## A Problem of Resistance

Bizarrely, last night I dreamt of this physics problem. This mean that I’ve seen it before somewhere, but if that’s the case then I’ve forgotten where. In the dream the problem of electrical resistance was muddled up with the problem of how to calculate the Euler Characteristic of a structure defined on a grid*, which is something I have used in the past. Anyway, with apologies for the poor quality of the drawing, here is the set up.

Twelve identical resistors R are arranged in four squares with common edges thus:

Yes, they’re meant to be identical squares!

What would be the effective resistance of this circuit measured between A and B?

Please post your answers through the comments box, with appropriate explanations. Bonus marks for elegant (i.e. short) solutions.

(In my dream this problem came up in contrast with the case where the four internal resistors and their connecting wires were absent, so the circuit was just a ring. The Euler Characteristic of the original connected set of squares is 1 while that of the ring is 0, not that it’s relevant to the problem in hand!)

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November 29, 2020 at 2:27 pm

By symmetry R

Michael Kane (long time lurker)

November 29, 2020 at 4:54 pm

I think you need to show more working than that!

November 29, 2020 at 5:27 pm

Same as below

November 29, 2020 at 3:24 pm

By symmetry, the entire midline is at the same potential (midway between the potentials at A and B), so no current flows through the two horizontal resistors in the middle. We can therefore take them away without affecting the current flow through the whole circuit. The remainder is equivalent to resistances 4R, 2R, 4R in parallel, which has equivalent resistance R.

November 29, 2020 at 6:33 pm

This looks like a similar problem from the Google Labs Aptitude Test, referenced on xkcd, where the resistor grid is infinite, and the question is to find the equivalent resistance to a knights move on the grid.

November 29, 2020 at 6:43 pm

For extra credit, I’m surprised you didn’t add the classic about the equivalent resistance of an infinite square grid of resistors – though I guess that would have required a larger sketch…

November 29, 2020 at 6:58 pm

I posted that already, but it was a long time ago (and apparently after a trip to Edinburgh!)

https://telescoper.wordpress.com/2010/02/18/talk-nosh-and-gridlock/