Question

A cube is painted Blue on all faces and is then cut into 125 cubes of equal size. How many cubes are not painted on any face?

• A. 8
• B. 16
• C. 27
• D. 36

Hint:

When dividing a cube into smaller cubes, the interior cubes will be those that are not on any face.

Answer 1

The Correct Option is C

To determine how many cubes are not painted on any face after the larger cube is divided into 125 smaller cubes, we need to follow these steps:

1. First, find the dimensions of the large cube. Since it is cut into 125 smaller cubes of equal size, we determine the side of the small cubes and calculate the larger cube's side.
2. The larger cube is divided into smaller cubes such that the total number of small cubes is \(125 = 5^3\). This means the larger cube was a 5x5x5 cube, because \(5 \times 5 \times 5 = 125\).
3. We are asked to find how many of these smaller cubes are not painted on any face. These are the cubes that are completely inside the cube, not on any surface.
4. To find such cubes, we need to exclude the outer layers of the cube which are painted. The non-painted (inner) cubes form a smaller cube inside.
5. The innermost layer is a 3x3x3 cube. The reason being that removing one cube layer from each side leaves \(5 - 2 = 3\) cubes on each side.
6. Therefore, the number of small cubes not painted on any face is \(3 \times 3 \times 3 = 27\).

Conclusion: 27 smaller cubes are not painted on any face of the original large cube.