I'll reword the problem slightly ... for convenience.
An cube is painted, then it is cut into unit cubes.
A unit cube can have: 0, 1, 2, or 3 painted faces.
Making some sketches, the table can be extended and generalized.
. . . . . . . .
We have an cube.
There are always 8 corner-cubes (3 painted faces).
There are 12 edges; each edge has edge-cubes (2 painted faces).
There are 6 faces; each face has face-cubes (1 painted face).
Inside, there is a smaller cube of side unpainted cubes.