An algorithm for computing indistinguishability quotients in misére impartial combinatorial games

Summary

This paper advances the theory of impartial misère octal games by developing an algorithm for finding certain infinite quotient monoids. The notion of a misère quotient monoid was introduced by Thane Plambeck, who also, together with Aaron Siegel, gave an algorithm for finding finite misère quotients. This paper examines the periodicity of outcomes when changing the number of heaps of various sizes. The quotient monoid for misère 0.3122 up to heaps of size 7 is found. It is the first example of an infinite misère quotient monoid.

This paper gives an algorithm for computing certain misère indistinguishability quotient monoids. The approach employed here is not the genus theory of [Berlekamp et al. 2003, Chapter 13], but rather the quotient monoid approach introduced by Thane Plambeck [2005].

The algorithm described here was initially designed to analyze octal games, but is also valid for a broader class of games which will be called “heap rulesets”.

The notion of a heap ruleset comes from Nim, which is played with heaps of beans. The rules of Nim, and its variations, specify how a player may remove beans from a heap. The terminology below is influenced greatly by play of Nim, and readers may wish to keep games like Nim in mind when reading this paper. However, the collection of heap rulesets includes many other impartial games whose standard descriptions do not involve heaps. Chomp and Cram are examples.