To answer this question, we'll need to find an expression for the value of the variable resistor that would make the voltmeter read zero.

First, it's important to realize what situation would result in a reading of zero from the voltmeter. For there to be no reading, that means that there cannot be any voltage difference between the top row of resistors and the bottom row. For this to happen, the voltage drop for the resistors on the left of the voltmeter must be equal, and the same is true for the two resistors to the right of the voltmeter. In other words, both rows of resistors will experience the same voltage decrease as current flows through, thus the difference of voltage drop in the top and bottom row will be identical.

So let's consider the top and bottom resistors on the left side first. In the top left corner, the voltage of the first resistor will be  from Ohm's law. Moreover, the voltage drop of the bottom left resistor will be . These two voltages will need to be equal to one another in order to have the voltmeter read zero.

Now let's take a look at the other resistors on the right. The voltage of the third resistor will be  and the voltage of the variable resistor will be . Just as before, these two resistors will also need to be equal in voltage.

Now that we have the two expressions shown above, we can isolate the term for the variable resistance in terms of the other three resistors to find our answer.

First we need to condense R3 and R4. They are in series, so we can simply add them to get:

Now we can condense R2 and R34. They are in parallel, so we will use the following equation:

Therefore:

The equivalent circuit now looks like:

Since everything is in series, we can simply add everything up:

The new circuit has two resistors in parallel: R2 and the new one attached. To find the equivalent resistance of these two branches, we use the following expression:

In this new equivalent circuit everything is in series, so we can simply add up the resistances:

Now we can use Ohm's law to calculate the total current through the circuit:

We will be working backwards on this problem, using the current to find the resistance. We know the voltage and desired current, so we can calculate the total necessary resistance:

Then we can calculate the equivalent resistance of the two resistors that are in parallel (R2 and our unknown):

Now we can calculate what the resistance between point A and B:

Rearranging for the desired resistance:

We can use the equation for equivalent resistance of parallel resistors to solve this equation:

We know the equivalent resistance, and we know that the resistance of each of the four resistors is equal:

Since we know the power loss and voltage of the circuit, we can calculate the equivalent resistance of the circuit using the following equations:

Substituting Ohm's law into the equation for power, we get:

Rearranging for resistance, we get:

This is the equivalent resistance of the entire circuit. Now we can calculate R4 using the expression for resistors in parallel:

We can use Ohm's law to calculate the equivalent resistance of the circuit:

Now we can use the expression for combining parallel resistors to calculate R1:

We will need to test the values of each answer to find the one that generates an equivalent resistance of .

We know that when condensing parallel resistors, the equivalent resistance will never be larger than the largest single resistance, and will always be smaller than the smallest resistance. Therefore, two of the answer options cen be eliminated immediately.

After we have narrowed our choices down to the other options answers, we just have to test them with the following formula:

We will test the incorrect answer first:

Now for the correct answer:

Because this circuit is neither purely series or purely parallel, we must simplify it before we solve it. Replace the right branch, which is purely series, with its equivalent resistance: 

Now we have a purely parallel circuit, each branch having a resistance of . Apply the parallel formula and solve:

For resistors all in series, the equivalent resistance is equal to the sum of the resistances.