A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted ?
- (a) 4
- (b) 6
- (c) 8 ✓ UPSC's answer
- (d) 10
Why the answer is (c)
• The volumes fix the cut: 4 big cubes and 32 small cubes can only come from a cube of side 4 units, cut into 4 big cubes of side 2 (4 × 8 = 32 unit volumes) and 32 small cubes of side 1 (32 unit volumes), 64 in all.
• The big cubes have no blue face, so they cannot touch the two opposite blue faces. Taking blue as top and bottom, the four 2 × 2 × 2 cubes fill the middle two layers completely, and the 32 small cubes form one 4 × 4 layer on top and one at the bottom.
• In each 4 × 4 layer only the 4 central cubes have just one painted face (the blue one); edge cubes have two painted faces and corner cubes three.
• Each big cube sits at a corner of the middle block and shows two painted side faces (yellow and black), so none of them has exactly one painted face.
• Cubes with only one face painted = 4 + 4 = 8, option (c).
Why the other options are wrong
- (a) 4
- 4 counts only one of the two layers of small cubes; each layer has 4 such cubes.
- (b) 6
- 6 does not match: each of the two small-cube layers gives exactly 4 such cubes.
- (d) 10
- 10 overcounts; the big cubes each have two painted faces, not one.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.