On Thu, 9 Oct 2003, Ed Pegg Jr wrote:
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}
Let me say that there should have been no lack of confidence up to 15, because I proved the above numbers best possible up to there long ago (I think in fact up to 16, but am not quite sure of that).
Other pertinent refs: http://mathworld.wolfram.com/MrsPerkinsQuilt.html http://mathworld.wolfram.com/PerfectSquareDissection.html
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. By the way, the name "Mrs Perkins's Quilt" comes from a problem in one of Dudeney's books, wherein he gives the answer for n = 13. John conway