Draw a polygon following grid paper lines. No crossings, no holes — in other words, a polyomino.

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:

**Conjecture**: it is impossible to draw a polyomino that does *not* have such a square inscribed in it. Try to do it!

(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.)

–Henri

Previous K-12 Unsolved problems on this blog: Heilbronn triangle | No three on a line

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