P-positions in Modular Extensions to Nim

TL;DR

This paper introduces and analyzes a new variant of Nim called m-Modular Nim, providing optimal strategies for specific cases, expanding the understanding of combinatorial game strategies.

Contribution

It develops a winning strategy for m-Modular Nim with two heaps and for odd m with any number of heaps, extending Nim theory to modular constraints.

Findings

Optimal strategy for 2-heap m-Modular Nim established.

Winning strategy for odd m with any number of heaps derived.

Enhanced understanding of Nim variants with modular rules.

Abstract

In this paper, we consider a modular extension to the game of Nim, which we call $m$-Modular Nim, and explore its optimal strategy. In $m$-Modular Nim, a player can either make a standard Nim move or remove a multiple of $m$ tokens in total. We develop a winning strategy for all $m$ with $2$ heaps and for odd $m$ with any number of heaps.