Current flowing through it is? Below is the question. I have seen many answers on it and some of them describes that the (3 ohm, 3 ohm) and (6 ohm,3 ohm) resistor are in parallel. So if they are in parallel it means that the point a and b has zero potential difference. If it has zero potential difference then the question directly dies here, the current must be zero. How's that possible? Please help me through this.

 A: I understand. The current flowing left goes through the two 3 ohm in parallel. $R= \tfrac{1}{1/3+1/3} =1.5$ for that step. Then through the 3 and 6 in parallel. R=2 for that step. So R=3.5 for the circuit. I = 2 amps total. So the voltage drop across the two 3’s is $V=IR= 2 \cdot 1.5 = 3 \text V$. So the next cross of resistors is 4v.
Because they are in parallel, the same amount flows through each, 1+1=2. Alternatively you can say the current through the top one is $I=V/R=3/3=1 \text{ amp}$. Same with the bottom resistor.
Then at 4v across the next one: $I=V/R=4/6=2/3 \text { amp}$ top and $4/3= 4/3$ bottom (and also they add to 2). So 1/3 amp has to get to the bottom of the right two resistors. It will flow there because there is no resistance to it flowing, $R=0$. It will go there and have no drop in voltage along the way, so the voltage will be the same $V=IR=(1/3) \cdot 0 =0$. There is no change in V across wires connected, but not zero current I.
A: The basic approach is to first consider the circuit without the conductor.

*

*Check where is plus and minus on the battery (Answer: plus is the longer bar, i.e on the left hand side)

*The voltage dividers cause point a to have higher potential than point b (Because 6 Ohms has larger voltage drop than the 3 Ohms resistor, so the upper, asymmetric voltage divider has a higher voltage than the lower, symmetric voltage divider which just splits the battery voltage in half)

*When "inserting" the conductor, current will flow from higher potential to lower potential. Utlimately it will perfectly equalize the potential if it has zero resistance. Of course this process is nearly instantaneous in reality, but for analysis we could imagine it as a process with finite duration. The voltage of the upper voltage divider will go down a bit and the voltage of the lower voltage divider will go up a bit (until they become equal) during that process. For that a permanent current flow is necessary.

*When checking the possible answers there is only one possibility. So we do not need to worry about calculating/estimating the amount of current, and stop here.

A: 
If it has zero potential difference then the question directly dies
here, the current must be zero.

Though this was addressed in the comment by gandalf61, I would like to expand on this a bit. If it were true that zero volts across implies zero current through, then how is there non-zero current through any of the (ideal) wires in the circuit shown?
In ideal circuit theory (which I assume is the context of this question), the wires shown in the schematic are ideal in the sense that they have zero volts across yet there is a current through. Look at this distinction between wires and resistors that I found here:

Yes, for non-zero resistance, zero volts across implies zero current through. However, for zero resistance (ideal wire), the current can be any finite value and satisfy Ohm's law:
$$V = I\cdot R = I\cdot 0 = 0\quad\mathrm{for\,any\,I} $$
UPDATE:
My intention here is to use KCL by inspection to show that the current through the wire from a to b cannot be zero if the series current $I$ is non-zero. Here's a scan of my hand drawn schematic:

I say by inspection because it is clear (I think) that the current $I$ entering the network from the left must divide equally between the two equal resistances $R$ on the left. Further, the current exiting the network from the right must divide as shown given the resistance ratio of the resistances on the right.
Then, at junction of the top two resistors, there must be non-zero current 'down' to satisfy KCL there. All that's left is to calculate the series current I for a numeric answer.
