A cube of side 6 cm is painted on all its 6 faces with red colour. It is then broken up into 216 smaller identical cubes . What is the ratio of `N_0 : N_1 : N_2` . Where `N_0 to` number of smaler cube with no colourd surface .
`N_1 to ` number of smaller cubes with 1 red face .
`N_2 to ` number of smaller cubes with 2 red faces :

Options

A. A)`3:4:6`
B. B)`3:4:5`
C. C)`4:6:3`
D. D)can't be determined

Answer

To solve the problem, we need to find the values of \(N_0\), \(N_1\), and \(N_2\) for the smaller cubes after the larger cube has been painted and divided. Let's break it down step by step.

### Step 1: Determine the side length of the smaller cubes
The larger cube has a side length of 6 cm and is divided into 216 smaller cubes.

To find the side length of each smaller cube, we use the formula:
\[ \text{Side of smaller cube} = \sqrt[3]{\frac{\text{Volume of larger cube}}{\text{Number of smaller cubes}}} \]
The volume of the larger cube is:
\[ \text{Volume} = \text{side}^3 = 6^3 = 216 \text{ cm}^3 \]
Thus, the side of the smaller cube is:
\[ a = \sqrt[3]{\frac{216}{216}} = \sqrt[3]{1} = 1 \text{ cm} \]

### Step 2: Calculate \(N_0\) (number of smaller cubes with no colored surface)
The smaller cubes with no colored surface are those that are completely inside the larger cube. To find this, we consider the inner cube that remains after removing the outer layer of cubes.

The side length of the inner cube is:
\[ \text{Side of inner cube} = \text{Side of larger cube} - 2 \times \text{Side of smaller cube} = 6 - 2 \times 1 = 4 \text{ cm} \]
The volume (and therefore the number of smaller cubes) of the inner cube is:
\[ N_0 = \text{Side of inner cube}^3 = 4^3 = 64 \]

### Step 3: Calculate \(N_1\) (number of smaller cubes with 1 red face)
The smaller cubes with one red face are located in the center of each face of the larger cube. Each face of the larger cube has a square of smaller cubes that are not on the edges.

The side length of the face is 6 cm, and the inner face (excluding the edges) has a side length of:
\[ \text{Side of face} - 2 \times \text{Side of smaller cube} = 6 - 2 \times 1 = 4 \text{ cm} \]
The number of smaller cubes on each face with one red face is:
\[ \text{Cubes per face} = \text{Inner face side}^2 = 4^2 = 16 \]
Since there are 6 faces, the total number of cubes with one red face is:
\[ N_1 = 6 \times 16 = 96 \]

### Step 4: Calculate \(N_2\) (number of smaller cubes with 2 red faces)
The smaller cubes with two red faces are located on the edges of the larger cube, excluding the corners.

Each edge of the larger cube has:
\[ \text{Cubes on edge} = \text{Total cubes on edge} - 2 \text{ (for corners)} = 6 - 2 = 4 \]
Since there are 12 edges in a cube, the total number of cubes with two red faces is:
\[ N_2 = 12 \times 4 = 48 \]

### Step 5: Find the ratio \(N_0 : N_1 : N_2\)
Now that we have all values:
- \(N_0 = 64\)
- \(N_1 = 96\)
- \(N_2 = 48\)

We can express the ratio:
\[ N_0 : N_1 : N_2 = 64 : 96 : 48 \]

### Step 6: Simplify the ratio
To simplify, we can divide each term by 16:
\[ \frac{64}{16} : \frac{96}{16} : \frac{48}{16} = 4 : 6 : 3 \]

Thus, the final ratio is:
\[ \boxed{4 : 6 : 3} \]