Consider a poorly designed robot vacuum cleaner which is in an enclosed, rectangular room. The robot is $1$x$1$ tiles wide. The room is mxn tiles wide, m$,$n $>2.$ The robot begins in the corner. The robot moves forward along the edge of the room until it collides with a wall, and then turns right ninety degrees. You are asked to prove that this robot never reaches the centre of the room. Which invariant would be most powerful in formulating this proof?