Search: a005842
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A005842
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a(n) = minimal integer m such that an m X m square contains non-overlapping squares of sides 1, ..., n (some values are only conjectures).
(Formerly M2401)
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+30
2
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1, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 33, 36, 39, 43, 47, 50, 54, 58, 62, 66, 71, 75, 80, 84, 89, 93, 98, 103, 108, 113, 118, 123, 128, 133, 139, 144, 150, 155, 161, 166, 172, 178, 184, 190, 196, 202, 208, 214, 221, 227, 233, 240, 246
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OFFSET
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1,2
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COMMENTS
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The entries for n=1, 2, 8, 15, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 35, 36, 37, 39, 41, 43, 44, 45, 46, 49, 50, 51, 54, and 56 all meet the lower bound in A092137 and are therefore correct. - Stuart E Anderson, Jan 05 2008
Simonis, H. and O'Sullivan showed that a(26) = 80. - Erich Friedman, May 27 2009
Houhardy S. showed a(32)=108, a(33)=113, a(34)=118, and a(47)=190. - Erich Friedman, Oct 11 2010
The values have been proved correct except those for n=38, 40, 42, 48, 52, 53 and 55, where they remain probable. - Erich Friedman, Oct 11 2010
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REFERENCES
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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, D5.
M. Gardner, Mathematical Carnival. Random House, NY, 1977, p. 147.
Simonis, H. and O'Sullivan, B., Search Strategies for Rectangle Packing, in Proceedings of the 14th international conference on Principles and Practice of Constraint Programming, Springer-Verlag Berlin, Heidelberg, 2008, pp. 52-66.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..56.
János Balogh, György Dósa, Lars Magnus Hvattum, Tomas Olaj, and Zsolt Tuza, Guillotine cutting is asymptotically optimal for packing consecutive squares, Optimization Letters (2022).
Erich Friedman, Math Magic.
S. Hougardy, A Scale Invariant Algorithm for Packing Rectangles Perfectly, 2012. - From N. J. A. Sloane, Oct 15 2012
S. Hougardy, A Scale Invariant Exact Algorithm for Dense Rectangle Packing Problems, 2012.
Minami Kawasaki, Catalogue of best known solutions
R. E. Korf, Optimal Rectangle Packing: New Results, Proceedings of the International Conference on Automated Planning and Scheduling (ICAPS04), Whistler, British Columbia, June 2004, pp. 142-149. [From Rob Pratt, Jun 10 2009]
Takehide Soh, Packing Consequtive Squares into a Sqaure (sic), Kobe University (Japan, 2019).
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CROSSREFS
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Cf. A092137 (lower bound).
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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STATUS
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approved
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1, 3, 4, 6, 8, 10, 12, 15, 17, 20, 23, 26, 29, 32, 36, 39, 43, 46, 50, 54, 58, 62, 66, 70, 75, 79, 84, 88, 93, 98, 103, 107, 112, 117, 123, 128, 133, 138, 144, 149, 155, 160, 166, 172, 178, 184, 189, 195, 202, 208
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Area of square must be large enough to contain all n squares without overlap.
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LINKS
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Table of n, a(n) for n=1..50.
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FORMULA
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a(n) = ceiling(sqrt(Sum_{k=1..n} k^2)).
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MATHEMATICA
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Table[Ceiling[Sqrt[Sum[k^2, {k, 1, n}]]], {n, 1, 50}]
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CROSSREFS
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Cf. A005842.
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KEYWORD
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easy,nonn
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AUTHOR
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Rob Pratt, Mar 30 2004
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STATUS
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approved
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A005670
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Mrs. Perkins's quilt: smallest coprime dissection of n X n square.
(Formerly M3267)
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+10
10
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1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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The problem is to dissect an n X n square into smaller integer squares, the GCD of whose sides is 1, using the smallest number of squares. The GCD condition excludes dissecting a 6 X 6 into four 3 X 3 squares.
The name "Mrs Perkins's Quilt" comes from a problem in one of Dudeney's books, wherein he gives the answer for n = 13. I gave the answers for low n and an upper bound of order n^(1/3) for general n, which Trustrum improved to order log(n). There's an obvious logarithmic lower bound. - J. H. Conway, Oct 11 2003
All entries shown are known to be correct - see Wynn, 2013. - N. J. A. Sloane, Nov 29 2013
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REFERENCES
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H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Ed Wynn, Table of n, a(n) for n = 1..120
J. H. Conway, Mrs. Perkins's quilt, Proc. Camb. Phil. Soc., 60 (1964), 363-368.
A. J. W. Duijvestijn, Table I
A. J. W. Duijvestijn, Table II
R. K. Guy, Letter to N. J. A. Sloane, 1987
Ed Pegg, Jr., Mrs Perkins's Quilts (best known values to 40000)
G. B. Trustrum, Mrs Perkins's quilt, Proc. Cambridge Philos. Soc., 61 1965 7-11.
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt
Ed Wynn, Exhaustive generation of 'Mrs Perkins's quilt' square dissections for low orders, arXiv:1308.5420 [math.CO], 2013-2014.
Ed Wynn, Exhaustive generation of 'Mrs. Perkins's quilt' square dissections for low orders, Discrete Math. 334 (2014), 38--47. MR3240464
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EXAMPLE
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Illustrating a(7) = 9: a dissection of a 7 X 7 square into 9 pieces, courtesy of Ed Pegg Jr:
.___.___.___.___.___.___.___
|...........|.......|.......|
|...........|.......|.......|
|...........|.......|.......|
|...........|___.___|___.___|
|...........|...|...|.......|
|___.___.___|___|___|.......|
|...............|...|.......|
|...............|___|___.___|
|...............|...........|
|...............|...........|
|...............|...........|
|...............|...........|
|...............|...........|
|___.___.___.___|___.___.___|
The Duijvestijn code for this is {{3,2,2},{1,1,2},{4,1},{3}}
Solutions for n = 1..10: 1 {{1}}
2 {{1, 1}, {1, 1}}
3 {{2, 1}, {1}, {1, 1, 1}}
4 {{2, 2}, {2, 1, 1}, {1, 1}}
5 {{3, 2}, {1, 1}, {2, 1, 2}, {1}}
6 {{3, 3}, {3, 2, 1}, {1}, {1, 1, 1}}
7 {{4, 3}, {1, 2}, {3, 1, 1}, {2, 2}}
8 {{4, 4}, {4, 2, 2}, {2, 1, 1}, {1, 1}}
9 {{5, 4}, {1, 1, 2}, {4, 2, 1}, {3}, {2}}
10 {{5, 5}, {5, 3, 2}, {1, 1}, {2, 1, 2}, {1}}
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CROSSREFS
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Cf. A005842, A089046, A089047.
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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b-file from Wynn 2013, added by N. J. A. Sloane, Nov 29 2013
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STATUS
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approved
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