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.)