Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

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)?


Also:
Dion Gijswijt wrote in part:
>Also I think that I have seen this (or something very related) somewhere,
>maybe in one of Gardners recreations... Does anyone know a reference?
>Thanks!

I HAVE read several of Gardner's books, as well as some Clifford Pickover 
books. So I may have indeed stold the idea from somewhere. I hope someone 
can tell us if this puzzle idea is original.

thanks,
Leroy Quet