SSC Cube and Cuboid Questions (Painting and Cutting) with Answers

Cube and cuboid questions ask how many small cubes have 3, 2, 1 or no painted faces after a painted cube is cut into smaller cubes. They appear in SSC CGL, CHSL, MTS and GD reasoning papers. With four simple formulas you can answer each question in about 15 seconds. This practice paper gives you 21 questions with answers and short solutions.

The Basic Idea of a Painted and Cut Cube

Imagine a big cube painted on all six faces and then cut into many small identical cubes. The small cubes are of four kinds depending on their position.

• Corner cubes have 3 painted faces.
• Edge cubes have 2 painted faces.
• Face cubes (inside a face) have 1 painted face.
• Inner cubes have no painted face.

Formulas and Shortcut Tricks

• For a cube cut into n x n x n small cubes: total = n^3.
• 3 faces painted = 8 (always). 2 faces painted = 12(n - 2). 1 face painted = 6(n - 2)^2. No face painted = (n - 2)^3.
• For a cuboid of l x b x h: 3 faces = 8, no face = (l - 2)(b - 2)(h - 2), 2 faces = 4[(l - 2) + (b - 2) + (h - 2)] and 1 face = 2[(l - 2)(b - 2) + (b - 2)(h - 2) + (l - 2)(h - 2)].
• n = side of the big cube divided by the side of the small cube.
• Cuts needed to make n pieces along one direction = n - 1.
• At least one face painted = total - unpainted cubes.

Set 1: Cube Cut into 27 and 64 Small Cubes (Q1–Q7)

Q1. A painted cube is cut into 27 equal small cubes. How many have no painted face?
Answer: 1
Solution: (3 - 2)^3 = 1.

Q2. In the same case, how many small cubes have 3 painted faces?
Answer: 8
Solution: The 8 corner cubes.

Q3. In the same case, how many small cubes have 2 painted faces?
Answer: 12
Solution: 12 x (3 - 2) = 12.

Q4. In the same case, how many small cubes have 1 painted face?
Answer: 6
Solution: 6 x (3 - 2)^2 = 6.

Q5. A painted cube is cut into 64 small cubes. How many have exactly 1 painted face?
Answer: 24
Trick: 6 x (4 - 2)^2 = 24.

Q6. In the same case, how many have exactly 2 painted faces?
Answer: 24
Solution: 12 x (4 - 2) = 24.

Q7. In the same case, how many have no painted face?
Answer: 8
Solution: (4 - 2)^3 = 8.

Set 2: Larger Cubes and Cuboids (Q8–Q14)

Q8. A painted cube is cut into 125 small cubes. How many have no painted face?
Answer: 27
Solution: n = 5, so (5 - 2)^3 = 27.

Q9. A cube of side 10 cm is painted and cut into 1 cm cubes. How many have exactly 2 painted faces?
Answer: 96
Solution: 12 x 8 = 96.

Q10. A cuboid 4 x 3 x 2 is cut into unit cubes. How many cubes are there in total?
Answer: 24
Solution: 4 x 3 x 2 = 24.

Q11. How many cuts are needed to divide a cube into 27 equal small cubes?
Answer: 6 cuts
Solution: 2 cuts in each of 3 directions.

Q12. A painted cuboid 6 x 4 x 3 is cut into unit cubes. How many have exactly 1 painted face?
Answer: 28
Trick: 2[(4)(2) + (2)(1) + (4)(1)] = 2 x 14 = 28.

Q13. In the same cuboid, how many cubes have exactly 3 painted faces?
Answer: 8
Solution: The 8 corners.

Q14. In the same cuboid, how many cubes have exactly 2 painted faces?
Answer: 28
Solution: 4[(4) + (2) + (1)] = 28.

Set 3: Exam-Level Questions (Q15–Q21)

Q15. In the same 6 x 4 x 3 cuboid, how many unit cubes have no painted face?
Answer: 8
Solution: (4)(2)(1) = 8.

Q16. A cube of side 6 cm is painted red and cut into cubes of side 2 cm. How many have exactly one face painted?
Answer: 6
Solution: n = 3, so 6 x (3 - 2)^2 = 6.

Q17. A cube of side 12 cm is cut into cubes of side 3 cm. How many small cubes are formed?
Answer: 64
Solution: (12 / 3)^3 = 64.

Q18. How many small cubes have at least one painted face when a painted cube is cut into 27 equal cubes?
Answer: 26
Solution: 27 - 1 = 26.

Q19. How many small cubes have at least one painted face when a painted cube is cut into 216 equal cubes?
Answer: 152
Solution: n = 6, unpainted = 4^3 = 64 and 216 - 64 = 152.

Q20. A cube of side 8 cm is painted and cut into 1 cm cubes. How many have exactly 2 painted faces?
Answer: 72
Solution: 12 x 6 = 72.

Q21. A cube of side 4 cm is cut into 64 cubes of side 1 cm. Find the ratio of the total surface area of all small cubes to that of the big cube.
Answer: 4 : 1
Trick: Small cubes: 64 x 6 = 384 sq cm. Big cube: 6 x 16 = 96 sq cm. Ratio = 4 : 1.

Common Mistakes in Cube and Cuboid Questions

• Using n as the side of the small cube instead of the number of cubes along one edge.
• Forgetting to subtract 2 from each dimension in cuboid formulas.
• Mixing up the 2-face and 1-face formulas.
• Counting the corner cubes as 6 instead of 8.
• Ignoring that a cube is not painted on a face cut as a new surface.
• Not checking that the total of all four types equals n^3.

How to Practice These 21 Questions

1. Memorise the four formulas for a cube.
2. Practise the cuboid formulas separately.
3. Check each answer by adding all the types.
4. Solve each set in 7 minutes.

Related Topics to Study Next

• SSC Reasoning Practice Set
• Mensuration Questions
• Venn Diagram Questions
• Odd One Out Questions

Frequently Asked Questions

How many small cubes have 3 painted faces?

Always 8, which are the corner cubes of the big cube or cuboid.

What is the formula for cubes with 2 painted faces?

12(n - 2) for a cube, where n is the number of small cubes along one edge.

What is the formula for cubes with 1 painted face?

6(n - 2)^2 for a cube.

What is the formula for cubes with no painted face?

(n - 2)^3 for a cube.

How do I find n?

Divide the side of the big cube by the side of the small cube.

How do I solve cuboid questions?

Subtract 2 from each dimension and apply the cuboid formulas.

How many cuts are needed to divide a cube into n^3 cubes?

3(n - 1) cuts, which means n - 1 cuts in each direction.

What are the cubes with at least one face painted?

Total cubes minus the unpainted inner cubes.

How many cube cutting questions come in SSC exams?

Usually one question per shift in the reasoning section, sometimes in maths.

Does the number of painted faces depend on the colour?

No. Only the position of the small cube matters, not the colour used.

How can I check my answer quickly?

Add the four counts (3-face, 2-face, 1-face and 0-face) and see if the total equals n^3.

How should I prepare cube questions for SSC exams?

Learn the formulas, solve 20 questions daily and revise with previous year papers.