Mathematics – Combinatorics
Scientific paper
2010-05-23
Mathematics
Combinatorics
11 pages, 2 figures
Scientific paper
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.
Fraenkel Aviezri S.
Hegarty Peter
Larsson Urban
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