Conjecture: the highest achievable outcome can be achieved by a king's-move-connected path through the grid. This seems very obvious to me intuitively, but I'm too scatterbrained to prove it at the moment. If it's true, it should speed up searches considerably. On Sun, Apr 14, 2013 at 4:14 PM, Fred W. Helenius <fredh@ix.netcom.com>wrote:
On 4/14/2013 2:00 PM, Hans Havermann wrote:
Eric Angelini wrote:
My best MAX is 40: is it the highest possible MAX in a 3x3 square?
I believe the highest MAX is 44.
I concur with Hans, and add that the highest for a 4x4 square seems to be 2473 (barring software or hardware errors). The filled-in grid looks like this:
1 1 1239 2473
2 4 419 815
6 18 100 296
6 30 48 148
It's easy to figure out the order in which the numbers are entered. The two 6s are a bit surprising.
My brute force approach handles 3x3 in under a second and 4x4 in under two hours; 5x5 would seem to be infeasible without better ideas.
-- Fred W. Helenius fredh@ix.netcom.com
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