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 given below :
- (a) I only
- (b) II only
- (c) Both I and II ✓ UPSC's answer
- (d) Neither I nor II
Why the answer is (c)
• To divide a cube into 60 equal smaller cubes, the dimensions must be factors of 60. The minimum number of cuts is achieved by maximizing the number of pieces per cut, which corresponds to the most balanced dimensions. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. The combination 3 x 4 x 5 = 60 is the most balanced.
• For a 3 x 4 x 5 grid, the number of cuts required along each dimension is (n-1). Thus, cuts = (3-1) + (4-1) + (5-1) = 2 + 3 + 4 = 9. This confirms Statement I is correct.
• The number of smaller cubes with no painted faces is given by the formula (l-2)(w-2)(h-2), where l, w, h are the dimensions of the grid.
• Substituting the dimensions: (3-2)(4-2)(5-2) = 1 * 2 * 3 = 6. This confirms Statement II is correct.
• Since both statements are correct, the correct option is (c).
Why the other options are wrong
- (a) I only
- Statement II is also correct, so selecting only I is incomplete.
- (b) II only
- Statement I is also correct, so selecting only II is incomplete.
- (d) Neither I nor II
- Both statements are correct, so neither being correct is false.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.