Cubes Reasoning Questions
Cube questions paint a large cube, cut it into equal small cubes, and ask how many small cubes have three, two, one or no painted faces. For an n × n × n cube painted on all sides the counts are 8, 12(n − 2), 6(n − 2)² and (n − 2)³.
This guide is about painted cubes cut into smaller cubes. For perfect cubes such as 27 and 64, see our Number Series guide. Cube questions sit alongside seating arrangements and direction sense as spatial reasoning topics.
What Are Cube Questions?
A typical question reads: “A cube of side 4 cm is painted on all its faces and then cut into 64 cubes of side 1 cm. How many of the small cubes have exactly two painted faces?” Every small cube sits in one of four groups, depending on how many of its faces were on the outside of the big cube: corner cubes, edge cubes, face cubes and inner cubes.
The Formulas for a Fully Painted Cube
For an n × n × n cube painted on all six faces and cut into n³ unit cubes (with n at least 2):
| Painted faces | Where they are | Count |
|---|---|---|
| Three | The 8 corners | 8 |
| Two | Along the 12 edges, between the corners | 12(n − 2) |
| One | In the middle of each of the 6 faces | 6(n − 2)² |
| None | The inner cube | (n − 2)³ |
The four counts always add up to n³. Each edge has n − 2 cubes between its two corners, each face has an (n − 2) × (n − 2) block in its middle, and the hidden core is an (n − 2) × (n − 2) × (n − 2) cube.
Quick reference:
| n | Three faces | Two faces | One face | None | Total |
|---|---|---|---|---|---|
| 2 | 8 | 0 | 0 | 0 | 8 |
| 3 | 8 | 12 | 6 | 1 | 27 |
| 4 | 8 | 24 | 24 | 8 | 64 |
| 5 | 8 | 36 | 54 | 27 | 125 |
| 6 | 8 | 48 | 96 | 64 | 216 |
| 7 | 8 | 60 | 150 | 125 | 343 |
Cuboids
For an a × b × c cuboid painted on all faces and cut into unit cubes, with every side at least 2:
| Painted faces | Count |
|---|---|
| Three | 8 |
| Two | 4 × [(a − 2) + (b − 2) + (c − 2)] |
| One | 2 × [(a − 2)(b − 2) + (b − 2)(c − 2) + (c − 2)(a − 2)] |
| None | (a − 2)(b − 2)(c − 2) |
When Only Some Faces Are Painted
If only some faces are painted, or different faces have different colours, the all-faces formulas no longer apply. Count by working through the cube layer by layer, or by listing which small cubes touch each painted face and checking where those groups overlap.
A Step-by-Step Method
- Find the side of the big cube, or the dimensions of the cuboid, from the number of small cubes. For example, 216 small cubes means a 6 × 6 × 6 cube.
- Check which faces are painted. If all of them, use the formulas. If not, count layer by layer.
- Apply the right count: corners for three faces, edges for two, face-middles for one, the core for none.
- Check the total. The groups should add up to the number of small cubes.
- For “at least one painted face”, subtract the unpainted inner cubes from the total.
Solved Examples
1. A 4 × 4 × 4 cube is painted on all faces and cut into 64 unit cubes. How many have three, two, one and no painted faces? Answer: 8, 24, 24 and 8. Three faces: 8. Two faces: 12 × (4 − 2) = 24. One face: 6 × (4 − 2)² = 24. None: (4 − 2)³ = 8. The total is 64.
2. A 5 × 5 × 5 cube is painted on all faces and cut into unit cubes. How many have exactly two painted faces, and how many have none? Answer: 36 and 27. Two faces: 12 × (5 − 2) = 36. None: (5 − 2)³ = 27.
3. A cube of side 6 is painted on all faces and cut into 216 unit cubes. How many have at least one painted face, and how many have exactly two? Answer: 152 and 48. The inner unpainted cubes number (6 − 2)³ = 64, so 216 − 64 = 152 have at least one painted face. Exactly two: 12 × (6 − 2) = 48.
4. A 3 × 3 × 3 cube has its top and bottom faces painted blue and its four side faces painted red. It is cut into 27 unit cubes. How many have both colours, and how many have no paint? Answer: 16 and 1. A cube has both colours if it touches a top or bottom face and also a side face. Each of the top and bottom layers has 9 cubes, of which only the centre one has no red, so 8 each, giving 16. The only unpainted cube is the one in the very centre of the big cube. (The remaining cubes are the 2 blue-only centres of the top and bottom, and 8 red-only cubes in the middle layer: 16 + 2 + 8 + 1 = 27.)
5. A 4 × 3 × 2 cuboid is painted on all faces and cut into unit cubes. How many have exactly two painted faces, and how many have exactly one? Answer: 12 and 4. Two faces: 4 × [(4 − 2) + (3 − 2) + (2 − 2)] = 4 × 3 = 12. One face: 2 × [(2)(1) + (1)(0) + (0)(2)] = 2 × 2 = 4. (There are also 8 corner cubes, and none with no paint, making 24 in all.)
6. What is the minimum number of cuts needed to cut a 4 × 4 × 4 cube into 64 unit cubes, if the pieces are not rearranged between cuts? Answer: 9. You need 3 cuts in each of the three directions to make 4 slices, so 3 × 3 = 9 cuts.
Common Mistakes
- Using the all-faces formulas when only some faces are painted. Those formulas assume every face is painted, so count layer by layer instead.
- Forgetting the minus 2. Each edge loses its two corner cubes, and each face loses its outer ring, so what is left to count is n − 2 along each side.
- Putting two-face cubes at the corners. Corners have three painted faces, and two-face cubes are only along the edges, between the corners.
- Applying the formulas to a side shorter than 2. The formulas need every side to be at least 2 units.
- Not checking the total. If your four counts don’t add up to the number of small cubes, one of them is wrong.
Practice Questions
Try each on paper, then check the answer.
1. A 3 × 3 × 3 cube is painted on all faces and cut into 27 unit cubes. How many have exactly one painted face? Answer: 6. 6 × (3 − 2)² = 6, one at the centre of each face.
2. A 5 × 5 × 5 cube is painted on all faces and cut into unit cubes. How many have exactly two painted faces? Answer: 36. 12 × (5 − 2) = 36.
3. A 6 × 6 × 6 cube is painted on all faces and cut into unit cubes. How many have no painted faces? Answer: 64. (6 − 2)³ = 4³ = 64.
4. A 4 × 4 × 4 cube is painted on all faces and cut into unit cubes. How many have at least one painted face? Answer: 56. The unpainted inner cubes number 8, so 64 − 8 = 56.
5. A 7 × 7 × 7 cube is painted on all faces and cut into unit cubes. How many have exactly one painted face, and how many have none? Answer: 150 and 125. One face: 6 × (7 − 2)² = 6 × 25 = 150. None: (7 − 2)³ = 125.
6. A 5 × 4 × 3 cuboid is painted on all faces and cut into unit cubes. How many have exactly one painted face, and how many have none? Answer: 22 and 6. One face: 2 × [(3)(2) + (2)(1) + (1)(3)] = 2 × 11 = 22. None: (3)(2)(1) = 6. (Three faces: 8, and two faces: 4 × [3 + 2 + 1] = 24, which total 60 with the 22 and 6.)
7. A painted cube is cut into 216 equal small cubes. How many of them have exactly two painted faces? Answer: 48. 216 = 6³, so the side is 6, and 12 × (6 − 2) = 48.
8. A 4 × 4 × 4 cube has only two opposite faces painted green, and is cut into 64 unit cubes. How many have exactly one green face, and how many have no green? Answer: 32 and 32. Each painted face has 4 × 4 = 16 unit squares, and no cube touches both opposite faces, so 16 + 16 = 32 have exactly one green face. The other 32 have none.
Frequently Asked Questions
What are cube questions in reasoning?
A large cube is painted, then cut into equal small cubes, and you count how many small cubes have a given number of painted faces. They test whether you can reason about corners, edges, faces and the inner core.
What is the formula for painted cube questions?
For an n × n × n cube painted on all faces: 8 cubes have three painted faces, 12(n − 2) have two, 6(n − 2)² have one, and (n − 2)³ have none.
How do I find the side of the big cube?
Take the cube root of the number of small cubes. For example, 216 small cubes means 6 × 6 × 6, because 6³ = 216.
What if only some faces of the cube are painted?
The all-faces formulas won’t work. Count layer by layer, or list the cubes touching each painted face and check where those groups overlap.
How many cuts are needed to cut a cube into unit cubes?
An n × n × n cube needs n − 1 cuts in each of the three directions, which is 3(n − 1) in total, if the pieces are not rearranged between cuts.
Which exams ask cube questions?
They appear in the reasoning sections of many bank and SSC exams and in some placement tests.