On the expansion of three-element subtraction sets

TL;DR

This paper investigates the periodicity of nim-sequences in three-element subtraction games, providing solutions for various cases, exploring how to augment subtraction sets without altering sequences, and proposing a conjecture on bipartite games.

Contribution

It offers new insights into the structure and augmentation of subtraction sets in combinatorial game theory, especially for three-element sets.

Findings

Identified conditions for periodicity in three-element subtraction games

Described methods to augment subtraction sets without changing nim-sequences

Proposed a conjecture on the nature of ultimately bipartite subtraction games

Abstract

We study the periodicity of nim-sequences for subtraction games having subtraction sets with three elements. In particular, we give solutions in several cases, and we describe how these subtraction sets can be augmented by additional numbers without changing the nim-sequences. The paper concludes with a conjecture on ultimately bipartite subtraction games.