I heard this puzzle from Bob Koca. Suppose we play misere tic-tac-toe (a.k.a. noughts and crosses) where both players are X. Who wins?
That particular puzzle is easy to solve, but more generally, has $n \times n$ impartial tic tac toe, in both normal and misere forms, been studied before?
EDIT: Thane Plambeck's paper, mentioned at the end of his answer below, coined the term Notakto for this game. That name seems to have caught on; for example, there is now a Wikipedia article on Notakto.