Can you cut a rectangle into 3 congruent non-rectangular pieces?
There are lots of variants of this question. The answer is also unknown if we ask for 5, 7, or 9 pieces. It is easy to see that the answer is "yes" if we ask for an even number of pieces, and for all odd integers other than 3, 5, 7, and 9 it is known that the answer is "yes". Another interesting version is to ask for the pieces all to be "polyominos", which are shapes made by connecting squares of the same size to each other along their edges. (I learned about this problem from "A dozen unsolved problems in geometry", a presentation given by Erich Friedman of Stetson University).