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

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.Comment: 11 pages, 2 figure

INVARIANT AND DUAL SUBTRACTION GAMES RESOLVING THE DUCHÊNE-RIGO CONJECTURE.

ABSTRACT. We prove a recent conjecture of Duchêne and Rigo, stating that every complementary pair of homogeneous Beatty sequences represents the solution to an 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 subtraction game. Given a pair of complementary sequences (an) and (bn) of positive integers, we define a game G by setting {{an, bn}} as invariant moves. We then introduce the invariant game G ⋆ , whose moves are all non-zero P-positions of G. Provided the set of non-zero P-positions of G ⋆ equals {{an, bn}}, this is the desired invariant game. We give sufficient conditions on the initial pair of sequences for this ’duality ’ to hold. 1. NOTATION, TERMINOLOGY AND STATEMENT OF RESULTS This note concerns 2-person, impartial games (see [BCG]) played under normal (as against misère) rules. Let N, N0 denote the positive and the non-negative integers respectively. For k ∈ N, let B = B(k): = (N k 0, ⊕, ≼) denote the partially-ordered semigrou