3 dimensional Wythoff Nim

TL;DR

This paper introduces a three-pile generalization of Wythoff Nim, analyzing its P-positions, their properties, and conjecturing their asymptotic behavior related to algebraic numbers of degree 5.

Contribution

It presents a novel three-pile version of Wythoff Nim, characterizes its P-positions, and conjectures their asymptotic distribution involving algebraic numbers.

Findings

P-positions have finite difference properties

P-positions produce a partition of positive integers

Conjecture on asymptotic approximation by a half-line with algebraic slope

Abstract

We introduce a new generalization of Wythoff Nim using three piles of stones. We show that its P-positions have finite difference properties and produce a partition of positive integers. Further, we give a conjecture that the P-positions approximate a half-line whose slope is described by algebraic numbers of degree 5.