Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

I wrote in part:

>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

It just occurred to me that because 64 is next to 1, and because 64 is 
horizontally adjacent to 1, that we can start anywhere in the grid and 
begin with either a diagonal or non-diagonal move.

So, (4n)-by-(4n) grids are completely fillable, starting anywhere, as 
long as we allow the path to cross itself.

thaanks,
Leroy Quet