**Solution:**

To find the charge that must be placed at the fourth corner of the square so that the total potential at the center is zero, we can use the principle of superposition. According to this principle, the total potential at a point due to a combination of charges is the algebraic sum of the potentials due to each individual charge.

**Step 1: Calculating the potential due to each charge:**

First, let's calculate the potential at the center of the square due to each individual charge. The potential due to a point charge is given by the formula:

V = k * (q / r)

where V is the potential, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), q is the charge, and r is the distance between the charge and the point where potential is being calculated.

The distances between the charges and the center of the square can be calculated using the Pythagorean theorem. Since the square has sides of length x, the distance from the center to each corner is x/√2.

Calculating the potential due to each charge:

- Charge at corner A: q1 = 2 μC, r1 = x/√2
- Charge at corner B: q2 = 4 μC, r2 = x/√2
- Charge at corner C: q3 = 6 μC, r3 = x/√2

V1 = (9 × 10^9 Nm^2/C^2) * (2 × 10^-6 C) / (x/√2)
V2 = (9 × 10^9 Nm^2/C^2) * (4 × 10^-6 C) / (x/√2)
V3 = (9 × 10^9 Nm^2/C^2) * (6 × 10^-6 C) / (x/√2)

**Step 2: Finding the charge at the fourth corner:**

To find the charge at the fourth corner, we need to ensure that the total potential at the center of the square is zero. Since the potential due to each charge is an algebraic sum, we can write the equation:

V1 + V2 + V3 + V4 = 0

Substituting the values of V1, V2, and V3, we get:

(9 × 10^9 Nm^2/C^2) * (2 × 10^-6 C) / (x/√2) + (9 × 10^9 Nm^2/C^2) * (4 × 10^-6 C) / (x/√2) + (9 × 10^9 Nm^2/C^2) * (6 × 10^-6 C) / (x/√2) + V4 = 0

Simplifying the equation:

(2 + 4 + 6) * (9 × 10^9 Nm^2/C^2) * (1 / (x/√2)) + V4 = 0

12 * (9 × 10^9 Nm^2/C^2) * (1 / (x/√2)) + V4 = 0

V4 = -12 * (9 ×