Grundy values of Fibonacci nim

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 (Fibonacci Quart 1:9–13, 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.