A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is :
- (a) 32
- (b) 24 ✓ UPSC's answer
- (c) 16
- (d) 8
Why the answer is (b)
• A side one-fourth of the big cube's side means the big cube is cut into 4 × 4 × 4 = 64 small cubes.
• Cubes with exactly one painted face lie in the interior of each face, away from edges: (4 – 2) × (4 – 2) = 4 per face.
• Six faces give 6 × 4 = 24 such cubes.
• Hence 24, option (b).
Why the other options are wrong
- (a) 32
- 32 is too many; each face has only 4 such cubes, 24 in all.
- (c) 16
- 16 undercounts; six faces have 4 such cubes each.
- (d) 8
- 8 is the number of corner cubes, which have three painted faces.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.