You place 4 charges of equal magnitude Q at the corners of a square of side length L, such that two of the charges are negative and two are positive, with the like charges opposite one another diagonally. What is the potential at the center?

Hello!
The electric potential of a point charge q at some another point is equal to  1/(4pi epsilon_0) q/r, where r is the distance from the point to the charge and epsilon_0 is an absolute constant (the permittivity of vacuum). Note that q is a signed value.
Also, it is known that the electric potential of point charges is additive, i.e. the potential of a system of charges is equal to the sum of the point's potentials. So we need to compute 4 potentials and sum them.
The distance r is the same for all four charges, and it is L/sqrt(2). The magnitudes of the charges are also the same, Q. Thus the sum is
1/(4pi epsilon_0) Q/r (1 + 1 - 1 - 1).
+1 is for positive charges, -1 is for negative.
We see that the result is zero, and it doesn't depend even on the rearrangement of the charges.