减法游戏中获胜位置的超多项式周期长度
Superpolynomial period lengths of the winning positions in the subtraction game
AI总结：
本文研究减法游戏中获胜位置序列的周期长度，证明初始种子在集合大小为1或2时不能改变周期长度，但在大小为3时可使周期长度变为超多项式函数。
摘要：
给定一个有限正整数集合 $A$，从一堆 $n$ 个筹码开始，Alice 和 Bob 轮流行动，每轮玩家选择一个 $x\in A$，要求 $x$ 小于或等于当前筹码数，并从堆中减去 $x$ 个筹码。当当前筹码数小于 $\min\{A\}$ 且无法继续行动时，游戏终止。最后一步行动的玩家获胜。我们可以定义 $w^A(n)$ 为 $1$，如果 Alice 在初始堆为 $n$ 个筹码时拥有获胜策略；为 $0$，如果 Bob 拥有获胜策略。根据鸽巢原理，$w^A(n)$ 会变成周期性的，并且容易看出周期长度至多是 $\max\{A\}$ 的指数函数。典型的周期长度是 $\max\{A\}$ 的线性函数，而指数周期长度是否可能是一个长期未解决的问题。我们考虑该游戏的一个轻微修改，引入一个初始种子 $S$，它告诉我们在少数初始筹码数下当前玩家还是对手玩家是获胜者。在本文中我们证明，如果 $A$ 的大小为 $1$ 或 $2$，初始种子不能改变 $w^A(n)$ 的周期长度，但当 $|A| \geq 3$ 时它可以改变周期长度。进一步，我们展示了一类大小为 $3$ 的集合 $A$ 以及相应的初始种子，使得周期长度成为 $\max\{A\}$ 的超多项式函数。
英文摘要：
Given a finite set of positive integers, $A$, and starting with a heap of $n$ chips, Alice and Bob alternate turns and on each turn a player chooses $x\in A$ with $x$ smaller or equal than the current number of chips and subtract $x$ chips from the heap. The game terminates when the current number of chips becomes smaller than $\min\{A\}$ and no moves are possible. The player who makes the last move is the winner. We can define $w^A(n)$ to be $1$ if Alice has a winning strategy with a starting heap of $n$ chips and $0$ if Bob has a winning strategy. By the Pigeonhole Principle, $w^A(n)$ becomes periodic, and it is easy to see that the period length is at most an exponential function of $\max\{A\}$. The typical period length is a linear function of $\max\{A\}$, and it is a long time open question if exponential period length is possible. We consider a slight modification of this game by introducing an initial seed $S$ that tells for the few initial numbers of chips whether the current or the opposite player is the winner. In this paper we show that the initial seed cannot change the period length of $w^A(n)$ if the size of $A$ is $1$ or $2$, but it can change the period length with $|A| \geq 3$. Further, we exhibit a class of sets $A$ of size $3$ and corresponding initial seeds such that the period length becomes a superpolynomial function of $\max\{A\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
