 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

\b[False.] The cube has eight vertices and twelve edges. <Euler formula for a connected planar graph> for its connected <plane graph> gives $8-12+F=2$, so
$$
\boxed{F=6.}
$$
The nested-square embedding has four faces between the squares, one inside the smaller square and the unbounded exterior face.

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