Cube of Resistor Problem - Nice one

[luiseduardo]

Homework Statement

Asymmetrical Cube - Determine the equivalent resistance between points A and C’ of the circuit below.

Homework Equations

U = R.i

The Attempt at a Solution

This problem was my professor that created. I know how to solve it, but I would like to know if other people have a better idea how to solve it. I tried to use Kirchhoff's Laws and I found 6 equations and was a bit complicate to find the final answer. I think it's a good challenge.
The answer is:

$$\frac{1}{4} \left(R_1+R_2+R_3+\frac{1}{R_1^{-1}+R_2^{-1}+R_3^{-1}} \right)$$

 

[mfb]

3 equations are sufficient, as the cube still has a symmetry. If you add the potential of opposite points in the cube, you always get the same value.

I like the result.

 

[rude man]

The easy way is to recognize that, by symmetry, the voltage at the three corners connecting A by one resistor each is the same. Similarly, the voltage at the three corners connecting C' by one resistor each is the same.

Nodes of equal voltage can be shorted together without affecting the circuit. This degenerates the cube into three parallel sections connected in series:
Requiv = R1||R2||R3 + R1||R1||R2||R2||R3||R3 + R1||R2||R3.
No KCL or KVL equations needed!

 

[mfb]

The easy way is to recognize that, by symmetry, the voltage at the three corners connecting A by one resistor each is the same. Similarly, the voltage at the three corners connecting C' by one resistor each is the same.

Nodes of equal voltage can be shorted together without affecting the circuit. This degenerates the cube into three parallel sections connected in series:
Requiv = R1||R2||R3 + R1||R1||R2||R2||R3||R3 + R1||R2||R3.
No KCL or KVL equations needed!

That symmetry does not exist with different resistors.

 

[rude man]

That symmetry does not exist with different resistors.

Yes, I hadn't noticed the resistors could be of different values. Such an old problem, never seen it with differing values before.

 

[Borek]

Makes me think of http://xkcd.com/356/

 

[rude man]

Makes me think of http://xkcd.com/356/

OK, but I resent a physicist being worth only 2/3 of a mathematician ... I suppose only 1 point per EE, or less even? :-)

 

[luiseduardo]

My solution:

http://www.luiseduardo.com.br/electricity/electrodynamics/cubeofresistors.htm