Dan, I believe that the point were symmetry in a classical sense has the most influence (or the clearest) is when playing for worst score instead of best. For instance, on the 3x3 square grid, the "smallest" greatest cell you can have is 8 I believe. But on a 3x3 torus, every cell is the neighbour of every other so all paths are equivalent and you end up with 128 = 2^7. etc. The sequence corresponding to the worst score in a nxn grid for strict rules seems to be 1,4, 8, 32 ?, 28 ? , ... Olivier On Mon, Apr 15, 2013 at 2:16 AM, Dan Asimov <dasimov@earthlink.net> wrote:
As is my custom, I'm curious what happens if the NxN square is replaced by an NxN square torus. This has more symmetries than the square, so probably won't be any harder.
Also of interest might be the same question asked of hexagon tori, where the nth one is made of what Wikipedia calls a "centered hexagonal number" (or just a "hex number") of hexagons, where that's made of
H(n) = 1 + T(n-1)
hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n= 0,1,2,....
(Of course for the hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge.)
--Dan