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.

• MSC: Primary 91A05
• Secondary 20M14
• 91A46

Acknowledgments

The authors wish to thank an anonymous reviewer for several very insightful suggestions that helped us to correct some confusing points and to greatly clarify our exposition.

Additional information

Notes on contributors

Rachel Eaton

RACHEL EATON earned a B.S. in Applied Mathematics from the United States Air Force Academy in 2017. She then completed an M.S. in Information Security at Carnegie Mellon University. She currently serves in the Air Force as a cyber operations officer.

Kurt Herzinger

KURT HERZINGER earned his Ph.D. in mathematics from the University of Nebraska–Lincoln in 1996. He has been a faculty member at the United States Air Force Academy in the Department of Mathematical Sciences since 1998. His research interests include numerical semigroups and methods of Diophantine approximation, including continued fractions and Greek ladders.

Ian Pierce

IAN PIERCE earned his Ph.D. in mathematics from the University of Nebraska–Lincoln in 2011. He then held a visiting position at St. Olaf College, and joined the faculty at the United States Air Force Academy in the Department of Mathematical Sciences in 2013. He is happy to work on pretty much any problem that wanders by.

Jeremy Thompson

JEREMY THOMPSON earned his M.Sc. in applied mathematics from University of Washington in 2011 and is a Ph.D. student at University of Colorado Boulder. His research interests include numerical semigroups and high performance scientific computing.