Invariant and dual subtraction games resolving the Duch\^e-Rigo conjecture

TL;DR

This paper proves a conjecture linking complementary Beatty sequences to invariant impartial games, generalizing subtraction games and establishing a duality between sequences and game positions.

Contribution

It introduces a broader class of invariant subtraction games based on complementary sequences and proves a duality principle connecting sequences with game positions.

Findings

Proves the Duchêne-Rigo conjecture for a class of sequences.

Defines a generalized notion of subtraction games.

Establishes conditions for the duality between sequences and game positions.

Abstract

We prove a recent conjecture of Duch\^ene and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an \emph{invariant} impartial game. Here invariance means that each available move in a game can be played anywhere inside the game-board. In fact, we establish such a result for a wider class of pairs of complementary sequences, and in the process generalize the notion of a \emph{subtraction game}. Given a pair of complementary sequences $(a_n)$ and $(b_n)$ of positive integers, we define a game $G$ by setting $\{\{a_n, b_n\}\}$ as invariant moves. We then introduce the invariant game $G^\star $, whose moves are all non-zero $P$-positions of $G$. Provided the set of non-zero $P$-positions of $G^\star$ equals $\{\{a_n,b_n\}\}$, this \emph{is} the desired invariant game. We give sufficient conditions on the initial pair of sequences for this 'duality' to hold.