Logical Reasoning Questions for CLAT | QB Set 103

A science museum organised a puzzle exhibition where visitors were introduced to the concept of painted cube problems. A large wooden cube was painted on all six outer faces and then cut into several smaller cubes of equal size. The organisers explained that after cutting, the smaller cubes could be classified based on the number of painted faces they possessed. Cubes at the corners would have three painted faces because they touched three outer surfaces of the original cube. Cubes along the edges, excluding the corners, would have two painted faces. Cubes located at the centre of each face, but not on the edges, would have only one painted face. The cubes entirely inside the large cube would have no painted faces at all. Visitors learned that the number of cubes in each category depends on the number of equal divisions made along each edge of the original cube. Understanding these patterns helps solve many competitive examination questions quickly without physically drawing the cube, making logical reasoning both interesting and systematic.

Questions

Q1. A cube is painted on all six faces and then divided into 5 equal parts along each edge. How many small cubes will have exactly two painted faces?

A. 24
B. 30
C. 36
D. 48

Q2. A cube is painted on all six faces and cut into 4 equal parts along each edge. How many small cubes will have exactly one painted face?

A. 18
B. 24
C. 12
D. 30

Q3. A cube is painted on all six faces and divided into 6 equal parts along each edge. How many small cubes will have no painted face?

A. 64
B. 56
C. 96
D. 72

Q4. A cube is painted on all six faces and cut into 7 equal parts along each edge. How many small cubes will have exactly three painted faces?

A. 6
B. 12
C. 14
D. 8

Q5. A cube is painted on all six faces and divided into 8 equal parts along each edge. How many small cubes will have exactly one painted face?

A. 180
B. 216
C. 196
D. 240

Answers with Detailed Explanations

Q1. Correct Answer: C. 36

For a cube divided into n equal parts:

• Cubes with two painted faces = 12 × (n − 2)

Here, n = 5

= 12 × (5 − 2)

= 12 × 3

= 36

Hence, Option C is correct.

Q2. Correct Answer: B. 24

Cubes with one painted face are found at the centre of each face.

Formula:

6 × (n − 2)²

Here,

n = 4

= 6 × (4 − 2)²

= 6 × 2²

= 6 × 4

= 24

Therefore, Option B is correct.

Q3. Correct Answer: A. 64

Interior cubes have no painted face.

Formula:

(n − 2)³

Here,

n = 6

= (6 − 2)³

= 4³

= 64

Therefore, Option A is correct.

Q4. Correct Answer: D. 8

Only the corner cubes have three painted faces.

Every cube has 8 corners, irrespective of its size.

Therefore, the number of cubes with three painted faces is always 8.

Hence, Option D is correct.

Q5. Correct Answer: B. 216

Formula for cubes with exactly one painted face:

6 × (n − 2)²

Here,

n = 8

= 6 × (8 − 2)²

= 6 × 6²

= 6 × 36

= 216

Therefore, Option B is the correct answer.