[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.
Unlike many similar mathematical games, sylver coinage has not been completely solved, mainly because many positions have infinitely many possible moves.
After playing a prime number that is 5 or larger as a first move, the first player in a game of sylver coinage can always win by following this (non-constructive) ender strategy on their next turn.
[2][5] In 2017, Conway (2017) offered a $1000 prize for determining who wins in the first unsolved case, the opening move 16, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the thrackle conjecture.