Mixed-Radix Nim

Abstract

We present take-away games whose Sprague-Grundy functions are given by the Nim sum of heap sizes in a mixed base $\beta$. Let ${\Delta }_{\beta }$ be the set of such games. We give a necessary and sufficient condition for the existence of a minimum of ${\Delta }_{\beta }$, and a recursive construction of the maximum of ${\Delta }_{\beta }$.