I now have a copy of the 1925 article by Moron'
ON THE DIVISION OF RECTANGLES INTO SQUARES,
which was published in Poland in Polish, but I also have a translation into
English by a colleague of mine here at St. John's in Annapolis.
Moron' found two squared rectangles--all squares unequal--of order ten and
nine. It is clear that he did not know of the OTHER squared rectangle of
order nine.
He refers to a theorem by M. Dehn in Math. Annalen in 1903, which shows
that a necessary condition for tiling a rectangle with squares is that the
sides of the squares be commensurable. If I remember correctly, Sherman K.
Stein proves this theorem in
MATHEMATICS THE MAN-MADE UNIVERSE
in his chapter on tiling. I wonder whether or not Stein knew this theorem
from Max Dehn or from elsewhere.
Z. M. doesn't even know whether or not there might be a smaller squared
rectangle than order nine, and he also remarks
we don't know whether a square can be divided into unequal squares.
One should publish a nice pamphlet consisting of
Dehn's Theorem
The article by Moron'
The Sprague article--perhaps 1939--that gives the first squared-square.
It is imperfect, for it contains a sub-rectangle made up of squares.
The article (1940?) by four Englishmen, which has the first perfect
squared-square, and, I think, the proof by Tutte that there is no
cubed-cube (all cubes unequal).
The article(s) by A J W Duijvestijn that not only gives the smallest
squared-square, of order 21, but also proves that there is nothing.
smaller.
Best wishes,
Sam Kutler