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?
2026-03-27 21:32:56.1774647176
Tiling m by n rectangle game
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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".
Basically, the general answer is not known, although we have a lot of results for particular small boards.