Twelve wires, each of resistance r ohms are connected in the form of a skeleton cube. Find the equivalent resistance of the cube when the current enters at one corner and leaves at the diagonally  opposite  corner

Let a current of magnituide 6i enter  at A and leave at G. The distribution  of current in the various arms is shown in figure

R be the equivalent resistance of the cube between points A and G and r be the resistance of each side of the cube.

Applying Kirchhoff’s loop law into the closed loop DCGA:

We get –2ir – ir – 2ir = – E

Or  5ir = E                         … (i)

Or  E = (5i)r   

But E = (6i)R                   … (ii)

Therefore 6iR = 5ir

or, R = 5/6r