The Gray-White Claim:The path on the bottom always moves on gray squares. The path on the top always moves on gray squares if $c$ is odd; it always moves on white squares if $c$ is even. Proof: If {$c$} were {$0$}, the parity (gray/white) would change between bottom and top.  If {$c$} were {$1$}, and we're not near a corner, the parity wouldn't change.  This pattern continues for {$c=2, 3, 4, \ldots$}. The only difficulty is that going around a corner changes whether or not the parity shifts. But by the Two Corners Claim, the path always passes exactly two corners, removing the effect of corners on parity. |