Odd case of 2-pile divisor Nim
Takayuki Morisawa · International Journal of Game Theory · 2025
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.