The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. After 1 is named, all positive integers can be expressed in this way: 1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, etc., ending the game. The player who named 1 loses.
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. This is a special case of the Coin Problem.
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.
- A names 11. Now the only remaining numbers are 1, 2, 3, 6, and 7.
- B names 6. Now the only remaining numbers are 1, 2, 3, and 7.
- A names 7. Now the only remaining numbers are 1, 2, and 3.
- B names 2. Now the only remaining numbers are 1 and 3.
- A names 3, leaving only 1.
- B is forced to name 1 and loses.
Each of A's moves was to a winning position.
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, is nonconstructive: it guarantees that such a position has a winning strategy but does not identify it. Hutchings's Theorem states that any of the prime numbers 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known. Complete winning strategies are known for answering the losing openings 1, 2, 3, 4, 6, 8, 9, and 12.
- Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. C7. ISBN 978-0-387-20860-2. Zbl 1058.11001.
- Sylvester, James J. (1884). Mathematical Questions from the Educational Times. 41: 21. Question 7382 Missing or empty