Unfortunally, as I just created my account here, I cannot comment. So this is not a complete answer.

The way you need to look at it, is that the equivalent resistance from nodes a-b, a-c an b-c must be equal comparing between both designs, so that's where you start your math.

If you want to, you can start algebra from that and you will get the answer.

I'll edit this answer ASAP in order to give you the full explanation including algebra stuff, I'm at work now :)

Hope it helps.

My previous answer was unhelpful to the OP, so I will attempt a different approach and delete the old answer.

Wye-delta transformations are mathematical and built on the same principle as Thevenin and Norton Equivalents - the superposition theorem for electric circuits.

I won't get into the details of the proof, but if you like it is explained fairly well here. Fundamentally, it states that the response of a bilateral* linear** circuit with multiple sources can be determined by algebraically summing the responses of the circuit to each source, with the remaining sources replaced by characteristic impedances.

*response is independent of current direction

**containing no non-linear elements (capacitors, inductors, etc)

To do this transformation, we apply an ideal current to nodes 1, 2, and 3 (respectively) in each step of the transformation:

1) (I1-I2), -(I1-I2), 0

2) 0, (I2-I3), -(I2-I3)

3) -(I3-I1), 0, (I3-I1)

This is consistent with Kirchoff's Current Law, as I1 + I2 + I3 = 0.

By choosing this approach, we eliminate one of the nodes from each step, allowing us to compute the equivalent resistance across the remaining nodes as we would normal series and parallel resistances. We then linearly sum those results to give a final equivalence.

In short, it is a choice backed by mathematical principles of superposition to make the problem inherently easier to solve.

## Best Answer

If you're happy to trade accuracy for speed, I'd note that the circuit is a 10 ohm in parallel with everything else. Everything else is going to work out to be at least 10 ohm, so Req is approximately 6 ohm. Good to within an order of magnitude.

For accuracy and generality it has to be nodal analysis (with a dummy source). Apply at every node and stick into a big matrix to be solved by a linear algebra package.