Sylver coinage

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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 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.

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.

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.

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.

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