Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 of Winning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named after James Joseph Sylvester, 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. 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. 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. Example A sample game between A and B: A opens with 5. Now neither player can name 5, 10, 15, .... B names 4. Now neither player can name 4, 5, 8, 9, 10, or any number greater than 11. (Example: 27 = 4·3 + 5·3) A names 11. Now the only remaining numbers are 2, 3, 6, and 7. B names 6. Now the only remaining numbers are 2, 3, and 7. A names 7. Now the only remaining numbers are 2, and 3. B names 2. Now the only remaining numbers is 3. A names 3, leaving nothing for B, and wins. Each of A's moves was to a winning position. Analysis Unlike many similar mathematical games, sylver coinage has not been completely solved, mainly because many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions, due to R. L. Hutchings, guarantees that such a position has a winning strategy but does not identify the strategy.