A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is:

The correct option is 24.

Let us break down the problem step-by-step to understand how we arrive at this answer:

Step 1: Understand the dimensions of the cubes
We start with a large cube. Let the side length of this big cube be represented by S.
It is cut into smaller cubes of equal sizes such that the side length of each small cube, let's call it s, is one-fourth of the big cube's side length.
Therefore, we have:
$$s=\frac{1}{4}S$$
This means that along each edge of the big cube, there are exactly 4 small cubes. The total number of divisions along each dimension is n = 4.

Step 2: Identify where cubes with "only one side painted" are located
When a large painted cube is cut into smaller pieces:
1. The corner cubes have 3 faces painted.
2. The cubes along the edges (excluding the corners) have 2 faces painted.
3. The cubes in the center of each face (excluding the edges and corners) have exactly 1 face painted.
4. The inner cubes (completely inside the big cube) have 0 faces painted.

We are asked to find the number of small cubes with only one side (face) painted. These are located on the surfaces/faces of the big cube, but not along any of the outer edges of those faces.

Step 3: Calculate the number of single-painted cubes per face
Each face of the big cube is a square grid of 4 × 4 small cubes (since n = 4).
To find the center cubes that do not touch any edge, we subtract the outer rows and columns. This leaves a central square grid of size (n - 2) × (n - 2).
Substituting n = 4:
$$(4-2)\times (4-2)=2\times 2=4$$
So, there are 4 small cubes with only one side painted on each face of the large cube.

Step 4: Calculate the total number of single-painted cubes
A cube has exactly 6 faces.
Since each face contains 4 such small cubes, the total number of small cubes with only one side painted is:
$$\text{Total cubes}=6\times 4=24$$
Thus, the number of small cubes with only one of their sides painted is 24.