A089046 Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem.

1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 10, 14, 18, 24, 30, 40, 54, 71, 92, 121, 155, 210, 266, 360, 476, 642, 833, 1117, 1485, 1967, 2595, 3465, 4534, 5995

Offset: 1

Author

R. K. Guy, Dec 03 2003

Keywords

• nonn
• hard
• more

Comments

An inverse to A005670.

More precisely, a(n) = least k such that A005670(k) >= n. - Peter Munn, Mar 14 2018

It is not clear which terms have been proved to be correct and which are just conjectures. - Geoffrey H. Morley, Aug 29 2012; N. J. A. Sloane, Jul 06 2017

n <= 15 (and possibly 16) proved minimal by J. H. Conway (Conway, J. H. "Re: [math-fun] Mrs. Perkins Quilt - Orders 89, 90 improved over UPIG." math-fun mailing list. October 10, 2003.). The conjectures are best currently known values of a(n) for n > 16. - Stuart E Anderson, Apr 21 2013

A089046 and A089047 are almost certainly correct up to 5000. - Ed Pegg Jr, Jul 06 2017

Deleted terms above 5000. - N. J. A. Sloane, Jul 06 2017

Upper bounds for the next terms in the sequence (which may well be the true values) are 7907, 10293, 13505, 17785, 23239, 31035, 39571, ... - Ed Pegg Jr, Jul 06 2017

References

• H. T. Croft, K. J. Falconer, and R. K. Guy, Section C3 in Unsolved Problems in Geometry, New York: Springer, 1991.
• M. Gardner, "Mrs. Perkins's Quilt and Other Square-Packing Problems," Mathematical Carnival, New York: Vintage, 1977.

Links

• Stuart E. Anderson, Mrs Perkins's Quilts
• J. H. Conway, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 60, 363-368, 1964.
• Ed Pegg, Jr., Mrs. Perkin's Quilts
• Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts (Wolfram Demonstrations Project)
• Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts Notebook source code
• G. B. Trustrum, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 61, 7-11, 1965.
• Eric W. 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.

Crossrefs

Cf. A005670, A089046, A089047.

Extensions

More terms from Ed Pegg Jr, Dec 03 2003

Corrected and extended by Ed Pegg Jr, Apr 18 2010

a(24)-a(27) (from Ed Pegg Jr, Jun 15 2010) added by Geoffrey H. Morley, Aug 29 2012

a(28)-a(30) from Stuart E Anderson, Nov 22 2012

Confirmed a(30) as best known, added a(31) as best known. - Stuart E Anderson, Apr 21 2013

Using James Williams recent discoveries of 15 million simple perfect squared squares in orders 31 to 44 I was able to extend the sequence of best currently known values for optimal quilts from a(32) to a(44). - Stuart E Anderson, Apr 21 2013

Using Anderson and Milla's enumeration of order 31 and 32 perfect squared squares, improved conjectures for a(32) and a(33) were obtained - Stuart E Anderson, Sep 16 2013

a(1)-a(19) confirmed by Ed Wynn, 2013. - N. J. A. Sloane, Nov 29 2013

a(29) corrected and further terms added by Ed Pegg Jr, Jul 06 2017