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

  • $\begingroup$ For people who don't know the definition: en.wikipedia.org/wiki/Polyomino $\endgroup$ – Giovanni De Gaetano Jul 5 '11 at 13:18
  • 2
    $\begingroup$ 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. $\endgroup$ – joriki Jul 5 '11 at 13:26
  • 1
    $\begingroup$ 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. $\endgroup$ – Giovanni De Gaetano Jul 5 '11 at 13:27
  • 1
    $\begingroup$ @Student73: I think you can do the proof for the rectangle by enumeration, starting in a corner. $\endgroup$ – joriki Jul 5 '11 at 13:44

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.