Circular Nim Games

TL;DR

This paper studies the structure of losing positions in Circular Nim, a two-player game involving token stacks arranged in a circle, providing initial results and open questions for future research.

Contribution

It characterizes losing positions for small n and k in Circular Nim and introduces open problems for further exploration.

Findings

Identified structure of losing positions for small n and k

Proposed open questions for larger n and k

Analyzed the game's combinatorial properties

Abstract

A circular Nim game is a two player impartial combinatorial game consisting of n stacks of tokens placed in a circle. A move consists of choosing k consecutive stacks, and taking at least one token from one or more of the k stacks. The last player able to make a move wins. We prove results on the structure of the losing positions for small n and k and pose some open questions for further investigations.