Home > Mathematica, Mathematics > Rooks Filling Space

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 n-1 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 (n-1) from (1,1) to (n,n) for even n as well as the distribution of “states” after n-1 steps.

nytrook

 

See update here.

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Categories: Mathematica, Mathematics
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  1. December 28, 2013 at 4:42 pm

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