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On an infinite grid of ideal one-ohm resistors, what's the equivalant resistance between two nodes a knights move away?

I first saw this problem on the Google Labs Aptitude Test.  A professor and I filled a blackboard without getting anywhere.  Have fun.

(please fix the tags, I didn't really know where to put it)

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Does anyone else think of Mass Ave. and MIT when looking at the cartoon? –  Pete L. Clark Dec 3 '10 at 5:35
You might also awnt to ask this on physics.stackexchange.com –  Frédéric Grosshans Dec 3 '10 at 10:08
Has anyone generalized this to a triangular hex-lattice yet? –  deoxygerbe Dec 4 '10 at 23:46
It's really a combinatorial problem more than a physics problem. –  Raskolnikov Dec 21 '10 at 14:19

2 Answers 2

up vote 11 down vote accepted

After Google asked this in their aptitude test, this has become a famous problem.

You can find a nice discussion and a more general solution here: http://www.mathpages.com/home/kmath668/kmath668.htm

I believe the answer for your case is $\displaystyle \dfrac{4}{\pi} - \dfrac{1}{2}$.

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I think it's been a famous problem a lot longer than that. (-: –  Tobin Fricke Dec 3 '10 at 18:29
@TObin: I guess... Hard to debate either way :-) –  Aryabhata Dec 3 '10 at 18:41

The good answer is indeed $\frac{4}{\pi}-\frac{1}{2}$. You can find a complete solution in the book of R. Lyons and Y. Peres "Probability on trees and networks", section 4.3, p. 124-127. This mainly uses Fourier analysis and the symmetry of the grid.

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