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:
Answer & explanation
Answer: (c) Both I and II
The fewest cuts come from dividing the cube into 3 × 4 × 5 slices, which takes 2 + 3 + 4 = 9 cuts. The inner block untouched by paint is (3 − 2) × (4 − 2) × (5 − 2) = 6 pieces, so both statements are correct.
- If the cube is cut into a × b × c pieces, it needs (a − 1) + (b − 1) + (c − 1) cuts. With a × b × c = 60, make a + b + c as small as possible.
- Factors closest to one another give the smallest sum: 3 × 4 × 5 = 60 with sum 12 (2 × 5 × 6 gives 13, 2 × 2 × 15 gives 19). Minimum cuts = 2 + 3 + 4 = 9.
- Unpainted pieces are the ones strictly inside: (3 − 2) × (4 − 2) × (5 − 2) = 1 × 2 × 3 = 6.
- Both statements are correct.
- ✓ I The 3 × 4 × 5 split needs 2 + 3 + 4 = 9 cuts; every other way of making 60 pieces needs more.
- ✓ II Removing the painted outer layer leaves an inner block of 1 × 2 × 3 = 6 pieces with no paint.
Remember · a × b × c pieces need (a + b + c − 3) cuts; unpainted pieces = (a − 2)(b − 2)(c − 2).
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·