Question

Twelve equal wires, each of resistance $6\Omega$ are connected so as to form a cube-frame. Find the equivalent resistance( in $\Omega$) between the diagonally opposite points 1 and 7.

Answer 1

Connect a source between points 1 and 7.
The network is symmetrical (path symmetric) about the diagonal 1-7. Current in resistors are distributed symmetrically about the diagonal. All paths from one point to another which have the same setting of resistances have the same amount of currents.

The current distribution is shown in figure.
Applying KVL for closed loop 1-2-3-7--9-10-1, we have
$-r\frac{i}{3}-r\frac{i}{6}-r\frac{i}{3}+V=0$
$\frac{V}{i}=\frac{5}{6}r$
As $r=6\Omega$
$R_{17}=\frac{V}{i}=\frac{5}{6}r=5\Omega$

Why this Question? Concept of symmetry, solves complex circuits in a simple manner.