On Nim-like games whose Sprague-Grundy functions are the same
Yuki Irie · arXiv (Cornell University) · 2018
Some games have the same Sprague-Grundy functions. For a mixed radix numeral system $b$, let $\sigma^b$ be the function that maps $(x^0, \ldots, x^{m - 1}) \in \mathbb{N}^m$ to $x^0 \oplus_b \cdots \oplus_b x^{m - 1}$, where $\oplus_b$ is addition without carry in $b$. We present variants of Nim whose Sprague-Grundy functions equal $\sigma^b$. Let $\Delta^b$ be the set of such games. When $b$ is the binary numeral system, we determine $\Delta^{b}$. In general, we give a characterization of $b$ such that $\Delta^{b}$ has a unique minimal element, and a construction of the maximum element of $\Delta^{b}$.