Generalized Algorithm for Wythoff's Game with Basis Vector $(2^b,2^b)$

TL;DR

This paper introduces a generalized algorithm for identifying losing positions in Wythoff's Game variants where players can remove multiples of a fixed power of two, extending previous work on the (2,2) case.

Contribution

The paper develops an algorithm to generate P-Positions for the (a,a) game where a is a power of two, generalizing prior results and providing an inductive proof for its correctness.

Findings

The algorithm works for the first a^2 terms in the (a,a) game.

Constructs from smaller cases can generate positions for larger games.

Conjectures that the structure applies only when a is a power of two.

Abstract

Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), $C$, for Wythoff's Game is given by $C := \left\{ (\lfloor k\phi \rfloor, \lfloor k\phi^2 \rfloor), (\lfloor k\phi^2 \rfloor, \lfloor k\phi \rfloor) : k \in \mathbb Z_{\geq 0} \right\}$. An open Wythoff problem remains where players make the valid Nim moves or remove $kb$ stones from each pile, where $b$ is a fixed integer. We denote this as the $(b,b)$ game. For example, regular Wythoff's Game is just the $(1,1)$ game. In 2009, Duch${\^e}$ne and Gravier proved an algorithm to generate the set of P-Positions for the $(2,2)$ game by exploiting the periodic nature of the differences of stones between the two piles modulo $4$. We observe similar cyclic behaviour for any $b$, where $b$ is a power of $2$, modulo $b^2$, and construct an algorithm to generate the set of P-Positions for this game. Let $a$ be a power of $2$. We prove our algorithm works by first showing that it holds for the first $a^2$ terms in the $(a,a)$ game. Next, we construct an ordered multiset for the $(2a,2a)$ game from the $a^2$ terms, and an inductive proof follows. Moreover, we conjecture that all cyclic games require $a$ to be a power of $2$, suggesting that there is no similar structure in the generalised $(b,b)$ game where $b$ isn't a power of $2$. Future directions for generalising this result would likely utilise numeration systems, particularly the PV numbers.