Following a discussion in Reasonable Deviations, I spend some three hours solving a (quoting) “little school-level problem” posed in a weird Russian CAPTCHA system. Not that it is extra-difficult, but some people may find it interesting to see the solution of its generalized formulation, so here it is.

Problem: we have an electric circuit as depicted in the following figure:

The task is to find the altogether resistance from A to B.

The answer is (numerator and denominator split by “-”):
(n2+n4)*(n1*y+n5*x)+(n1+n2)*n4*x
—————————————
(n1+n2)*y

, where:

x = n3*n2-n1*n4
y = (n1+n2)*(n3+n4)+(n1+n2+n3+n4)*n5

The solution is after the jump.

Let’s depict the currents in this circuit:

We marked each current as a proportion to the one coming from A (and to B), e.g. I3=i*i3. Using Kirchhoff’s circuit laws, let’s write down the system of equations for this circuit:

x = n3*n2 – n1*n4
y = (n1+n2)*(n3+n4) + (n1+n2+n3+n4)*n5

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URL really smart!

PS happy 2008!