The Geometry of Nim

TL;DR

This paper explores the geometric relationship between the Sierpinski triangle and Nim, introducing a high-dimensional analog and showing their equivalence in finite dimensions.

Contribution

It introduces a new high-dimensional Sierpinski analog and establishes a geometric interpretation of Nim's losing positions, proving their equivalence.

Findings

High-dimensional Sierpinski analog defined

Geometric interpretation of Nim's losing positions established

Equivalence shown in finite dimensions

Abstract

We relate the Sierpinski triangle and the game of Nim. We begin by defining both a new high-dimensional analog of the Sierpinski triangle and a natural geometric interpretation of the losing positions in Nim, and then, in a new result, show that these are equivalent in each finite dimension.