4 cm × 4 cm × 4 cm के एक घन के बाह्य पृष्ठ को पूरी तरह लाल रंग में रंगा गया है। तत्पश्चात् इसे फलकों के समान्तर 1 cm × 1 cm × 1 cm के चौंसठ छोटे घनों में काटा गया है। कितने छोटे घनों की फलकें रंगी हुई नहीं होंगी?
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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?
(a) 8 · (b) 16 · (c) 24 · (d) 36
उत्तर और व्याख्या · explanation in English
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). ·