Sure, this is fun! My program is written to take any box dimensions, so doing non-primes is easy. I wondered about that myself, but figured odd primes may have some sort of “primitive” quality that non-prime dimensions would obscure. Another observation is that none of the boxes with a dimension (necessarily the smallest) = 3 are on the list of variable game lengths. I wonder whether there is something about 3 that precludes the “shortcut” phenomenon? I am thinking about running cases of 3 x p2 x p3 with p2 and/or p3 higher than 31. Currently, the max box volume is 21000, to accommodate the 23 x 29 x 31. My program can allow boxes with 2 or 3 dimensions equal. Are those of interest? Are there any box sizes or ranges of box sizes you are particularly interested in? I’m happy to run anything that executes in less than a day or so. Run time seems to be roughly proportional to p1^2 * p2^2 * p3. 23 x 29 x 31 takes 23 minutes for 1000 games. — Mike
On Mar 14, 2016, at 11:18 AM, James Propp <jamespropp@gmail.com> wrote:
Thanks Mike! Do you have a feeling for why primeness might play a role here? It might be illuminating to look at n1-by-n2-by-n3 boxes with n1 < n2 < n3 where n1, n2, and n3 are not required to be odd primes. Delving into an example smaller than the 5-by-11-by-13 block (the smallest block on your list) might give us some insight into what's going on in the general case.
Meanwhile, I'll try to find out who's responsible for the problem. It's quite possible that the author submitted a valid solution but that the editor of the column didn't appreciate the subtlety of the problem and therefore published a shorter invalid solution.
Jim