标题： 论组合博弈中最优策略的存在性 显示英文标题

标题： On the Existence of Optimal Strategies in a Combinatorial Game

摘要： 我们研究了一个从德国全国数学竞赛问题中衍生出的组合游戏。 在这个游戏中，两名玩家轮流从一个有限的自然数集合中移除数字，以满足某种可除性条件。 我们引入了原始游戏的一个推广版本，该版本依赖于两个参数：初始数字集合的大小和一个固定除数。 对于两位玩家，我们确定了一大类游戏变体，在这些变体中他们可以强制获胜。 特别是，我们证明了对于偶数大小的集合，后手玩家总能获胜，而对于许多奇数大小的情况，先手玩家有一个获胜策略。 该游戏的网络实现展示了我们的一些实际结果。 显示英文摘要

摘要： We study a combinatorial game derived from a problem in the German National Mathematics Competition. In this game, two players take turns removing numbers from a finite set of natural numbers, aiming to satisfy a certain divisibility condition. We introduce a generalized version of the original game, which depends on two parameters: the size of the initial number set and a fixed divisor. For both players, we identify a broad range of game variants in which they can force a win. In particular, we show that for even-sized sets, the second player to move can always win, while for many odd-sized cases, the first player to move has a winning strategy. A web implementation of the game demonstrates some of our results in practice.