Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

hv at crypt.org wrote:
>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.

And I wrote yesterday:
>>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.

It occurred to me last night while lying awake in bed that the 
(4n)-by-(4n) proof-by-example can be easily modified to get any 
(4n+2)-by-(4n+2) example. (duh!)

Hugo's example for 6-by-6 shows how to do this.

So, any even-by-even grid can be completely filled, starting anywhere, 
and starting either with a diagonal or a non-diagonal move. (as long as 
crossings are allowed)


thanks,
Leroy Quet