Invariant and dual subtraction games resolving the Duchê-Rigo conjecture

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.