Solving Cram using Combinatorial Game Theory

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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 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.
Original languageEnglish
Title of host publicationAdvances in Computer Games: 16th International Conference, ACG 2019
EditorsTristan Cazenave, Jaap van den Herik, Abdallah Saffidine, I-Chen Wu
PublisherSpringer
Pages91-105
Volume12516
Publication statusPublished - Dec 2020
Event16th International Conference, ACG 2019 - Macao, China, Macao, China
Duration: 11 Aug 201913 Aug 2019

Publication series

SeriesLecture Notes in Computer Science
Volume12516
ISSN0302-9743

Conference

Conference16th International Conference, ACG 2019
Abbreviated titleACG
Country/TerritoryChina
CityMacao
Period11/08/1913/08/19

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