On Aperiodic Subtraction Games with Bounded Nim Sequence

TL;DR

This paper constructs a subtraction game with a ternary, aperiodic Sprague-Grundy sequence and develops a theory that could generalize this construction, advancing understanding of complex subtraction game behaviors.

Contribution

It introduces a novel subtraction game with an aperiodic, ternary Sprague-Grundy sequence and proposes a theoretical framework for potential generalization.

Findings

Constructed an example of an aperiodic ternary Sprague-Grundy sequence

Developed a theory for analyzing such sequences in subtraction games

Potential for generalizing the construction to broader classes of games

Abstract

Subtraction games are a class of impartial combinatorial games whose positions correspond to nonnegative integers and whose moves correspond to subtracting one of a fixed set of numbers from the current position. Though they are easy to define, sub- traction games have proven difficult to analyze. In particular, few general results about their Sprague-Grundy values are known. In this paper, we construct an example of a subtraction game whose sequence of Sprague-Grundy values is ternary and aperiodic, and we develop a theory that might lead to a generalization of our construction.