Cube & Dice

The picture to keep in mind

A big cube of side n (made of n × n × n unit cubes) is painted on the outside and then cut into n³ unit cubes. A small cube keeps paint only on the faces that were on the surface of the big cube. Its position decides how many of its faces are painted.

The four counts

• 3 faces painted — corners. A corner cube touches three faces of the big cube. There are always 8, whatever n is.
• 2 faces painted — edges. Each of the 12 edges holds n cubes, but the 2 ends are corners, leaving n − 2 per edge: 12(n − 2).
• 1 face painted — face centres. On each face, the cubes not on any edge form an (n − 2) × (n − 2) block: 6(n − 2)².
• 0 faces painted — the core. Peeling one layer from every side leaves an (n − 2) × (n − 2) × (n − 2) block: (n − 2)³.

Why (n − 2) everywhere

A cube on an edge but not at a corner has lost 2 of its edge-mates to the corners; a face block has lost its border ring. Always subtract 2 before squaring or cubing — using n instead of (n − 2) is the classic error.

The total check

8 + 12(n − 2) + 6(n − 2)² + (n − 2)³ = n³. Test with n = 3: 8 + 12 + 6 + 1 = 27. Test with n = 4: 8 + 24 + 24 + 8 = 64. If your four counts do not add to n³, something is wrong.

Combined questions

• At least one face painted: everything except the core, n³ − (n − 2)³.
• At least two faces painted: corners plus edges, 8 + 12(n − 2).
• Adding the counts of two classes is enough — no new formulas are needed.

Working backwards

Sometimes a count is given and n is asked. Divide: if exactly-two cubes number 12(n − 2) = 24, then n − 2 = 2 and n = 4. If exactly-one cubes number 6(n − 2)² = 54, then (n − 2)² = 9 and n = 5. Then answer the real question with n.

Cuboid a × b × c

The same positions, with each edge keeping its own leftover:

• corners: 8;
• edges: 4[(a − 2) + (b − 2) + (c − 2)];
• face centres: 2[(a − 2)(b − 2) + (b − 2)(c − 2) + (c − 2)(a − 2)];
• core: (a − 2)(b − 2)(c − 2). These hold when every side is at least 2. Check: 5 × 4 × 3 gives edges 4(3 + 2 + 1) = 24 and core 3 × 2 × 1 = 6.

Common mistakes

• Writing n instead of (n − 2) inside the formulas.
• Counting 8 corners as 12(n − 2) + 8 twice, or forgetting that corners are never counted among edge cubes.
• Adding the four counts and expecting n³ − something: they must total exactly n³.
• For a cuboid, using n − 2 with a single n instead of each side's own (side − 2).

Quick revision

• 3 painted: 8 (corners, any size).
• 2 painted: 12(n − 2) (edges).
• 1 painted: 6(n − 2)² (face centres).
• 0 painted: (n − 2)³ (core).
• Check: the four counts add to n³.
• At least one: n³ − (n − 2)³; at least two: 8 + 12(n − 2).