Find the equivalent resistances between the points a and c of the network shown in figure. Each resistance is equal to `r`.

Explanation

Suppose a potential difference V is applied between a and c so that a current I enters at a and the same current leaves at c. The current I divides in three parts at a. By symmetry, the part in ad and in ab will be equal. Let each of these currents be`i_(1).`The current through ao is `i-2i_(1).`Similarly , surrent i. Since the situation at c is equivalent to that at a, by symmetry the currents in dc and bc will be `i_(1)` and that in oc will be `i-2i_(1).`
`Applying Kirchhoff's junction law at d, we see that the current in do in zero. Similarly,the current in ob is zero. We can remove do and ob for further analysis. It is then equivalent to three resistances, each of value 2r,in parallel.the equivalent resistance is, therefore,`2r/3`.