Perfect Rectangle
PerfectRectangles
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 maintains an online database of perfect rectangles at http://www.squaring.net/.
See also
Mrs. Perkins's Quilt, No-Touch Dissection, Nowhere-Neat Dissection, Perfect Square Dissection, Rectangle TilingExplore with Wolfram|Alpha
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References
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-fizyczny 3, 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.Update a linkWolf, T. "The 70^2 Puzzle." http://home.tiscalinet.ch/t_wolf/tw/misc/squares.htmlReferenced on Wolfram|Alpha
Perfect RectangleCite this as:
Weisstein, Eric W. "Perfect Rectangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PerfectRectangle.html