A Variant $\beta$-Wythoff Nim on Beatty's Theorem

TL;DR

This paper explores a variant of $eta$-Wythoff Nim, establishing conditions under which two-pile take-away games have P-positions characterized by complementary Beatty sequences, linking combinatorial game theory with number theory.

Contribution

It introduces a new variant of $eta$-Wythoff Nim and provides simple rules for identifying P-positions using Beatty sequences, expanding understanding of combinatorial game structures.

Findings

Characterization of P-positions via Beatty sequences

Conditions for two-pile take-away games to satisfy the Beatty property

Connection between game positions and number theory

Abstract

We give short rules for two-pile take-away games satisfying that a pair of complementary homogeneous Beatty sequences together with $(0,0)$ constitute a complete set of $P$-positions.