A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?
Answer & explanation
Answer: (a) 12
Small cubes with exactly two painted faces sit along the edges of the big cube, excluding the corners. A cube has 12 edges, each with 3 − 2 = 1 such cube here, giving 12.
- The cube splits into 3 × 3 × 3 = 27 small cubes.
- Corner cubes have 3 painted faces; edge cubes other than corners have exactly 2.
- Each edge has 3 small cubes, 2 of them corners, leaving 1 per edge.
- 12 edges × 1 = 12 cubes with exactly two painted faces.
- Check: 8 (three faces) + 12 (two) + 6 (one) + 1 (none) = 27.
Remember · Painted n-cube: 8 corners with three faces, 12(n − 2) edge cubes with two, 6(n − 2)² with one.
Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·