Abstract
In this paper we investigate the board game Cram, which is an impartial combinatorial game, using an \(\alpha \beta \) solver. Since Cram is a notoriously hard game in the sense that it is difficult to obtain reliable and useful domain knowledge to use in the search process, we decided to rely on knowledge obtained from Combinatorial Game Theory (CGT).
The first and most effective addition to our solver is to incorporate endgame databases pre-filled with CGT values (nimbers) for all positions fitting on boards with at most 30 squares. This together with two efficient move-ordering heuristics aiming at early splitting positions into fragments fitting in the available databases gives a large improvement of solving power. Next we define five more heuristics based on CGT that can be used to further reduce the sizes of the search trees considerably.
In the final version of our program we were able to solve all odd \(\times \) odd Cram boards for which results were available from the literature (even \(\times \) even and odd \(\times \) even boards are trivially solved). Investigating new boards led to solving two boards not solved before, namely the \(3 \times 21\) board, a first-player win, and the \(5 \times 11\) board, a second-player win.
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Notes
- 1.
By a heuristic we mean a method that does not impact the final results, but only (hopefully) obtains the results faster.
- 2.
We use chess-like notation for moves, where squares are indicated by their column (‘a’, ‘b’, etc, from left to right) and their row (‘1’, ‘2’, etc, from bottom to top). A move consists of the two squares covered separated by a colon.
- 3.
The fact that these move-ordering heuristics are so successful does not contradict our statement that it is hard to find criteria indicating which moves are probably good or bad. It just leads to investigating first moves with small search trees (including hopefully “accidentally” winning moves).
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Uiterwijk, J.W.H.M. (2020). Solving Cram Using Combinatorial Game Theory. In: Cazenave, T., van den Herik, J., Saffidine, A., Wu, IC. (eds) Advances in Computer Games. ACG 2019. Lecture Notes in Computer Science(), vol 12516. Springer, Cham. https://doi.org/10.1007/978-3-030-65883-0_8
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