A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces?
- (a) 12 ✓ UPSC's answer
- (b) 8
- (c) 6
- (d) 4
Why the answer is (a)
['- A 3 cm cube cut into 1 cm cubes results in a 3x3x3 arrangement, totaling 27 small cubes.', '- Small cubes with exactly two painted faces are located along the edges of the large cube, excluding the corners.', '- A cube has 12 edges in total.', '- On each edge of length 3 cm, there are 3 small cubes: the two at the ends are corners (3 painted faces) and the one in the middle has exactly 2 painted faces.', '- Therefore, there is 1 such cube per edge, and with 12 edges, the total count is 12 x 1 = 12.', '- Hence, 12 small cubes have exactly two painted faces.']
Why the other options are wrong
- (b) 8
- 8 is the number of corner cubes, which have three painted faces, not two.
- (c) 6
- 6 is the number of faces on the cube, not the count of edge-center cubes.
- (d) 4
- 4 is the number of cubes on one face of the 3x3 grid excluding the center, which is irrelevant to the edge calculation.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.