Consider an $m$ by $n$ rectangle. On this rectangle, two players take turns placing either $1$ by $2$ tiles or $3$ by $4$ tiles. The player who is able to place the last tile wins. Which player has a winning strategy and when?
To expand on Henry's comment, the game has a name and has been studied a bit, even without the option of placing $3\times4$ tiles. If you can only place dominoes, the game is called "[normal play] Cram".