A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct? I. The minimum number of cuts is 9. II. The number of smaller pieces which are not painted on any face is 6. Select the correct answer using the code below:

1. A. I only
2. B. II only
3. C. Both I and II
4. D. Neither I nor II

Answer: C

Explanation

To cut a cube into N smaller equal pieces, if N = x × y × z, the minimum number of cuts required is (x-1) + (y-1) + (z-1). To minimize the number of cuts, the values of x, y, and z should be as close to each other as possible.

Given total pieces = 60. We need to find three factors (x, y, z) of 60 that are as close as possible.
Prime factorization of 60 = 2 × 2 × 3 × 5.
The closest factors are 5, 4, and 3 (5 × 4 × 3 = 60).

Statement I: The minimum number of cuts is 9.
Number of cuts = (x-1) + (y-1) + (z-1) = (5-1) + (4-1) + (3-1) = 4 + 3 + 2 = 9.
So, Statement I is correct.

Statement II: The number of smaller pieces which are not painted on any face is 6.
Pieces with no painted faces are the interior pieces. For a cuboid of dimensions x, y, z, the number of unpainted pieces is given by the formula (x-2)(y-2)(z-2).
Using x=5, y=4, z=3:
Number of unpainted pieces = (5-2)(4-2)(3-2) = 3 × 2 × 1 = 6.
So, Statement II is also correct.

Since both statements I and II are correct, the answer is (c). This question tests spatial reasoning and problem-solving skills related to cubes and cuboids, a common topic in the visual reasoning section of UPSC CSAT.