A game discussed in "Winning Ways". The players alternately name a positive integer, which represents a new denomination of coin. It is forbidden to name any denomination which can be made as a combination of existing coins - for example, once 4 and 5 have been named, 8 (4+4) and 14 (4+5+5) are forbidden - or to name the number 1.

2 and 3 are both losing first moves, because if the first player names one of them, the second will name the other, and as any number greater than 3 can be written as a sum of 2s and 3s, no legal moves will remain. 4 is also a losing first move, although it's a bit harder to prove this. However, not all first moves lose.

As stated, it's not even clear that every game of Sylver Coinage must terminate (i.e. that one player must lose at some stage). Perhaps the two players can just continue naming huge integers, each inexpressible as sums of previous moves?

That's a highly undesirable property in a game. When I play, I want to know that the game will be over at some stage. For instance, chess and Go both have this problem potentially. In chess, the 50 move rule prevents it (triple repetition doesn't quite cut it, as it only allows a player to draw, but doesn't force the draw). In Go, the Ko rule(s) (and some refereeing decisions) are supposed to prevent nontermination. Does Sylver Coinage also require some such rule? What could it possibly be?

It doesn't. See a proof that every game of Sylver Coinage ends.

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