A holiday puzzle

I am afraid of gifts, both receiving and giving.  Luckily, I have been largely spared having to confront this challenge.  I am often (rightly) criticized that in rare occasions when I give, the gifts are often what I like, not what the receivers would.  People say gifting is an art — no wonder I’m bad at it.   It is therefore a pleasant surprise that I received a holiday gift a few days ago, and it is a fun puzzle.

Consider an infinite grid where each branch (solid blue line segment) has a resistance of 1 ohm, as shown in the figure below. fig-grid

What is the equivalent resistance between any pair of adjacent nodes?   In other words, take an arbitrary pair of adjacent nodes, labeled + and − in the figure, and apply an 1-volt voltage source to the pair (the dotted line connecting the voltage source to the grid is idealized and has zero resistance). Denote the current through the voltage source by I_0.  What is the value of the equivalent resistance R := 1/I_0?

Chances are such an interesting problem must have been solved.  But instead of researching on prior work and its history, why not have some fun with it.  We don’t have to worry about (nor claim any) credits or novelty with a holiday puzzle!

…. But I would appreciate any pointer to its history or solution methods if you do know.  Even a random guess of the answer will be welcome.

In the next post, I’d describe two methods: one is a simple symmetry argument for a special case, and the other a numerical solution for the general case.   Meanwhile, have fun and happy holidays!

5 thoughts on “A holiday puzzle

  1. Hi Adam,

    It looks like you’ve been nerd sniped:


    See also Doyle & Snell’s 1984 textbook “Random Walks and Electrical Networks” for a discussion of many related problems (though I don’t know if they discuss the precise effective resistance puzzle you’re posing):

    Click to access walks.pdf

    Happy new year,

  2. Purcell (http://cds.cern.ch/record/102855) applies superposition and symmetry to show that the equivalent resistance between adjacent nodes in the infinite grid is 1/2 ohm.
    A survey of different approaches is done by Harley Flanders in 1972 (infinite Networks :Il-Resistance in an Infinite Grid). Apparently Kirchhoff’s laws are not sufficient to analyze the infinite networks of resistors: Infinite networks: I–Resistive networks, Flanders, H. , 1971

  3. Pingback: Rigor + Relevance | A holiday puzzle: solution

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