A005670 Mrs. Perkins's quilt: smallest coprime dissection of n X n square. (Formerly M3267)

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

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

Cf. A005842, A089046, A089047.

Sequence in context: A296183 A021876 A261491 * A234948 A123860 A122817

Adjacent sequences: A005667 A005668 A005669 * A005671 A005672 A005673

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

b-file from Wynn 2013, added by N. J. A. Sloane, Nov 29 2013

Status

approved