Sylver coinage is a mathematical game for two players, invented by John H. Conway.[1] The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. The player who names 1 loses. For instance, if player A opens with 2, B can win by naming 3 as A is forced to name 1.[2] Sylver coinage is an example of a game using misère play because the player who is last able to move loses.
Sylver coinage is named after James Joseph Sylvester,[2][3] who proved that if a and b are relatively prime positive integers, then (a − 1)(b − 1) − 1 is the largest number that is not a sum of nonnegative multiples of a and b.[4] Thus, if a and b are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the greatest common divisor of the moves played so far is g, then only finitely many multiples of g can remain to be played, and after they are all played then g must decrease on the next move. Therefore, every game of sylver coinage must eventually end.[2] When a sylver coinage game has only a finite number of remaining moves, the largest number that can still be played is called the Frobenius number, and finding this number is called the coin problem.[5]
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