Numerical Semigroups and the Game of Sylver Coinage

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.