electrical resistance - Will any charge flow in this circuit?

I had to find the resistance of this circuit between A and B. My teacher said that since the blank wires have zero resistance, the potential difference across the ends of the blank wire is zero(according to v = ir) and thus 1,4 are equipotent and 2,5 are equipotent and the circuit is thus simplified.The equivalent resistance comes out to be 2r/5.

MY QUESTION:If the points 1,4 and 2,5 are equipotent how can current flow through the horizontal circuit.

MY REASONING: The potential difference across each branch of a parallel circuit is same and in this case it is 0 and hence no current can flow through the circuit.

Answer

You can explicitly compute the currents to see this.

Let us conventionally assume the currents are positive when they flow from left to right, let $V_i$ denote the potential at node $i$, and $I_1, I_2, I_3$ the currents respectively between the nodes 1-2, 2-4 and 4-5.

The blank wires shortcircuit some of the nodes, so that we have

$$V_1 = V_4 \qquad \text{ and } \qquad V_2 = V_5.$$

Applying the usual Ohm's law we see that:

$$V_2 - V_1 = r I_1, \\ V_4 - V_2 = V_1 - V_2 = 2r I_2, \\ V_5 - V_4 = V_2 - V_1 = r I_3$$ from which it follows that $I_1 = I_3 = - 2 I_2.$

As you can see the shortcircuits in the example do not prevent a flow of current in the circuit.