Now try to inscribe a square in it, with all its vertices at lattice points on the perimeter of the polyomino.
Here are two examples:
(On the other hand, it is not too difficult to find a polyomino that has only one such square. The above examples, by virtue of their symmetry, have more than one.)
This is another problem from K-12 Unsolved. It is a marvelous extension of the Geoboard Squares exploration, itself a great foundation for a proof of the Pythagorean theorem. (See Geometry Labs, Lab 8.5, a free download on my Web site.)
Previous K-12 Unsolved problems on this blog: Heilbronn triangle | No three on a line