From:           "Matthew C. Clarke" <>
Newsgroups:     sci.math,rec.puzzles
Subject:        Re: Fake dissection
Date:           31 Jan 1996 11:21:30 GMT
Organization:   Computer Science and Information Systems UNP (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 :^)