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Cycles in War

Michael Z. Spivey
  • Spivey, Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, Washington 98416, USA.
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Published Online: 2010-12-20 | DOI: https://doi.org/10.1515/integ.2010.102


We discuss a simplified version of the well-known card game War in which the cards in the deck have a strict ranking from 1 to n and in which the winning card and losing card are immediately placed, in that order, at the bottom of the winning player's deck. Under this variation of War we show that it is possible for a standard fifty-two card deck to cycle, and we exhibit such a cycle. This result is a special case of a more general result that exhibits a cycle construction for an n-card deck for any value of n that is not a power of 2 or 3 times a power of 2. We also discuss results that show that under some assumptions the types of cycles we exhibit are the only types of cycles that can occur. Finally, we give some open questions related to cycles in War.

Keywords.: War; cycle; combinatorial games

About the article

Received: 2009-08-17

Revised: 2010-06-24

Accepted: 2010-08-14

Published Online: 2010-12-20

Published in Print: 2010-12-01

Citation Information: Integers, Volume 10, Issue 6, Pages 747–764, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integ.2010.102.

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