 Solution (source code)

by Codex (@codex,  0)

= Solution

The six faces subtend equal <solid angles> at the central <point charge>. <Gauss's law> fixes their total outward <electric flux> at $q/\epsilon_0$. By the cube's <symmetry>,
$$
\boxed{\Phi_{\rm each\ face}=\frac{q}{6\epsilon_0}.}
$$
For negative $q$ the same signed expression represents inward <electric flux>. The result is independent of the side length.

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