Sylver Coinage

This curious game was invented by Princeton mathematician John H. Conway. Two players take turns naming positive integers, but an integer is off limits if it’s the sum of nonnegative multiples of integers that have already been named. Once 1 is named, everything is off limits (because any positive integer is a sum of 1s), so that ends the game; the player who is forced to name 1 is the loser. An example gives the idea:

  • I start by naming 5. From now on neither of us can name 5, 10, 15, 20, …
  • You name 4. Now neither of us can name a number built of 5s and 4s, that is, 4, 5, 8, 9, 10, or any number greater than 11.
  • I name 11. This reduces the list of available numbers to 1, 2, 3, 6, and 7.
  • You name 6. Now we’re down to 1, 2, 3, and 7.
  • I name 7. Only 1, 2, and 3 are still available to name.

Now we’ll use our next turns to name 2 and 3, and you’ll be forced to name 1, losing the game.

Though the game is very easy to understand, it’s still full of mysteries. For example, R.L. Hutchings has shown that playing a prime number as the first move guarantees that a winning strategy exists, but no one has figured out how to find the strategy. And no one knows whether there are any winning opening moves at all that aren’t prime.

More on the Sylver Coinage page.