Michael Kleber wrote: << There are clearly lots of choices for what restrictions you put on the pieces. Personally, I'd be happy to learn anything: a proof that the obvious trisection is the only one possible with polygonal pieces would be great, as would an abstract, AC-dependent construction that created three congruent point-sets whose union is the square.
My hunch is that under modest restrictions on the kind of pieces, the following line of attack will help: Keep your eye on where the 4 corners and the 8 one-third and two-thirds points on each edge go among the 3 pieces. Also, the problem should be defined loosely enough that boundary points of the pieces can be ignored, and care must be taken to avoid showing that no such decomposition is possible! A RELATED PROBLEM that some people could've predicted I'd ask is: In what ways can the square torus be decomposed into n congruent pieces for any positive n (other than the obvious n parallel annuli decomposition [whose slopes, btw, can be any fixed rational number]) ? --Dan