Self-similarity of $\mathcal{P}$-positions of $(2n+1)$-dimensional Wythoff's game

TL;DR

This paper analyzes the structure of $\\mathcal{P}$-positions in odd-dimensional Wythoff's game, revealing self-similar fractal patterns, including the Sierpinski sponge in three dimensions.

Contribution

It explicitly characterizes $\\mathcal{P}$-positions for $(2n+1)$-dimensional Wythoff's game and uncovers their self-similar, fractal nature.

Findings

$\\mathcal{P}$-positions exhibit self-similarity.

3D $\\mathcal{P}$-positions form the Sierpinski sponge.

Explicit characterization of $\\mathcal{P}$-positions in odd dimensions.

Abstract

Wythoff's game as a classic combinatorial game has been well studied. In this paper, we focus on $(2n+1)$-dimensional Wythoff's game; that is the Wythoff's game with $(2n+1)$ heaps. We characterize their $\mathcal{P}$-positions explicitly and show that they have self-similar structures. In particular, the set of all $\mathcal{P}$-positions of $3$-dimensional Wythoff's game generates the well-known fractal set---the Sierpinski sponge.