1,2

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 exclude 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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. H. Conway, Mrs. Perkins's quilt, Proc. Camb. Phil. Soc., 60 (1964), 363-368.

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C3.

Trustrum, G. B., Mrs Perkins's quilt, Proc. Cambridge Philos. Soc., 61 1965 7-11.

A. J. W. Duijvestijn, Table I

A. J. W. Duijvestijn, Table II

Ed Pegg, Jr., Mrs Perkin's Quilt

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt

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}}

Cf. A005842.

Sequence in context: A011275 A006185 A021876 this_sequence A123860 A122817 A074764

Adjacent sequences: A005667 A005668 A005669 this_sequence A005671 A005672 A005673

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Extended using values from Ed Pegg's web site.

It is not clear how many of these terms have been proved to be correct and how many are just conjectures. To be safe, regard all the entries in this sequences as conjectures, unless stated otherwise.