TITLE: Number of grid points outside a rook circuit
QUESTION [17 upvotes]: When I draw a circuit on a 8x8 chess board that passes through all squares, no matter how many turns I draw, I always end up with 18 grid points outside my circuit (and 31 grid point inside). It looks to me that I get also constant values for any $n$ x $m$ board. Has anybody ever seen a formula of how many grid points are inside/outside for an $n$ x $n$ board? It looks obvious that there are (n/2 - 1) * (n - 2) points outside if $n$ is even, but I am searching for a proof.
If you could point me to some literature, that would be great!

REPLY [14 votes]: For an $m$ by $n$ array with $mn$ even, consider a lattice on the centres of squares. The path passes through $mn$ lattice points and contains no internal points. So, by Pick's theorem it contains an area of $\frac{mn}{2}-1$.
This area consists of unit squares centred on the 'grid points' of the question and so there are $\frac{mn}{2}-1$ enclosed grid points.