An Aperiodic Subtraction Game of Nim-dimension Two

TL;DR

This paper introduces a new infinite subtraction game with an aperiodic Sprague-Grundy function, demonstrating a specific example with unique nim-dimension properties related to Fibonacci numbers.

Contribution

It provides an elementary example of an aperiodic subtraction game with a specified nim-dimension, expanding understanding of such games and their properties.

Findings

The game has an aperiodic Sprague-Grundy function.

The group of nim-values has order four, indicating a nim-dimension of two.

The example clarifies the relationship between Fibonacci-based subtraction sets and game properties.

Abstract

In a recent arXiv-manuscript Fox studies infinite subtraction games with a finite (ternary) and aperiodic Sprague-Grundy function. Here we provide an elementary example of a game with the given properties, namely the game given by the subtraction set $\{F_{2n+1}-1\}$, where $F_i$ is the $i$th Fibonacci number, and where $n$ ranges over the positive integers. Our definition of nim-dimension reflects the precise number of power-of-two-components generated by the games; the group of nim-values is of order four so the dimension is two (in the classical definition this dimension would have been one). Thanks to Carlos Santos for an enlightening discussion on this matter.