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)
• A cubical block made of 125 identical cubes has a side length of 5 cubes, since $5^3 = 125$.
• Cubes that are surrounded by other cubes from each side are the internal cubes, which do not touch any outer face of the large block.
• To find the number of internal cubes, subtract 2 from the side length to exclude the outer layers on both ends: $5 - 2 = 3$.
• The number of such internal cubes is the volume of this smaller inner cube: $3 \times 3 \times 3 = 27$.
• Therefore, 27 cubes are completely surrounded by other cubes.
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.