Correction to my last post: I misstated the size of the nth "hex number" H(n). This is correct (rest of post repeated with annotation for context): ----- 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 + 6*T(n-1) hexagons, where T(k) = k(k+1)/2 is the kth triangular number, n=0,1,2,.... The sequence of H(n) is 1, 7, 19, 37, 61, 91, 127,.... But in fact, there are many other-sized hexagonal groups of hexagons that can have opposite sides identified consistently to make a hexagonal torus. The most natural of these is to iterate making a rosette of the previous pre-identification configuration, giving the sequence of powers 7^n of 7 as the cell counts: 1, 7, 49, 343, 2401,.... (Of course for any hexagonal torus (or, OK, grid), the analogue of a king's move would just be a move to any cell sharing an edge. ----- --Dan On 2013-04-14, at 5:19 PM, Allan Wechsler 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.)