$m$ -Wythoff Nim和$p$ -互补Beatty序列的限制
Restrictions of $m$-Wythoff Nim and $p$-complementary Beatty Sequences
摘要：
修正正整数$m$。游戏的\emph{$m$-Wythoff Nim}（A.S. Fraenkel，1982）是\emph{Wythoff Nim}的一个著名的扩展，又名“角落的女王”。它的$P$ -位置的集合可以由一对非负整数的递增序列表示。众所周知，这些序列是所谓的\emph{complementary homogeneous} \emph{Beatty sequences}，即它们满足Beatty定理。对于正整数$p$，我们将$m$-Wythoff Nim的解推广到一对\emph{$p$-complementary} --每个正整数恰好出现$p$次--齐次Beatty序列$a = (a_n)_{n\in \M}$和$b = (b_n)_{n\in \M}$，对于所有的$n$，它们满足$b_n - a_n = mn$ .通过后者的性质，我们表明，$a$和$b$是唯一的\emph{all}对非递减$p$互补序列。我们证明了这样的对可以被划分为$p$对互补的贝蒂序列。我们的主要结果是，$\{\{a_n,b_n\}\mid n\in \M\}$代表的解决方案，三个新的“$p$ -限制”的$m$ -Wythoff尼姆--其中之一有一个\emph{blocking maneuver}的\emph{rook-type}选项。C. Kimberling证明了Wythoff Nim的解满足\emph{complementary equation} $x_{x_n}=y_n - 1$。我们将这个公式推广到我们的对$a$和$b$所满足的某个“$p$ -互补方程”。我们还表明，一个可以获得我们的新的序列对由三个所谓的\emph{Minimal EXclusive}算法。我们以Aviezri Fraenkel的附录结束。
英文摘要：
Fix a positive integer $m$. The game of \emph{$m$-Wythoff Nim} (A.S. Fraenkel, 1982) is a well-known extension of \emph{Wythoff Nim}, a.k.a 'Corner the Queen'. Its set of $P$-positions may be represented by a pair of increasing sequences of non-negative integers. It is well-known that these sequences are so-called \emph{complementary homogeneous} \emph{Beatty sequences}, that is they satisfy Beatty's theorem. For a positive integer $p$, we generalize the solution of $m$-Wythoff Nim to a pair of \emph{$p$-complementary}---each positive integer occurs exactly $p$ times---homogeneous Beatty sequences $a = (a_n)_{n\in \M}$ and $b = (b_n)_{n\in \M}$, which, for all $n$, satisfies $b_n - a_n = mn$. By the latter property, we show that $a$ and $b$ are unique among \emph{all} pairs of non-decreasing $p$-complementary sequences. We prove that such pairs can be partitioned into $p$ pairs of complementary Beatty sequences. Our main results are that $\{\{a_n,b_n\}\mid n\in \M\}$ represents the solution to three new '$p$-restrictions' of $m$-Wythoff Nim---of which one has a \emph{blocking maneuver} on the \emph{rook-type} options. C. Kimberling has shown that the solution of Wythoff Nim satisfies the \emph{complementary equation} $x_{x_n}=y_n - 1$. We generalize this formula to a certain '$p$-complementary equation' satisfied by our pair $a$ and $b$. We also show that one may obtain our new pair of sequences by three so-called \emph{Minimal EXclusive} algorithms. We conclude with an Appendix by Aviezri Fraenkel.
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