Mrs.
Perkins Quilts
Ed Pegg Jr
Dividing a square into smaller squares is
often called the Mrs. Perkins'
Quilt problem. I mentioned a while ago that Erich Friedman found a
way to divide a side67
square into 20 smaller squares all with doubledigit sides. Can
you find his solution? Answer. The only
solver of this puzzle was Lance Gay. Lance followed up by finding
better solutions for squares of size 88 to 90. Robert Wainwright
sent me better answers for sizes 53 and 91. All of these new
solutions were graciously accepted by Richard Guy, for the next edition
of Unsolved Problems in Geometry.
John Conway let me know that Dudeney named the puzzle. Geoffrey
Morley sent in transcriptions of Duijvestijn's solutions. Erich
Friedman sent me this list
of best
solutions (Antony Boucher and Lance Gay filled in the blanks). The
current best known solutions are as follows: {1

1}, {4
 2}, {6
 3}, {7  4}, {8  5}, {9  6,7}, {10  8,9},
{11 
1013}, {12

1417}, {13  1823}, {14
 2429}, {15 
3039,41}, {16  40,4253}, {17  5470}, {18  7191}, {19 
92108}. Lance Gay: "Since seeing your writeup on
mathpuzzle.com last month, I have written some Mrs. Perkins quilt
search software to look for solutions. For fun, I added a routine to
print out the bitmaps of solutions. I run my searches on a pair of 1.8
GHz MSWindows machines. It appears that computers have gotten faster
since people have last done some serious searching. For now, I have
completed all searching for sides up to 100. I am now concentrating on
larger squares (such as attempting to beat the order20 side154
square). My search algorithm is not exhaustive so the trick is reducing
the search space to the most fruitful areas." John Conway: "You
might add the two 1950s papers both called "Mrs Perkins's Quilt", one
by me and one by G.B.Trustrum, in Proc. Camb. Phil. Soc. 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). Since there's an
obvious logarithmic lower bound, all that remains is to find the best
constant. I don't know if there's been any progress on that
problem since those two papers." Antony Boucher sent me some
corrections for the list, and also sent solving code.