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From article <9105242046.AA15308@athos.cs.ua.edu>, by borie@ATHOS.CS.UA.EDU (Richard Borie):
> Is it possible to partition the interior of a square into
> smaller squares, no two of which have the same size?
> (Notice that it is trivial to partition a square with side 2
> into 4 squares each with side 1, hence the requirement of distinct sizes.)
> The closest I have been able to get is to partition a 32-by-33
> rectangle into squares with sides 1, 4, 7, 8, 9, 10, 14, 15, and 18.
> 

In F. Le Lyonnais' book: "Les Nombres Remarquables" (Hermann Ed.),
such a square is called "perfectly perfect".
According to this book, the smallest such square is 112 x 112 and is of 
"order" 21, that is, contains 21 squares. This is the smallest possible 
order.
It has been discovered by A. W. Duijvestijn (1976 or 1978) by computer.
Le Lyonnais gives the list of the sizes of the smaller squares:
(2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50), but no
drawing nor precise reference.
He quotes other known perfectly perfect squares: one of order 24 and
size 175, and one of order 26 and size 608 (Brooks, Smith, Tutte, 1938).

A. Jean-Marie