Solving Cram using Combinatorial Game Theory

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 prefilled 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 x odd
Cram boards for which results were available from the literature (even
x even and odd x even boards are trivially solved). Investigating new
boards led to solving two boards not solved before, namely the 3 x 21
board, a first-player win, and the 5 x 11 board, a second-player win.