Rooks Filling Space
This post is motivated by this New York Times NumberPlay. The comments provide the answer that one cannot go from (1,1) to (n,n) with a space filling curve when n is even. This is a parity argument.
I thought I would look at this from the viewpoint of paths from (1,1) to (n,n) and random “rook” walk from (1,1) and determine the probability of getting to (n,n) afer n1 moves. The sace fillign curves are a subset of these, showing how rare they are.
You can either take powers of adjacency matrix to get number of paths from i to j or look at is a Markov chain.
The following is derived shows the effect of n and demonstrates that there are no paths of length (n1) from (1,1) to (n,n) for even n as well as the distribution of “states” after n1 steps.
See update here.

December 28, 2013 at 4:42 pmMore Rooks on a Boards  Unknown Blogger Mathematica