By signing in you confirm you are 18 or older, or have your parent’s or guardian’s consent.

UPSC Prelims 2025 CSAT, question 15

Topic: Arrangements and puzzles

Q15

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 :

  1. (a) I only
  2. (b) II only
  3. (c) Both I and IICorrect answer
  4. (d) Neither I nor II

Solution.

Answer: (c) Both I and II

  1. To divide a cube into 6060 equal smaller cuboids with the minimum number of cuts, factor 60=3×4×560 = 3 \times 4 \times 5 as closely as possible.
  2. Statement I is correct: making 3−1=23 - 1 = 2, 4−1=34 - 1 = 3, and 5−1=45 - 1 = 4 cuts along the three mutually perpendicular axes requires 2+3+4=92 + 3 + 4 = 9 cuts.
  3. Statement II is correct: the number of inner pieces with no painted face is (3−2)(4−2)(5−2)=1×2×3=6(3 - 2)(4 - 2)(5 - 2) = 1 \times 2 \times 3 = 6.