Let us fix the power source and let n vary. By analogy to pressure, the voltage everywhere in the circuit is inversely proportional to the total length of wire, which is n(n-1)/2. Meanwhile, the effective current between any pair of nodes is inversely proportional to the average travel time between the two nodes. Since the current forks in n-1 equally attractive directions at each junction, this travel time is proportional to n-1.

So, between any pair of nodes, V/I is in linear proportion to n-1/[n(n-1)/2] = 2/n.

If the resistance in each wire is R, then the resistance between any pair of nodes is 2R/n (since this gives the right constant of proportionality when n=2).

I imagine a real physics maven could explain this more succinctly and persuasively!

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