for AI research and game theory.

Abstract

The game of Othello is one of the world's most complex and popular games that has yet to be computationally solved. Othello has roughly ten octodecillion (10 to the 58th power) possible game records and ten octillion (10 to the 28th power) possible game positions. The challenge of solving Othello, determining the outcome of a game with no mistake made by either player, has long been a grand challenge in computer science. This paper announces a significant milestone: Othello is now solved. It is computationally proved that perfect play by both players lead to a draw. Strong Othello software has long been built using heuristically designed search techniques. Solving a game provides a solution that enables the software to play the game perfectly.

Figures (4)

• Figure 1: (Left) The initial board position of $8\times 8$ Othello. (Right) A diagram of an optimal game record designated by our study. The game record is "F5D6C3D3 C4F4F6F3 E6E7D7C5 B6D8C6C7 D2B5A5A6 A7G5E3B4 C8G6G4C2 E8D1F7E2 G3H4F1E1 F2G1B1F8 G8B3H3B2 H5B7A3A4 A1A2C1H2 H1G2B8A8 G7H8H7H6". The numbers in the stones indicate the order of moves, and the colors of stones indicate the final result. Our study confirms that if a deviation from this record occurs at any point, our software, playing as the opponent, is guaranteed a draw or win.
• Figure 2: Positions with 36 empty squares that were solved to prove the initial position were sorted in descending order by the number of search phases reported by Edax, and the cumulative number (orange) and number of search phases for a single particular question (blue) were plotted for each $1/1000$.
• Figure 3: Positions with 36 empty squares that were solved to prove the initial position were classified according to the value of Algorithm \ref{['algo:Eval']} (horizontal axis) and the sum of the numbers of searched positions reported by Edax was calculated for each (vertical axis).
• Figure 4: A graphical representation of results about opening of Othello. The bold black moves show the optimal game record. Our perfect player always chooses the bold (black or gray) moves in the corresponding position. Right five positions are proved that all those game-theoretic values are draws. Center one with asterisk is the progress of Figure \ref{['fig:fig_game_record']}.