Count cubes by painted faces

All six outside faces of the large cube are painted first, including the bottom and the faces hidden from view. Then it is cut into equal small cubes. New cut faces have no paint.

Let n be the number of small cubes along each edge. Removing one cube at each end leaves n − 2 interior positions.

Group cubes by where they sit. These counts apply when n is at least 2:

• No paint, fully inside: (n − 2)³.
• One painted face, away from all edges: 6 × (n − 2)².
• Two painted faces, on edges but not corners: 12 × (n − 2).
• Three painted faces, at the corners: 8.

With two cubes per edge, all eight small cubes are corners. An uncut cube, with one per edge, has six painted faces; that case is not used in the questions.

For a 4 × 4 × 4 cube, count small cubes with exactly two painted faces. Each of the twelve edges has 4 − 2 = 2 cubes after excluding both corners. So 12 × 2 = 24.

Check every horizontal layer from above. Dots mark matching cubes, not painted cut surfaces. The bottom and top each contribute 8; the two middle layers each contribute 4. Thus 8 + 4 + 4 + 8 = 24.

Now use a 5 × 5 × 5 cube, painted on all six outside faces before cutting.

Count only cubes with one painted face. Leave out the edges and corners, then include all six face interiors.