Puzzle: A Cube Of Resistors, Updated With Links To Solutions

I posted this puzzle a few days ago:

Consider 12 identical resistors connected to form a cube. That is, each resistor lies on the edge of a cube, and there is an electrical connection joining three resistors at each of the cube’s eight vertices.

What is the effective resistance between two vertices across a body diagonal of the cube?

One way to solve the problem is to write down many equations, based on Kirchhoff’s laws, and then solve them. This is rather brute-force, and is effective but a pain. Another way is to use symmetry; because the resistances are all equal, one can immediately infer that certain currents in the cube are also equal. This turns out to be effective, and a lot simpler than the first method.

Both methods are illustrated very well in these solutions by David Randall which you can find here (scroll down and look for “resistor cube problem”).

In terms of problem-solving strategy, the second solution reminds us of two important points:

1. Be on the lookout for symmetry, and when you identify some symmetry in a problem, make use of it.

2. When solving circuit problems, if you can’t think of anything better to do, or the other options are painfully complicated, you can always fall back on labelling each vertex of the circuit with a potential. This is not exactly what Randall does in his second solution, but it’s close enough. To be effective in the cube problem, one still has to use symmetry, but the idea of labelling each vertex with a potential is more general, and is often helpful.

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Finally, one can read a different kind of solution here, which involves redrawing the circuit in a way that allows one to calculate the net resistance without considering current flows, but simply by using the rules for combining resistances in series and parallel. In other words, the method is to redraw the circuit so that it is clearly a combination of series and parallel branches.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)