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For reference, here are the five tetrominoes:

All five tetrominoes

Suppose you join ten dominoes to make a polyomino made of twenty unit squares. Is it possible that you can tile the polyomino using all the five given tetrominoes (each exactly once)? You can rotate and/or turn over any of the tetrominoes.

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  • $\begingroup$ The question is a little bit unclear to me: when I first read it I understood that the task was to figure out whether it is always possible to tile the large polyminos no matter its shape. Looking at the accepted answer I understood you were asking whether it is possible to find at least one "tilable" large polymino. $\endgroup$
    – melfnt
    Commented Jul 21, 2022 at 2:40

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The answer is:

No. To see that it is impossible, use a checkerboard coloring. Each domino covers one black and one red square, but the five tetrominoes have two more of one color than the other (because of the T).

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