A cube of edge 6 cm is painted on all sides and then cut into units cubes. The number of unit cubes with no sides painted is

Options

A. A)0
B. B)64
C. C)186
D. D)108

Answer

To solve the problem, we need to find the number of unit cubes that have no sides painted after a cube of edge 6 cm is painted on all sides and then cut into unit cubes (1 cm x 1 cm x 1 cm).

### Step-by-step Solution:

1. **Understand the dimensions of the cube**:
The cube has an edge length of 6 cm.

2. **Calculate the total number of unit cubes**:
When the cube is cut into unit cubes, each unit cube has an edge length of 1 cm. The total number of unit cubes can be calculated using the formula for the volume of the cube:
\[ \text{Total unit cubes} = \text{edge length}^3 = 6^3 = 216 \]

3. **Determine the dimensions of the inner cube**:
The unit cubes that have no sides painted are those that are not on the surface of the larger cube. To find the edge length of the inner cube (the part that remains unpainted), we subtract 2 cm from each dimension (1 cm for each side that is painted):
\[ \text{Edge length of inner cube} = 6 - 2 = 4 \text{ cm} \]

4. **Calculate the number of unit cubes in the inner cube**:
Now we can calculate the number of unit cubes in the inner cube:
\[ \text{Number of unit cubes with no sides painted} = \text{edge length of inner cube}^3 = 4^3 = 64 \]

5. **Conclusion**:
The number of unit cubes with no sides painted is 64.

### Final Answer:
The number of unit cubes with no sides painted is **64**.

---