Mixed-Radix Nim

TL;DR

This paper explores take-away games with Sprague-Grundy functions based on Nim sums in mixed bases, providing conditions for minimal and maximum elements within this class.

Contribution

It introduces a framework for mixed-base Nim games, offering a necessary and sufficient condition for minimality and a recursive method to construct the maximum.

Findings

Characterization of the set of mixed-base Nim games

Necessary and sufficient condition for the existence of minimal games

Recursive construction of the maximum game in the set

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 $\Delta_\beta$.