解决Duchê-Rigo猜想的不变和对偶减法博弈
Invariant and dual subtraction games resolving the Duchê-Rigo conjecture
摘要：
我们证明了Duchêne和Rigo最近提出的一个猜想，即每一对互补的齐次Beatty序列代表了\emph{invariant}公平博弈的解。在这里，不变性意味着游戏中的每一步都可以在游戏板内的任何地方进行。事实上，我们建立这样的结果，为更广泛的一类互补序列对，并在此过程中推广的概念，\emph{subtraction game}。给定一对正整数互补序列$(a_n)$和$(b_n)$，通过将$\{\{a_n, b_n\}\}$设为不变移动，定义了一个博弈$G$.然后，我们介绍了不变的游戏$G^\star $，其移动都是非零的$P$ -位置的$G$。假设$G^\star$的非零$P$ -位置的集合等于$\{\{a_n,b_n\}\}$，则这\emph{is}是期望的不变博弈。我们给出了充分条件的初始序列对这种“对偶”举行。
英文摘要：
We prove a recent conjecture of Duchêne 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.
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