Question

How do you solve this?
Do you use kirchoffs laws?

Answer 1

     Let there be no current in the 6V & 6 Ω branch. Then there is current in the 12V, 12 Ω and R Ω.

    Current i = 12/(12+R)  Amp
   Voltage across R = V = R i = 12R/(12+R)

   If the voltage V = 6 Volts, then there will not be any current through the 6 Ohms resistor.

      So   12 R/ (12+R) = 6 
              12 + R = 2 R
              R = 12 Ω.
Answer.
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This problem can also be solved by assuming the currents in the two branches with batteries as i1 and i2.  Then we solve for i1 and equate it to 0 to find R.  we get 12Ω.

Answer 2

the equivalent resistance is 12 ohm