I actually found time to work through the trig. I find this quite counter-
intuitive, but it must be.
On the cube (assuming all +ve charges), the center is unstable towards each
face. Intuitively, I would have thought that as the (+ve) charge
approaches a plane (side of the cube), it would be repelled more strongly
than from the opposite side and be pushed back to the middle.
Instead, because of the steepening angle, the relevant component from the
closer charges is smaller than that from the distant charges.
There are not many days when I find such a neat result. Oh, happy day!
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.