# How would one prove that the electric flux through any particular face of a cube is constant regardless of charge distribution within the cube?

My physics teacher claimed in class today that for a 3-dimensional cube with discrete point charges, it matters not where any of the charges are distributed when finding the electric flux through any one of the six faces of the cube. This seemed... off to me, but I couldn't come up with any counterexamples on the fly, so I went with it.

I understand Gauss' Law pretty well (at least I think I do), and while it makes total sense to me that the sum of the flux through the the six faces is constant wherever the charges are located, it doesn't make sense to me to say that the flux through each face is equal to every other face regardless of how the charges are distributed.

Mathematically, what she was saying was that for a 3-dimensional cube with arbitrary charge distribution, it must be true that the flux through any particular face is $$\Phi_{face}=\frac{Q_{enclosed}}{\epsilon_0*6}$$

and that $$\Phi_{face}$$ is constant regardless of which face is selected.

Is this true and how would one show it?

Edit: one fairly easy counterexample is putting a point charge in the corner of a cube, and then the flux is $$\frac{Q_{enclosed}}{\epsilon_0*24}$$ for some sides and 0 for others. However, this isn't a true counterexample since the charge isn't inside the cube, just on it.