125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side ?
- (a) 27 ✓ UPSC's answer
- (b) 25
- (c) 21
- (d) 18
Why the answer is (a)
• 125 = 5 × 5 × 5, so the block is 5 cubes along each edge.
• A cube is surrounded on every side only if it does not lie on any outer face.
• Removing the outer layer on each side leaves an inner block of (5 − 2) × (5 − 2) × (5 − 2) = 3 × 3 × 3.
• That inner block has 27 cubes, option (a).
Why the other options are wrong
- (b) 25
- 25 is the number of cubes in a single 5x5 face layer, not the count of fully internal cubes.
- (c) 21
- 21 is an incorrect calculation that does not correspond to the volume of the inner 3x3x3 core.
- (d) 18
- 18 is an incorrect calculation that does not correspond to the volume of the inner 3x3x3 core.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.