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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?

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$mn$ should be even, to even have a winner! –  Yiorgos S. Smyrlis Dec 22 '13 at 9:42
The player who is able to place the last tile wins. –  okarin Dec 22 '13 at 9:45
Presumably that means if all the remaining gaps are isolated 1 by 1 squares, the next player loses. This is not simple even ignoring the 3 by 4 tiles. –  Henry Dec 22 '13 at 10:00

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up vote 1 down vote accepted

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.

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