A solid cube is coloured red on all faces. It is cut into 64 smaller cubes of equal size. How many small cubes have no face coloured ?

Question

Accepted Answer

To determine how many smaller cubes have no face colored, we need to analyze the structure of the original cube and how it is divided.

1. **Understanding the Cube Division:**
   - The original cube is a solid cube colored red on all its faces.
   - It is cut into 64 smaller cubes of equal size. Since 64 is $4^3$, the original cube is divided into a 4x4x4 grid of smaller cubes.

2. **Identifying Cubes with No Face Colored:**
   - The cubes with no face colored are those that are completely inside the larger cube, not exposed to any outer face.
   - In a 4x4x4 cube, the inner cubes that have no face colored are those that are not on the outermost layer.
   - To find these, we consider the inner 2x2x2 section of the cube, as the outermost layers (first and last rows/columns) are exposed to the surface.

3. **Calculating the Number of Inner Cubes:**
   - The inner section is a smaller cube of size 2x2x2.
   - Therefore, the number of smaller cubes with no face colored is $2 \times 2 \times 2 = 8$.

4. **Explanation of Incorrect Options:**
   - **Option 1 (24):** This would imply a much larger number of cubes with no face colored, which is not possible given the structure of a 4x4x4 cube.
   - **Option 2 (16):** This would suggest a 2x2x4 or similar configuration, which does not fit the inner cube structure.
   - **Option 4 (4):** This would imply a 1x1x4 or similar configuration, which is incorrect as it underestimates the number of inner cubes.

Thus, the correct answer is Option 3, which accurately reflects the number of smaller cubes with no face colored in the given scenario.