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!)
Follow @telescoper
In the Dark 