11 Oct
2003
11 Oct
'03
7:06 p.m.
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).
Up to 15, you are definitely correct, and Gardner mentions the size 41 square as a particularly difficult case in his column. For order 16, the order 53 square seems best, and that seems to be a more recent discovery. Here's the pertinent square {{29,24},{6,5,13},{24,4,1},{5},{3,4},{7},{6,3},{16},{13}} If that's familiar, you are correct about 16. --Ed Pegg Jr.