Cube Cutting and Painting: Formulas for Painted Faces, Number of Cuts and Solved Examples for SSC Reasoning

In cube cutting and painting questions, a big cube or cuboid is painted on its outer faces and then cut into many small cubes of equal size. You are asked how many small cubes have one, two, three or no painted faces. Once you know the formulas, the answer takes less than 20 seconds. This guide on cube cutting and painting for SSC gives the formulas for cubes and cuboids, with solved examples, a table of results and practice questions.

Basic Idea

A cube has 8 corners, 12 edges and 6 faces. When it is cut into n × n × n small cubes (n pieces along each side), the position of each small cube decides how many painted faces it has:

• Corner cubes have 3 painted faces.
• Edge cubes (not corners) have 2 painted faces.
• Face cubes (not on edges) have 1 painted face.
• Inner cubes have no painted face.

Formulas for a Cube (n × n × n, All Faces Painted)

Table for Common Values of n

Formulas for a Cuboid (a × b × c, All Faces Painted)

Number of Cuts

To cut a cube into n × n × n small cubes, you need (n − 1) cuts in each of the three directions, so the total number of cuts is 3(n − 1). Each cut, if you do not rearrange the pieces, increases the number of pieces along that direction by one.

If Only Some Faces Are Painted

When only some faces are painted, count with the position method. For example, if only the top and bottom faces of a 4 × 4 × 4 cube are painted, cubes with exactly one painted face = 2 × (4 × 4) = 32 (all cubes on the top and bottom layers), and the cubes with no paint = 64 − 32 = 32.

Solved Examples

Example 1. A cube of side 4 cm is painted on all faces and cut into 1 cm cubes. How many cubes have exactly two faces painted?
12 × (4 − 2) = 24.

Example 2. The same cube: how many have no paint?
(4 − 2)³ = 8.

Example 3. A cube of side 6 cm is painted and cut into 1 cm cubes. How many have one face painted?
6 × (6 − 2)² = 6 × 16 = 96.

Example 4. A 5 × 4 × 3 cuboid is painted and cut into unit cubes. Find the cubes with 2 painted faces, 1 painted face and no paint.
2 faces: 4 (3 + 2 + 1) = 24. 1 face: 2 (3 × 2 + 2 × 1 + 1 × 3) = 2 (6 + 2 + 3) = 22. No paint: 3 × 2 × 1 = 6. Check: 8 + 24 + 22 + 6 = 60 = 5 × 4 × 3.

Example 5. How many cuts are needed to cut a cube into 64 equal small cubes?
n = 4, so 3 (4 − 1) = 9 cuts.

Tricks

• Remember that corners are always 8 in any cube or cuboid.
• Use (n − 2) in all formulas, since the outer layer is removed on both sides.
• Check the total: the four categories must add up to n³.
• If the question asks for cubes with at least one painted face, subtract the unpainted cubes from the total.
• Read carefully which faces are painted: all faces or only some.

Quick Tips

• Memorise the four formulas: 8, 12(n−2), 6(n−2)², (n−2)³.
• Verify that the totals add to n³.
• Use (n−2) for the cuboid formulas.
• Check which faces are painted.

Quick Revision Points

• 3 faces = 8; 2 faces = 12(n−2); 1 face = 6(n−2)²; 0 faces = (n−2)³.
• Cuts needed = 3(n−1) for an n × n × n cube.
• For cuboids, use (a−2), (b−2) and (c−2).
• At least one painted face = n³ − (n−2)³.
• Corners are always 8.

Practice Questions With Answers

1. A 5 cm cube is painted and cut into 1 cm cubes. How many have 3 faces painted?
   Answer: 8
2. For the same cube, how many have 2 faces painted?
   Answer: 36
3. For the same cube, how many have 1 face painted?
   Answer: 54
4. For the same cube, how many have no paint?
   Answer: 27
5. A 3 cm cube is cut into 1 cm cubes after painting. How many have no paint?
   Answer: 1
6. A 6 cm cube is painted and cut into 1 cm cubes. How many have at least one face painted?
   Answer: 152
7. How many cuts are needed to make 125 small cubes from a cube?
   Answer: 12 cuts
8. A 4 × 3 × 2 cuboid is painted and cut into unit cubes. How many have no paint?
   Answer: 0
9. A 5 × 4 × 3 cuboid is painted and cut into unit cubes. How many have 3 faces painted?
   Answer: 8
10. A 4 cm cube has only the top and bottom painted and is cut into 1 cm cubes. How many have exactly one face painted?
    Answer: 32

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Reasoning notes page, read the previous topic Eligibility Test and the next topic Geometrical Seating Arrangement.

Frequently Asked Questions

How many small cubes have 3 painted faces?

8, which are the corner cubes.

How many small cubes have 2 painted faces?

12 (n − 2) for an n × n × n cube.

How many small cubes have 1 painted face?

6 (n − 2)².

How many small cubes have no painted face?

(n − 2)³.

What is the total number of small cubes?

n³ for a cube, and a × b × c for a cuboid.

What is the formula for the number of cuts?

3 (n − 1) for cutting a cube into n × n × n pieces.

How do I find at least one painted face?

Subtract the unpainted cubes from the total: n³ − (n − 2)³.

Do the formulas work for a cuboid?

Yes, with the formulas in terms of (a − 2), (b − 2) and (c − 2).

Why do we use n − 2?

Because the outer layers on both opposite sides are painted and are removed from the middle part.

Is this topic asked in SSC?

Yes. Cube cutting and painting questions are common in SSC CGL and CHSL reasoning.

What if only some faces are painted?

Count by position, layer by layer, instead of using the all-faces formulas.

What are common mistakes?

Using n instead of n − 2 and forgetting to check the total.