OFFSET
1,2
COMMENTS
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
REFERENCES
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).
LINKS
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
EXAMPLE
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}}
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
b-file from Wynn 2013, added by N. J. A. Sloane, Nov 29 2013
STATUS
approved