Hard physics electricity problem

Question

I apologize for my poor English.

12 resistors, each with resistance R, are connected into a cube-shaped framework, which has one resistor on each side.

How to create scheme of equivalent circuit if joint resistivity of the circuit is measured on one of the resistors ends (on two adjacent vertexes of the cube) and find the resistivity of the circuit

Answer 1

Since there are 12 edges in a cube and just 11 resistors ( each of resistance R) in the problem, it can be concluded that one edge of the cube lacks a resistor. Call this edge the unique edge. Take this unique edge to be the front left vertical. The problem then becomes to figure the current in each edge (except the unique one) when a voltage, V, is imposed across the unique edge.

It appears that this circuit cannot be simply decomposed. However, Kirkoff's junction rule and the symmetry of the circuit can be used to show that there are just two independent currents.

These can be taken as the he currents in the vertical back right edge - call that current value e , and the equal currents in the other two vertical edges - call this current b.

Two Kirkoff;'s voltage loop equations can be set up and solved to find

e = (1/7) V/R and b = (2/7) V/R. From this it is easy to see that the total current flowing is

e + 2b = (5/7) V/R . Since the equivalent resistance is V/(total current)

the equivalent resistance is (7/5) R.