How Do Resistors in Series and Parallel Work?

If a resistor network is a combination of series and parallel resistors, then it can be broken up into smaller parts which are either one or the other. For instance, if R1 and R2 are in parallel and R3 is in series with them, then their total resistance (Req) is given by:

Req = (R1xR2 / R1+R2) + R3

Conversely, many resistor networks can't be broken up this way. For instance a cube; where each edge has been replaced by a resistor. Finding the resistance between two opposite vertices requires additional transforms.