The resistance in each of the three arms of the network is 4 Ω. Hence, to balance the network, the resistance in the fourth arm must also be 4 Ω.
Hence, the resistance (R) to be connected across. i.e., in parallel to, the 12 Ω resistance should be such that their equivalent resistance is 4Ω.
∴ \(\cfrac14\) = \(\cfrac1{12}\) + \(\cfrac1R\)
∴ \(\cfrac1R\) = \(\cfrac14\) - \(\cfrac1{12}\) = \(\cfrac{3-1}{12}\) = \(\cfrac2{12}\) = \(\cfrac16\)
∴ R = 6 Ω