Abstract In Conway’s game, Sylver Coinage, the set of legal plays forms the complement of a numerical semigroup after a finite number of turns. Our goal is to show how the tools and techniques of numerical semigroups can be brought to bear on questions related to Sylver Coinage. We begin by formally connecting the definitions and concepts related to the game of Sylver Coinage with those of numerical semigroups. Then we reframe a number of previously known results about the play and strategy of Sylver Coinage in terms of basic numerical semigroup theory, culminating with a semigroup-based proof of the quiet end theorem. We conclude by suggesting how several of R. Guy’s twenty questions about Sylver Coinage may be approached using this new framework.

中文翻译：

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数字半群和西尔弗造币游戏

摘要 在 Conway 的游戏 Sylver Coinage 中，合法游戏集在有限轮次后形成了数值半群的补集。我们的目标是展示如何使用数值半群的工具和技术来解决与 Sylver Coinage 相关的问题。我们首先将与 Sylver Coinage 游戏相关的定义和概念与数值半群的定义和概念正式联系起来。然后，我们根据基本的数值半群理论重新构建了一些先前已知的有关 Sylver Coinage 的游戏和策略的结果，最终以基于半群的静端定理证明。最后，我们建议如何使用这个新框架来解决 R. Guy 提出的关于 Sylver Coinage 的 20 个问题中的几个。