Grundy values of Fibonacci nim

TL;DR

This paper analyzes the Grundy values in Fibonacci nim, extending previous work by computing positions with Grundy values up to 3 and examining the growth of Grundy values after removing Fibonacci numbers.

Contribution

It extends Whinihan's analysis by explicitly computing Grundy values up to 3 and studying the growth behavior of Grundy values when Fibonacci numbers are excluded.

Findings

Positions with Grundy value at most 3 are characterized.

Removing Fibonacci numbers causes Grundy values to increase.

Bounds on the growth rate of Grundy values are established.

Abstract

In this article, we investigate the Grundy values of the popular game of Fibonacci nim. The winning strategy, which amounts to understanding positions of Grundy value 0, was known since Whinihan in 1963. In this paper, we extend Whinihan's analysis by computing all the positions of Grundy value at most 3. Furthermore, we show that, when we delete the Fibonacci numbers (which have Grundy value 0), the Grundy values of the starting positions are increasing, and we give upper and lower bounds on the growth rate.