Sylver coinage
From Wikipedia, the free encyclopedia
Sylver Coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in Winning Ways for your Mathematical Plays.
The two players take it in turn to name a positive integer that is not the sum of 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. This makes Sylver Coinage a misère game, since by the usual convention in combinatorial game theory the last player able to move is the winner.
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.
(A must have known what he was doing here: each one of his moves was to a winning position!)
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.
Unlike many mathematical games of its ilk, Sylver Coinage has not been completely solved. Part of the difficulty arises from the fact that many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions is nonconstructive, so that even though you may know that you have a winning move, you are not told what it is. For instance, it is known that any of the prime numbers 5, 7, 11, 13, etc. are winning openings for the first player, but very little is known about the subsequent moves which will consummate the win. Complete winning strategies are known to respond to the losing openings 1, 2, 3, 4, 6, 8, 9, and 12.
[edit] References
James J. Sylvester, Mathematical Questions from the Educational Times 41 (1884), 21 (question 7382).
[edit] External links
Some recent findings about the game are detailed at http://www.monmouth.com/~colonel/sylver/.
