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This question is motivated by this xkcd comic strip
.

The problem is indeed interesting, and my first recollection upon reading this was a similar problem in the book Problems in General Physics by I.E.Irodov(which, in my humble opinion,is a masterpiece).

The question I wanted to ask is, are there any generalizations of the problem known? Can be find the resistance between any two nodes of the grid as a function of the distance between the nodes?

Many thanks in advance!

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    $\begingroup$ possible duplicate of On this infinite grid of resistors, what's the equivalent resistance? $\endgroup$ May 24, 2011 at 6:42
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    $\begingroup$ @mark: This question is regarding the resistance between any two nodes. $\endgroup$ May 24, 2011 at 7:13
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    $\begingroup$ @Kound It is functionally the same question. If you read the answers to the previous question in detail, you would have known the answer to this one. In fact, both questions were answered with a link to exactly the same web page. $\endgroup$ May 24, 2011 at 11:28
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    $\begingroup$ There are more generalizations, how about 3D cubic grid? n-cubic grid? $\endgroup$ May 24, 2011 at 12:02
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    $\begingroup$ I have taken the liberty of changing your title to something more informative. "Nerd sniping" is an activity which involves any problem. The problem you have asked about is not widely known under the title "the Nerd Sniping Problem." $\endgroup$ Jan 14, 2017 at 3:52

2 Answers 2

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Yes, it is possible. For example Kevin Brown did here and here including this table.

enter image description here

so for the xkcd problem the answer is $-\frac{1}{2}+\frac{4}{\pi} \approx 0.773$.

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  • $\begingroup$ Who is this Kevin Brown? I bought his book and I don't know anything about him. $\endgroup$
    – Kasper
    May 24, 2011 at 12:42
  • $\begingroup$ @kmm: He is deliberately inconspicuous. In MathPages, his name appears on pictures of the covers of his books and in one other place. $\endgroup$
    – Henry
    May 24, 2011 at 13:14
  • $\begingroup$ That's strange. I'd just want to e-mail him saying I immensely enjoyed his book. $\endgroup$
    – Kasper
    May 24, 2011 at 20:17
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As far as I know, the first solution to the general problem is given by Cserti,

József Cserti. Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors. Am. J. Phys. 68 no. 10, pp. 896 (2000). doi:10.1119/1.1285881, arXiv:cond-mat/9909120 [cond-mat.mes-hall])

using lattice Green's functions (and there are references to previous partial solutions). For your first question, the recurrence relation that gives the resistance between nodes in a square lattice is equation 32. The paper also describes how to solve or derive asymptotics for rectangular lattices, triangular lattices, honeycombs, and cubic lattices in higher dimensions.

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