Odd case of 2-pile divisor Nim

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2025年6月3日

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Takayuki Morisawa

摘要

We study the Sprague–Grundy value g(m, n) of 2-pile divisor Nim which is the variation of 2-pile Nim. In particular, we fix an odd integer m and focus our attention on the periodicity of g(m, n) with respect to a non-negative integer n. The aim of the present paper is to show that g(m, n) is periodic under some assumptions on the periodicity of \(g( m', n )\) for any \(m' < m\) . Moreover, this result gives us the fundamental period without concrete calculation of g(m, n). For example, we obtain that the fundamental period of g(7, n) with respect to n is equal to 120.