However, he made a reference player avaible for those who want to play against a perfect opponent. However, he did not not publish any deeper analysis such as which moves win by which margin, so his results are of limited importance for real play. After creation of 39 GB of endgame databases (all positions with 34 or fewer seeds), 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.Īnders Carstensen solved four variants of Kalah(6/6) including the standard game in 2011. Mark Rawlings, of Gaithersburg, MD, has quantified the magnitude of the first player win in Kalah(6/6) with the 'empty capture' rule (October 2015). Game-theoretic values for different instances of Kalah However, this research was based on the 'Empty Capture' variant. If played perfectly, the game-theoretic value depends on the number of seeds in each hole and the number of holes per row. Minor mistakes in the programming, which nevertheless can give quite different results, will usually go unnoticed.ĭonkers strongly solved (according to Allis definition) Kalah for small instances using full-game databases and weakly for larger instances by Geoffrey Irving et al. This reveals a major problem: Most research done in solving games is not really peer-reviewed. However, both sites ( and the Oracle) have been taken off the internet and no further research appears to be possible.

Therefore it seems that the programming of Romein and Bal had major flaws and that their results are not valid. Víktor Bautista i Roca claimed on his now defunct homepage that several Awari endgames were incorrectly analyzed by the ' Awari Oracle' based on Romein's and Bal's research. The only perfect move to start a game is pit 6 (the last pit). Romein and Bal proved in 2002 that the game-theoretic value of Awari is a draw. Many algorithms rely on a huge pre-generated database, and are effectively nothing more than that. 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. 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. even if mistakes have already been made on one or both sides. Strong: The strongest sense of solution requires an algorithm which can produce perfect play from any position, i.e.

Weak: More typically, solving a game means providing 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.

Any game using the pie rule is either a draw or a second-player win. This can be a non-constructive proof (possibly involving a strategy stealing argument) that may not actually help determine this perfect play. Ultra-weak: In the weakest sense, solving a game means proving whether the first player will win, lose, or draw from the initial position, given perfect play on both sides. Some trivial games have also been completely analyzed: Micro-Wari and Nano-Wari.Ī two-player game can be solved on several levels: The game-theoretic value is known for Awari (draw), Kalah (depending on the instance), MiniMancala (draw), Ohvalhu (first-player win). In a two-player game it can be a win (W), loss (L) or draw (D) for the first player.Ī few mancala games have been solved. The game-theoretic value is the outcome of a game, when played perfectly from its initial position. Games which have not been solved are said to be 'unsolved'. A solved game is a game whose outcome can be correctly predicted from any position when each side plays optimally.

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