In the given figure,  $R=3\Omega .$ Determine equivalent resistance between points A and B 

Step-I: Attach a 100 V battery and assign potential at each node. 

Step-II: By symmetry  $100-y=y-0ory=50V$

Step-III: Now apply  KCL at node c

$\frac{x-100}{R}+\frac{x+50}{R}+\frac{x-0}{R}=0$

Or x= 50 V

Step-IV: If $R_{eq}$  is equivalent resistance of this circuit, we have

$\frac{100}{R_{eq}}=\frac{x}{R}+\frac{50}{R}+\frac{50}{12}or{R}_{eq}=\frac{2}{3}R$

Alternative method

Making 2D projection

 There is perpendicular axis of symmetry, i..e, the points COP will be equipotential then the circuit will be