Rich wrote:
There are some interesting possibilities for dissecting a cube into three congruent pieces. Select a triagonal, and split the cube in half with a plane perp-bisecting the triagonal.[...]
This 3-fold axis of symmetry means you can get away with almost anything -- make any 2d surface in 3-space whose boundary is a line and which doesn't intersect itself when rotated by a multiple of 120 degrees, and cut the cube according to that.
I believe one of Gardner's columns was about dissections, and said the square-into-5-congruent- pieces problem had only the trivial solution, and that a proof existed.
I just found this last night (thanks to the MAA's wonderful all-Gardner-columns-on-a-CD set!). It took me a while, because it's *not* in his column on dissections into congruent pieces. It's one of his "Nine Problems" cols, reproduced as chapter 15 in The Unexpected Hanging. He shows two nontrivial congruent-4-sections to mislead the reader, then asks for a congruent-5-section of a square. The answer shows the obvious one, with a little ha-ha-you- took-a-long-time-to-think-of-this-didn't-you, and states that it's unique without even any mention of the notion of a proof. Obviously it'd be nicest to prove there's no other trisection in a way which works for 5, and as many other numbers as possible, as well. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.