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 ✓ UPSC's answer
- (b) 16
- (c) 24
- (d) 36
Why the answer is (a)
• The original cube has dimensions 4 cm × 4 cm × 4 cm, which is divided into 1 cm × 1 cm × 1 cm cubes, resulting in a 4 × 4 × 4 grid of 64 small cubes.
• Small cubes with no painted faces are located strictly in the interior of the large cube, meaning they are not on any of the six outer surfaces.
• To find the number of interior cubes, subtract 2 from each dimension of the large cube to remove the outer layer of cubes on both sides: (4 - 2) × (4 - 2) × (4 - 2).
• This calculation yields 2 × 2 × 2 = 8 small cubes.
• Therefore, exactly 8 small cubes have no painted faces, which corresponds to option (a).
Why the other options are wrong
- (b) 16
- Option (b) is incorrect because 16 is the total number of cubes in a single 4x4 face layer, not the count of interior cubes.
- (c) 24
- Option (c) is incorrect because 24 represents the number of edge cubes (excluding corners) in a 4x4x4 cube, which have two painted faces.
- (d) 36
- Option (d) is incorrect because 36 is the number of face-center cubes (excluding edges and corners) in a 4x4x4 cube, which have one painted face.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.