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.