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Mrs. Perkins's Quilt

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A Mrs. Perkins's quilt is a dissection of an n×n square, for a positive integer n, into squares with positive integer side lengths whose greatest common divisor is 1. Such a dissection is called prime. For n=1, the undivided unit square is included as the trivial quilt. The name comes from Dudeney's problem 173 (Dudeney 1917), which asks for a solution with n=13. Unlike a perfect square dissection, the component squares need not have different sizes.

MrsPerkinsQuilts

The smallest numbers of squares needed to create prime dissections of an n×n quilt for n=1, 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.

MrsPerkinsQuiltsGay

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 best known numbers of squares s(n) for various side lengths n. The values for n<=120 are proved minimal (Wynn 2014). Conway (1964) gave an upper bound of order n^(1/3), which Trustrum (1965) improved to order lnn. More explicitly, log_2n<s(n)<6log_2n for n>1 (Wynn 2014).

s(n)n
11
42
63
74
85
96, 7
108, 9
1110-13
1214-17
1318-23
1424-29
1530-39,41
1640, 42-53
1754-70
1871-91
1992-120, 122, 126
20121, 123-125, 127-154, 157, 158
21155, 156, 159-197, 199-204, 209, 216
22198, 205-208, 210-215, 217-252, 254-257, 260, 262, 263, 265
23253, 258, 259, 261, 264, 266-332, 334-339, 341, 342, 344, 346, 349, 352, 364
24333, 340, 343, 345, 347, 348, 350, 351, 353-363, 365-432, 434-436, 438, 440-444, 450, 453, 456

A squared torus gives a related problem with opposite sides of a rectangle identified, allowing squares to cross the boundary. Such a squaring is perfect when all its squares have different side lengths.

See also

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

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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. "MathPuzzle: List of Best Solutions." Jan. 13, 2004. https://www.mathpuzzle.com/perkinsbestquilts.txt.Pegg, E. Jr. "Math Games: Square Packing." Dec. 1, 2003. https://www.mathpuzzle.com/MAA/06-Square%20Packing/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.Wynn, E. "Exhaustive Generation of 'Mrs. Perkins's Quilt' Square Dissections for Low Orders." Disc. Math. 334, 38-47, 2014. https://doi.org/10.1016/j.disc.2014.06.022.

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Mrs. Perkins's Quilt

Cite this as:

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

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