Lance Gay has cracked a problem in Unsolved Problems in Geometry. Quoting RK Guy -- known records with "some slightly increasing lack of confidence" are as follows: {1 >> 1}, {4 >> 2}, {6 >> 3}, {7 >> 4}, {8 >> 5}, {9 >> 6,7}, {10 >> 8,9}, {11 >> 10-13}, {12 >> 14-17}, {13 >> 18-23}, {14 >> 24-29}, {15 >> 30-39,41}, {16 >> 40,42-50}, {17 >> 51-66,68,70}, {18 >> 67,69,71-87}, {19 >> 88-100} RK can pencil in a correction for that last bit. Lance Gay (Lance.Gay@ngc.com) found order-18 solutions for side 89 and 90 Mrs. Perkins Quilts. These appear to be better than the current known values. Side=89 Order=18 - [48 41][5 5 11 20][2 8][41 9][2 9][3 7][12][8 28][20] Side=90 Order=18 - [49 41][8 12 21][41 9 7][3 9][2 8][11][2 28][5 5][21] Lance also asks for a source for the best know solutions for side 100-250. The only source I knew of was Duijvestijn, A. J. W. ftp://ftp.cs.utwente.nl/pub/doc/dvs/TableI. -- but this seems to be down. Other pertinent refs: http://mathworld.wolfram.com/MrsPerkinsQuilt.html http://mathworld.wolfram.com/PerfectSquareDissection.html I must humbly take blame for botching a diagram on Eric's site -- the order 7 square there is wrong (but the sequence right under is correct). Nick Gardner pointed out the mistake to me. For an exercise, try to dissect the order 7 square into 9 smaller squares. An Ascii solution is below, along with the Duijvestijn code. --Ed Pegg Jr, www.mathpuzzle.com (Fixed width Ascii art) ___ ___ ___ ___ ___ ___ ___ | | | | | | | | | | | | | |___ ___|___ ___| | | | | | |___ ___ ___|___|___| | | | | | | |___|___ ___| | | | | | | | | | | | | | | | |___ ___ ___ ___|___ ___ ___| [3 2 2][1 1 2][4 1][3]