Nim on hypercubes

TL;DR

This paper completely solves the game of Nim played on hypercube graphs with unit weights, extending classical Nim to a graph-theoretic setting and discussing the more complex weighted case.

Contribution

It provides a complete solution for Nim on hypercubes with unit weights and explores the connection to known results in weighted cases.

Findings

Solved Nim on hypercubes with unit weights

Discussed the arbitrary weight case and its relation to existing results

Extended classical Nim to a graph-theoretic context

Abstract

The ordinary game of Nim has a long history and is well-known in the area of combinatorial game theory. The solution to the ordinary game of Nim has been known for many years and lends itself to numerous other solutions to combinatorial games. Nim was extended to graphs by taking a fixed graph with a playing piece on a given vertex and assigning positive integer weight to the edges that correspond to a pile of stones in the ordinary game of Nim. Players move alternately from the playing piece across incident edges, removing weight from edges as they move. This paper solves Nim on hypercubes in the unit weight case completely. We briefly discuss the arbitrary weight case and its ties to known results.