Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

By the way, I heard that there were some problems with seq.fan, so others 
probably discovered the following result and posted it before me, I bet. 
But I have not seen a proof for all even n's.

I have a simple proof that shows that any grid of (2n)-by-(2n) can be 
filled in completely, assuming that the path is allowed to cross itself 
(which it can only do on diagonal sections).

It can be best illustrated with an example:
(For n = 8)

1  2 17 18 33 34 49 50
3 16 19 32 35 48 51 64
4 15 20 31 36 47 52 63
14 5 30 21 46 37 62 53
13 6 29 22 45 38 61 54
7 12 23 28 39 44 55 60
8 11 24 27 40 43 56 59
10 9 26 25 42 41 58 57

Generally, we can have as many "towers" (of 2 squares width each, and of 
any even number of squares high) as we want.

Which grids are possible to fill in completely, however, if the path is 
not allowed to cross itself?

thanks,
Leroy Quet