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.
An example 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, 12, 13, 14, 15, ...
- 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!)
A theorem of James Joseph Sylvester, after whom the game is named, shows that every game must eventually end, i.e. the players cannot avoid having to say 1 forever.
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 N-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] External link
Some recent findings about the game are detailed at http://www.monmouth.com/~colonel/sylver/.
