125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side?
Answer & explanation
Answer: (a) 27
125 cubes make a 5 × 5 × 5 block. A cube surrounded on every side cannot lie on any outer face, so remove one layer from each side: the hidden core is 3 × 3 × 3 = 27 cubes.
- 125 = 5 × 5 × 5, so the block is 5 cubes long, wide and high.
- Every cube in the outer layer has at least one face exposed.
- Removing one layer from each side leaves (5 − 2) × (5 − 2) × (5 − 2) = 3 × 3 × 3 = 27 cubes, each touched by cubes on all sides.
Remember · Fully hidden cubes in an n × n × n block = (n − 2)³.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·