Not all circuits can be reduced to sets of series and parallel resistors. (Notice that the three circuit diagrams above all describe the same circuit).
How can we handle a circuit like this which is more complex than being a combination of series and parallel pieces?
Assign a current In through each resistor.
At each junction, the sum of the currents in must equal the sum of the currents out; this is just charge conservation.
Around any circuit loop, apply energy conservation in the form of
For our example,
Watch the signs!
With both of Kirchoff's Rules, there will be enough information,
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(c) Doug Davis, 2002; all rights reserve