Find the resistance between points `A` and `B` of the circuit shown in Fig.

Explanation

From the symmetry of the problem , current flow is indicated, as shown in the figure.
Now, `varphi_(A) - varphi_(B) = i_(1) r + (i - i_(1)) R`....(1)
In the loop `12561`, from `Delta varphi = 0`
`(i - i_(1)) R + (i - 2i_(1))r - i_(1) r = 0`
or, `i_(1) = ((R + r))/(3r + R)i` .....(2)
Equivalent resistance between the terminals `A` and `B` using (1) and (2),
`R_(0) = (varphi_(A) - varphi_(B))/(i) = ([(R + r)/(3r + R) (r - R) + R]i)/(i) = (r (3R + r))/(3r + R)`