Aperiodic Subtraction Games

Abstract

Periodicity is a fundamental property of many combinatorial games. It is sought vigorously, yet remains elusive in important cases, such as for some octal games, notably Grundy's game. Periodicity is important, because it provides poly-time winning strategies for many games. In particular, subtraction games, impartial and partizan, have been proved to be periodic. Our main purpose here is to exhibit constructively a class of subtraction games which is demonstratively aperiodic and yet is shown to have linear-time winning strategies.