Cube Cutting and Painting: Counting Cubes with 0, 1, 2 and 3 Painted Faces

The classic question takes a large cube, paints all six faces, cuts it into n × n × n equal small cubes, and asks how many small cubes have three, two, one or no painted faces. You do not need to picture all the small cubes. Every small cube is in exactly one of four places, and each place has a fixed number of painted faces.

• Corners touch three outer faces, so they have 3 painted faces. A cube always has 8 corners.
• Edges, excluding the corners, touch two faces: 2 painted faces. Each of the 12 edges has n − 2 such cubes.
• Face centres, away from edges, touch one face: 1 painted face. Each of the 6 faces has (n − 2)² of them.
• Core cubes are hidden inside: 0 painted faces. They form an inner cube of side n − 2.

Here n is the number of small cubes along one edge. If the question gives the total number of small cubes, take its cube root to find n: 64 small cubes means n = 4. If it gives lengths, divide: a 6 cm cube cut into 1 cm cubes has n = 6.

One face: 6 × 2² = 24.

No face: 2³ = 8.

Check: 8 + 24 + 24 + 8 = 64, the total number of small cubes. Always make this check; it catches most slips.

It also has 6 × 3² = 54 cubes with exactly one painted face.

To divide each edge into n equal parts you need n − 1 cuts, and you must cut in three directions. So the minimum number of straight cuts is 3(n − 1), assuming you do not rearrange the pieces between cuts.

For a painted cuboid of l × b × h small cubes, the same four positions apply, but the edges and faces have different lengths. There are still 8 corners. The 12 edges come in three groups of four, of lengths l − 2, b − 2 and h − 2. The six faces come in three pairs.

One face: 2 × (2 × 1 + 1 × 0 + 2 × 0) = 4. With 8 corners and no core, 8 + 12 + 4 + 0 = 24, the full count of small cubes.

If only some faces are painted, the four formulas no longer apply. Count by layers instead: decide which layers touch a painted face, and count the cubes in each layer.

• ✘ Using n as the total number of cubes instead of the number along one edge.
• ✘ Forgetting to subtract the two corners from each edge: the edge count is n − 2, not n.
• ✘ Applying the all-faces formulas when only some faces are painted.
• ✘ Skipping the check that the four counts add up to the total.

Spatial questions also include dice problems and counting figures. Cube roots of the totals come quickly with the cube root of a perfect cube trick, and more methods are on the study hub.

Do the formulas work if the cube is cut into only 8 small cubes?

Yes. With n = 2, every small cube is a corner, so all 8 have three painted faces, and the formulas for two, one and zero faces all give 0 because n − 2 = 0.

What if the small cubes are not all the same size?

The formulas assume equal cuts along every edge. If the cuts are uneven, draw the cube layer by layer and count each kind of small cube directly, because the corner, edge, face and core pattern no longer has equal pieces.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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