You have four resistors, each of which has a resistance r. it is possible to connect these four together so that the equivalent resistance of the combination is also r. how many ways can you do it? there is more than one way.

[tex] {\text{Consider four resistors, each of which has a resistance }}r\,{\text{ohm}}{\text{.}} \hfill \\ {\text{Case I:}} \hfill \\
{\text{Let pairs of two resistors connected in parallel, while each pair is in series}}{\text{.}} \hfill \\
{\text{Let }}R'\,{\text{ohm be the resultant of each pair of resistors combined in series}} \hfill \\ {\text{and }}R\,{\text{ohm be the resultant of whole combinations}}{\text{.}} \hfill \\R' = r + r \hfill \\ \Rightarrow R' = 2r \hfill \\ [/tex][tex] \therefore \frac{1}{R} = \frac{1}{{2r}} + \frac{1}{{2r}} \hfill \\
\Rightarrow \frac{1}{R} = \frac{2}{{2r}} \hfill \\
\Rightarrow \frac{1}{R} = \frac{1}{r} \hfill \\
\Rightarrow R = r \hfill \\
{\text{Case II:}} \hfill \\
{\text{Let pairs of two resistors connected in series, while each pair is in parallel}}{\text{.}} \hfill \\
{\text{Let }}R'\,{\text{ohm be the resultant of each pair of resistors combined in parallel }} \hfill \\ [/tex][tex] {\text{and }}R\,{\text{ohm be the resultant of whole combinations}}{\text{.}} \hfill \\
\frac{1}{{R'}} = \frac{1}{r} + \frac{1}{r} = \frac{2}{r} \hfill \\
\Rightarrow R' = \frac{r}{2} \hfill \\
\therefore R = \frac{r}{2} + \frac{r}{2} \hfill \\
\Rightarrow R = 2 \times \frac{r}{2} \hfill \\
\therefore R = r \hfill \\
{\text{Hence, there is two ways using which when four resistors, each of which has a resistance }}r\,{\text{ohm combined,}} \hfill \\ [/tex][tex] {\text{their resultant is again comes out to be }}r\,{\text{ohm.}}\hfill \\ {\text{Thus, there are more than one way to achieve this goal}}{\text{.}}\hfill \\ [/tex]