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# Mrs. Perkins's Quilt

A Mrs. Perkins's quilt is a dissection of a square of side into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect square dissection, however, the smaller squares need not be all different sizes. In addition, only prime dissections are considered so that patterns which can be dissected into lower-order squares are not permitted.

The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (OEIS A005670), the first few of which are illustrated above.

On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly 16) proved minimal by J. H. Conway (pers. comm., Oct. 10, 2003). More specifically, Conway (1964) gave an upper bound of order for general , which Trustrum (1965) improved to order . However, the coefficient of the upper bound does not appear to be known.

 1 1 4 2 6 3 7 4 8 5 9 6, 7 10 8, 9 11 10-13 12 14-17 13 18-23 14 24-29 15 30-39,41 16 40, 42-53 17 54-70 18 71-91 19 92-120, 122, 126 20 121, 123-125, 127-154, 157, 158 21 155, 156, 159-197, 199-204, 209, 216 22 198, 205-208, 210-215, 217-252, 254-257, 260, 262, 263, 265 23 253, 258, 259, 261, 264, 266-332, 334-339, 341, 342, 344, 346, 349, 352, 364 24 333, 340, 343, 345, 347, 348, 350, 351, 353-363, 365-432, 434-436, 438, 440-444, 450, 453, 456

No-Touch Dissection, Nowhere-Neat Dissection, Perfect Rectangle, Perfect Square Dissection

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## References

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 80-81, 2003.Conway, J. H. "Mrs. Perkins's Quilt." Proc. Cambridge Phil. Soc. 60, 363-368, 1964.Conway, J. H. "Re: [math-fun] Mrs. Perkins Quilt - Orders 89, 90 improved over UPIG." math-fun@mailman.xmission.com mailing list. October 10, 2003.Croft, H. T.; Falconer, K. J.; and Guy, R. K. §C3 in Unsolved Problems in Geometry. New York: Springer-Verlag, 1991.Devincentis, J. "Square the Square." http://members.bellatlantic.net/~devjoe/sqsq/.Dudeney, H. E. Problem 173 in Amusements in Mathematics. New York: Dover, 1917.Dudeney, H. E. Problem 177 in 536 Puzzles & Curious Problems. New York: Scribner, 1967.Duijvestijn, A. J. W. "Electronic Computation of Squared Rectangles." Philips Res. Reports 17, 523-613, 1962.Gardner, M. "Mrs. Perkins' Quilt and Other Square-Packing Problems." Ch. 11 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage, 1977.Guy, R. K. "Mrs. Perkins's Quilt." Draft, Dec. 4. 2003.Littlewood, J. E. Littlewood's Miscellany. Cambridge, England: Cambridge University Press, p. 28, 1986.Pegg, E. Jr. "List of Best Solutions." http://mathpuzzle.com/perkinsbestquilts.txt.Pegg, E. Jr. "Mathematical Games: Square Packings." http://www.maa.org/editorial/mathgames/mathgames_12_01_03.html.Scherer, K. "Square the Square." http://karl.kiwi.gen.nz/prosqtsq.html.Sloane, N. J. A. Sequence A005670/M3267 in "The On-Line Encyclopedia of Integer Sequences."Trustrum, G. B. "Mrs. Perkins's Quilt." Proc. Cambridge Phil. Soc. 61, 7-11, 1965.

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Mrs. Perkins's Quilt

## Cite this as:

Weisstein, Eric W. "Mrs. Perkins's Quilt." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MrsPerkinssQuilt.html