All the faces of a cube are painted with blue colour. Then it is cut into 125 small equal cubes. How many small cubes having no coloured face?

Options

A. 27
B. 8
C. 16
D. 24

Explanation

To solve the problem step by step, we need to determine how many small cubes have no colored faces after a larger cube, painted on all its faces, is cut into smaller equal cubes.

### Step 1: Understand the Problem
We have a large cube that is painted on all its faces and then cut into 125 smaller cubes. We need to find out how many of these smaller cubes have no paint on them.

### Step 2: Determine the Size of the Large Cube
Since the large cube is cut into 125 smaller cubes, we can find the dimensions of the large cube. The number of smaller cubes (125) is equal to \( n^3 \), where \( n \) is the number of smaller cubes along one edge of the larger cube.

\[ n^3 = 125 \implies n = \sqrt[3]{125} = 5 \]

Thus, the large cube has dimensions of \( 5 \times 5 \times 5 \).

### Step 3: Identify the Small Cubes with No Colored Faces
The small cubes with no colored faces are those that are completely inside the larger cube, not touching any of the outer faces. To find these, we can visualize that the inner cubes form a smaller cube inside the larger cube.

The inner cube will have dimensions of \( (n - 2) \times (n - 2) \times (n - 2) \) because we remove one layer of cubes from each face of the larger cube.

\[ \text{Inner cube dimension} = (5 - 2) \times (5 - 2) \times (5 - 2) = 3 \times 3 \times 3 \]

### Step 4: Calculate the Number of Inner Cubes
Now, we calculate the number of small cubes in the inner cube:

\[ \text{Number of inner cubes} = 3^3 = 27 \]

### Step 5: Conclusion
Thus, the number of small cubes that have no colored faces is \( 27 \).

### Final Answer:
**27 small cubes have no colored face.**

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