A few days ago I saw this comic reposted, and I thought: wait! Unlike every prior time I have seen this comic, I actually know how to solve this now!

One thing which I often find really cool, and which I don’t think comes up a lot in most people's mathematical education, is when you can learn something about some thing non-random by analysing a random process. So, let me show you a way to find out the resistance between two points in a circuit by instead finding out the number of times someone randomly walking around that circuit will step on those points.

Equivalent Resistance

Here's a closeup of the problem:

You may be familiar with the rules for series and parallel resistors from high school: if there are some resistors in series (one after another), this is “equivalent to” having a single resistor in the circuit which has a resistanceequal to the sum of their resistances. That is, if you replace the resistors with just one of resistance , the behaviour of the rest of the circuit will be the same.

Some resistors in series

If there are some resistors in parallel, this is “equivalent to” having [...]

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Outline:

(00:54) Equivalent Resistance

(02:20) Circuit Laws

(05:57) A System of Equations

(08:44) Random Walks

(11:22) The Potential Function

(14:14) Fourier Analysis

(18:00) Recap

The original text contained 5 footnotes which were omitted from this narration.

“The Problem in the “Nerd Sniping” xkcd Comic” by peralice