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The polyomino XX XX (two blocks of two squares with a gap) does not tile any rectangle, how to prove/disprove that it tiles the plane?

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For people who don't know the definition: – Giovanni De Gaetano Jul 5 '11 at 13:18
My first inclination was to try to prove that it doesn't, but I've now managed to fill up quite a bit of graph paper and there's no indication how I'd run into trouble if I go on, so my guess is now that it does. – joriki Jul 5 '11 at 13:26
My guess too is that it does tile the plane. But I'm curious too look at the proof that it doesn't tile any rectangle. – Giovanni De Gaetano Jul 5 '11 at 13:27
@Student73: I think you can do the proof for the rectangle by enumeration, starting in a corner. – joriki Jul 5 '11 at 13:44
up vote 18 down vote accepted

You can fill the plane, by forming rows repeating horizontally the following figure:
enter image description here
then attaching the rows with a shift of $1$ square.

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