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 4 cm cube is cut into 4 × 4 × 4 = 64 cubes of 1 cm.
• Cubes with no painted face lie in the interior, after removing the outer layer on every side.
• The interior block is (4 − 2) × (4 − 2) × (4 − 2) = 2 × 2 × 2 = 8 cubes, option (a).
Why the other options are wrong
- (b) 16
- 16 is not the size of the inner 2 × 2 × 2 block, which has 8 cubes.
- (c) 24
- 24 is the number of cubes with exactly one painted face (6 × 2 × 2), and also those with exactly two (12 × 2).
- (d) 36
- 36 does not match any group of cubes in a painted 4 × 4 × 4 cube.
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.