The outer surface of a 4 cm × 4 cm × 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm × 1 cm × 1 cm small cubes. How many small cubes do not have painted faces?
Answer & explanation
Answer: (a) 8
Only the small cubes that touch no outer face stay unpainted. Peeling a 1 cm layer off every side leaves a 2 × 2 × 2 core, so 8 small cubes have no painted face.
- A 4 × 4 × 4 cube gives 64 small cubes.
- Cubes with no painted face form the inner block: (4 − 2) × (4 − 2) × (4 − 2).
- (4 − 2)³ = 2³ = 8.
- Check: 8 corner cubes + 24 edge cubes + 24 face-centre cubes = 56 painted, and 56 + 8 = 64.
Remember · For an n × n × n painted cube, the unpainted small cubes number (n − 2)³.
Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·