Resistance Cube- Interesting Question

Calculate the equivalent resistance between points A & B.

If you ended up with 5/6 ohms, you've nailed it! If not, no worries, continue reading below for a detailed explanation.

First things first, how do we go about this question?

Do we try to resolve it into simpler- series and parallel combinations ?
- Nope, If you would manage to do that, it would take forever.

Do we look for the shortest path current can take?
- Nope, because some current flows through all the paths, moreover, all resistances in this question are equal!

When all fails, Kirchhoff's laws always work!
Kirchoff's laws state that-

KCL- Kirchhoff's Current Law
At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

KVL- Kirchhoff's Voltage Law-

The directed sum of the electrical potential differences (voltage) around any closed network is zero, or:

More simply, the sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop, or:

The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.

Lets quickly take a look at a simpler question which uses the same concept, to give you an idea of how Kirchhoff's laws can be applied.

Consider the circuit above.
STEP1- Lets start by assigning currents to each of the paths by following KCL discussed above.

STEP2- Now lets select a loop! For instance-

Alright! Now lets quickly frame some equations from our closed loop by following  KVL as discussed above.

By forming multiple equations, we can simply add or subtract two or more equations and eliminate some variables to get a simpler equation which will help us get what we are looking for !

Back to the original question-

To start with lets write down how the current will get distributed if it enter the network. To simplify calculations lets assume that 3I current enters the circuit!

Golden Rule-1 - This circuit is symmetric! So the currents in the circuit will also be symmetric as in if 3I current splits into 3 ways , I current will flow through each of these 3 paths.

Golden Rule-2- At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node i.e if 3I current enters a network, 3I current must exit the network!

But we are talking about closed loops! Wouldn't it be a fair deal if I said- When a battery with voltage V is connected to the resistor network, 3I current flows through the network?

Alright! Now, lets select a loop as well!

Here's the beauty of Kirchoff's law- Anything that doesn't lie in this closed loop is as good as it doesn't exist!

Lets write a closed loop equation for our selected loop-

Now, lets use this result with Ohms law for the circuit as a whole, in order to calculate the overall resistance.

Hope the explanation helped you understand Kirchhoff's Epic Laws better!