Diagonal/Non-diagonal Grid Filling

[Leroy Quet]

We start with an n-by-n grid.
We can put 1 in any square.
We put the integer m in an empty square adjacent to (m-1).
But each even m is always put either above, below, right of, or left of 
the integer (m-1).
And each odd m (m>=3) is always put in any of the four main diagonal 
directions from (m-1).

What is the sequence where a(n)= the highest possible m that can be 
achieved for an n-by-n grid before we get stuck?

(For example, n=2 is easy.

1 2
3 4

a(2) = 4.)

There are several variations, such as:
1) Instead, odd m's are always diagonal from (m-1),
even m's are always non-diagonal from (m-1).
2) We can start with either a diagonal move or a non-diagonal move, 
whichever gets the highest m.
3) We are forced to start, say, in the upper-left square of the grid.
4) The grid has toroidal topology.

I might have posted a similar question to this group over a year ago, I 
believe. But at that time I was only concerned with the case of odd n and 
the path always starting at the center square, I think it was.
(In any case, I received no replies to my question at that time, I 
believe.)

thanks,
Leroy Quet