A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is :
- (a) 32
- (b) 24 ✓ UPSC's answer
- (c) 16
- (d) 8
Why the answer is (b)
• The big cube is cut into smaller cubes where the side of the small cube is 1/4th of the big cube, meaning the big cube is divided into 4x4x4 smaller cubes.
• Small cubes with only one face painted are located at the center of each face of the big cube, excluding the edges and corners.
• For a cube divided into n x n x n parts, the number of small cubes with exactly one face painted is given by the formula 6 * (n - 2)^2.
• Substituting n = 4 into the formula gives 6 * (4 - 2)^2 = 6 * (2)^2 = 6 * 4.
• The calculation results in 24 small cubes with only one side painted.
• Therefore, the correct option is (b) 24.
Why the other options are wrong
- (a) 32
- Option (a) 32 is incorrect because it does not match the calculation of 6 * (4-2)^2 = 24.
- (c) 16
- Option (c) 16 is incorrect because it likely miscalculates the number of cubes per face or the total faces involved.
- (d) 8
- Option (d) 8 is incorrect because it corresponds to the number of corner cubes (which have three faces painted), not the face-center cubes.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2016, held on 7 August 2016. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.