Sylver Coinage

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Sylver Coinage is a game invented by J. H. Conway in which players take turns choosing positive integers—but a player may only choose an integer that cannot be expressed as a sum of positive multiples of integers already chosen. The player who is forced to choose “1” loses the game. The name of the game is a nod to J. J. Sylvester, who proved that is the largest sum that cannot be paid exactly using coins of value and , if and are relatively prime. Gameplay is simple: just click the value you want to "coin", and illegal moves will be removed from the board automatically. Finding a winning strategy may not be so simple, however.


This implementation enforces some restrictions on the opening moves in order to keep gameplay within a finite (and reasonably small) range of values.


Contributed by: Jacob A. Siehler (March 2011)
Open content licensed under CC BY-NC-SA



The Sylver Coinage Page.

Sylver Coinageon Wikipedia.

E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for Your Mathematical Plays, Vol. 3, Natick, MA: AK Peters, 2003.

R. K. Guy, "Twenty Questions Concerning Conway's Sylver Coinage," American Mathematical Monthly, 83(8), 1976 pp. 634–637.

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