Diagonal/Non-diagonal Grid Filling

[hv at crypt.org]

Leroy Quet <qq-quet at mindspring.com> wrote:
:Here is a simple example showing that it is possible to fill a 
:(4n)-by-(4n) grid, starting at one of the 4 center squares, and moving 
:diagonally first then non-diagonally (and crossings are allowed).
:
:  45 44 37 36 29 28 21 20
:  43 46 35 38 27 30 19 22
:  42 47 34 39 26 31 18 23
:  48 41 40 33 32 25 24 17
:  49 56 57 64  1  8  9 16
:  55 50 63 58  7  2 15 10
:  54 51 62 59  6  3 14 11
:  52 53 60 61  4  5 12 13
:
:But is it possible to fill the (4n+2)-by-(4n+2) grid or the odd-by-odd 
:grid by starting in the (almost) center (crossings allowed, start with 
:either a diagonal or a non-diagonal move)?

Did you want an example or a rule? Here's a (looping) example for 6 x 6:

 26 27 22 23 18 19
 28 25 24 21 20 17
 29 36  1  8  9 16
 35 30  7  2 15 10
 34 31  6  3 14 11
 32 33  4  5 12 13

Since 36 connects legally to 1, this can start either diagonally or
orthogonally.

My code finds no solution for 5 x 5 starting either diagonally or
orthogonally; trying 7 x 7 now.

Hugo