This is question 2.30 from the 2nd edition of the Purcell book on Electricity and Magnetism. It's an interesting puzzle, and I've been thinking about it for a while, but I can't make any headway, so maybe you guys can do better. Suppose you have a uniformly charged cube (charge density ρ) of side length b. The electric potential is set to 0 at infinity. What is the ratio of the potential at the center of the cube to the potential at a corner? Purcell claims you can solve this using only superposition, without requiring any mathematics. He also suggests you first consider a bigger cube which is 2b on each side. - Here is what I have so far - due to symmetry, the potentials at the corners of a cube are identical. Due to superposition, this means that the potential at the center of any cube is equal to 8 times the potential at the corner of a cube which is half as long on each side. I can't prove that the corner/center potential ratio is independent of cube size, so I can't make any progress from here. Everything I tried ended up failing.