From:"Matthew C. Clarke" <clarkem@unpsun1.cc.unp.ac.za>Newsgroups:sci.math,rec.puzzlesSubject:Re: Fake dissectionDate:31 Jan 1996 11:21:30 GMTOrganization:Computer Science and Information Systems UNP

propp@math.mit.edu (Jim Propp) wrote: > > There's a well-known way of dissecting an 8-by-8 square into four pieces > that, when rearranged, seem to form a 5-by-13 rectangle, thereby "proving" > that 64=65. What actually happens is that the part of the rectangle > that's not covered by the four pieces forms a very acute (and nearly > undetectable) parallelogram of area 1. > > What I'd like to know is, can anyone think of a good way to dissect > a 5-by-5-by-5 cube and a 6-by-6-by-6 cube into pieces that, when > rearranged, seem to form a 7-by-7-by-7 cube, thereby "disproving" > Fermat's Last Theorem? (5^3 + 6^3 = 341, while 7^3 = 343: that's > pretty close!) I'm sorry not to suggest an answer, but the task reminds me of what is called the Barnach-Tarski Paradox. Using the process of decomposition into finite sets it is possible to divide the set of points on the surface of a sphere into partitions which can be rearranged into two spheres both of which have the same surface area as the original sphere. Needless to say, the nature of the partitioning is such that you couldn't actually cut the sphere and physically rearranging it. This is not a trick, but a proven result! The original paper includes a 16 page proof (in French), but I've lost the reference (Sorry). There is a more recent book devoted to the paradox and its implications but I don't have a reference to it either. Perhaps if one found the original paper, one could partition it and rearrange it to form the book :^) Matt.