Question

A network of 12 resistors each of value $R=6\Omega$ are interconnected as shown in figure, being placed along the sides of a regular dodecagon. Each of the terminals 1, 2, 3, ..., 12 has been connected to each of the 9 terminals (other than nearest) directly by insulated wires each of resistance R, there being 9 wires from each terminal making 108 wire connections totally. [Only one set of 9 wires, from terminal 1 have been shown]. Find the equivalent resistance of the network when the current enters at the terminal 1 and leaves at terminal 2.

Answer 1

1 \Omega

Answer 2

The network described is a complete graph ($K_{12}$) where all 12 vertices are interconnected by resistors of value $R$. The total number of edges (resistors) is $N(N-1)/2 = 12(11)/2 = 66$, which matches the sum of 12 dodecagon resistors and 54 internal wires. For a complete graph $K_N$ with identical edge resistances $R$, the equivalent resistance between any two vertices is given by $R_{eq} = \frac{2R}{N}$. Substituting $R=6\Omega$ and $N=12$, we get $R_{eq} = \frac{2 \times 6}{12} = 1\Omega$.