QUESTION

CSAT

Medium

Maths

Prelims 2025

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 $6$ .

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Explanation

We are told that the cube is divided into 60 smaller equal pieces.

Step 1: Determine the dimensions of division To get 60 smaller cubes, the large cube must be divided along its three edges in such a way that: $n₁ \times n₂ \times n₃ = 60$

For the minimum number of cuts, the divisions should be as evenly distributed as possible. Thus, take: $60 = 5 \times 4 \times 3$

Q25

Step 2: Calculate the number of cuts To make 5 pieces along one edge $\rightarrow$ 4 cuts To make 4 pieces along one edge $\rightarrow$ 3 cuts To make 3 pieces along one edge $\rightarrow$ 2 cuts

Total minimum cuts = $4 + 3 + 2 =$ 9

Statement I is correct.

Step 3: Find the number of unpainted (interior) pieces Only the cubes completely inside the larger cube will have no painted faces. So, interior cubes = $(5 - 2) \times (4 - 2) \times (3 - 2)$ $= 3 \times 2 \times 1$ $=$ 6