At a fun level, there is the two-player game of Chomp.
Briefly, you have an $m\times n$ chocolate bar, divided into squares as usual. The lower left-hand little square is poisoned. The two players, A and B, play alternately. At any move, a player picks the lower left-hand corner of a square, and eats all squares above and/or to the right of that corner. The objective is not to eat the poisoned square.
One can prove quite simply that A has a winning strategy for any chocolate bar except the $1\times1$. But the proof is indirect. It is clear that for any specific bar, one of the two players has a winning strategy. One then shows that if B had a winning strategy, then A could adapt that strategy and win, by taking the square in the upper right-hand corner.
However, for even modest-sized chocolate bars, say $19\times 19$, no winning strategy for A is known. I may be out of date on the $19$, but know that computer searches for strategies have not had great success.