Sylver coinage is a game for 2 players invented by John H. Conway.
The two players take turns naming positive integers that are not the sum of non-negative multiples of any previously named integers. The player who is forced to name 1 loses.
James Joseph Sylvester proved the following fact.
Lemma: 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.
Therefore, if a and b have no common prime factors and are the first two moves, this formula gives an upper bound on the next number that can still be played.
R. L. Hutchings proved the following fact.
Theorem: If the first player selects any prime number as a first move then he/she has a winning strategy.
Very little is known about the subsequent winning moves. That is, a winning strategy exists but it’s not know what it is!
Unsolved problem:Are there any non-prime winning opening moves in Sylver coinage?
For further info, Sicherman maintains a Sylver coinage game webpage.