AI 中文总结
本文证明除数尼姆游戏中SG值受最小堆大小约束，改进周期界并推导公共周期，提出两个关于SG上界和周期的猜想。
AI 中文摘要
在除数尼姆游戏中，玩家从一个堆中移除的正数石子数必须整除其他每个堆的大小。我们证明 Sprague--Grundy (SG) 值满足 $g(P)\le 2h_{\min}+\lfloor\log_2h_{\min}\rfloor$，其中 $h_{\min}$ 是最小的非空堆，该界与其它堆的数量和大小无关。证明从奇数堆开始，并根据移除数中因子 $2$ 的个数对移除进行分类。初始棋盘中的最大堆随后为整个后继链提供有效的 SG 界。当某个堆变化时，我们改进了 Morisawa 的最终周期界：最小的固定堆控制 SG 值范围、比较的连续值数量，以及变化堆中重复合法移动所需的增量。对固定棋盘的剩余依赖性体现在直接后继的 SG 周期中。利用最大的固定堆，我们为所有这些递归层推导出一个显式的公共周期。其可能的素因子属于一个固定的有限集合，与堆的数量无关；它们的指数受固定石子总数的控制。最后，在共享除数规则下合并棋盘保持并可能收紧 SG 上限。最小 2-adic 深度及其计数奇偶性决定每个合并结果，且包含至少两个奇数堆的合并具有精确的 SG 值 $0$ 或 $1$。我们以两个猜想结束：$g(P)\le2h_{\min}$，以及当一个堆变化时，最小最终 SG 周期整除 $2\operatorname{lcm}(1,\ldots,h_{\max})$。这里 $h_{\max}$ 是最大的固定堆，不包括变化堆。第二个主张将同时从周期界中去除对堆数量的依赖。
英文摘要
In Divisor Nim, a player removes from one heap a positive number of stones that divides the size of every other heap. We prove that the Sprague--Grundy (SG) value satisfies $g(P)\le 2h_{\min}+\lfloor\log_2h_{\min}\rfloor$, where $h_{\min}$ is the smallest nonempty heap, independently of the number and sizes of the other heaps. The proof starts with odd heaps and classifies removals by their factors of $2$. The largest heap in a starting board then supplies an SG bound valid throughout every successor chain. When one heap varies, we sharpen Morisawa's eventual-period bound: the smallest fixed heap controls the SG value range, the number of consecutive values compared, and an increase in the varying heap that repeats legal moves. The remaining dependence on the fixed board lies in the SG periods of direct successors. Using the largest fixed heap, we derive an explicit common period for all these recursive layers. Its possible prime divisors belong to a fixed finite set, independent of heap count; their exponents remain controlled by the total fixed stones. Finally, joining boards under one shared divisor rule preserves and can tighten the SG ceiling. Minimum 2-adic depth and its count parity determine every joined outcome, and a join with at least two odd heaps has exact SG value $0$ or $1$. We close with two conjectures: $g(P)\le2h_{\min}$, and that the least eventual SG period, when one heap varies, divides $2\operatorname{lcm}(1,\ldots,h_{\max})$. Here $h_{\max}$ is the largest fixed heap, excluding the varying heap. The second claim would remove heap-count dependence from the period bound as well.
Comments23 pages, 15 tables
