A rectangle which cannot be built up of squares all of different sizes is called an imperfect rectangle. A rectangle
which can be built up of squares all of different sizes
is called perfect. The number of perfect rectangles of orders 8, 9, 10, ... are 0,
2, 6, 22, 67, 213, 744, 2609, ... (OEIS A002839)
and the corresponding numbers of imperfect rectangles are 0, 1, 0, 0, 9, 34, 103,
283, ... (OEIS A002881).
Anderson, S. "Perfect Rectangles, Perfect Squares." http://www.squaring.net/.Bouwkamp,
C. J. "On the Dissection of Rectangles into Squares. I." Indag.
Math.8, 724-736, 1946.Bouwkamp, C. J. "On the
Dissection of Rectangles into Squares. II." Indag. Math.9, 43-56,
1947.Bouwkamp, C. J. "On the Dissection of Rectangles into
Squares. III." Indag. Math.9, 57-63, 1947.Brooks,
R. L.; Smith, C. A. B.; Stone, A. H.; and Tutte, W. T. "The
Dissection of Rectangles into Squares." Duke Math. J.7, 312-340,
1940.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Squaring
the Square." §C2 in Unsolved
Problems in Geometry. New York: Springer-Verlag, pp. 81-83, 1991.Descartes,
B. "Division of a Square into Rectangles." Eureka, No. 34,
31-35, 1971.Duijvestijn, A. J. W. Electronic Computation
of Squared Rectangles. Thesis. Eindhoven, Netherlands: Technische Hogeschool,
1962.Moroń, Z. "O rozkładach prostokatów na kwadraty."
Przeglad matematyczno-fizyczny3, 152-153, 1925.Scherer,
K. "Square the Square." http://karl.kiwi.gen.nz/prosqtsq.html.Sloane,
N. J. A. Sequences A002839/M1658
and A002881/M4614 in "The On-Line Encyclopedia
of Integer Sequences."Stewart, I. "Squaring the Square."
Sci. Amer.277, 94-96, July 1997.Wells, D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, p. 73, 1986.Wolf, T. "The Puzzle." http://home.tiscalinet.ch/t_wolf/tw/misc/squares.html.
Puzzle."