[THE PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL]
The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the right side and $2$ on the left side. The top level of the tower model has no doorways. The front of the tower model is signified by a disk symbol on the top level.
We paint the tower model with three colors: Blue, Yellow and Brown by fulfilling the following requirements:
1. The top level is painted with only one color.
2. In the second level, the $3$ doorways at the front are painted with the same color which is different from the one used for the center doorway at the back. Besides, any two adjacent doorways, including the pairs at the same corners, are painted with different colors.
3. For the first level, we apply the same rules as for the second level.
[img]https://cdn.artofproblemsolving.com/attachments/2/3/18ee062b79693c4ccc26bf922a7f54e9f352ee.png[/img]
(a) In how many ways the first level can be painted?
(b) In how many ways the whole tower model can be painted?
### Part (a): In how many ways can the first level be painted?

1. **Choosing the color for the three doorways at the front:**
   - There are 3 colors available: Blue, Yellow, and Brown.
   - Therefore, there are \(3\) ways to choose the color for the three doorways at the front.

2. **Choosing the color for the center doorway at the back:**
   - The color of the center doorway at the back must be different from the color of the three doorways at the front.
   - Therefore, there are \(3 - 1 = 2\) ways to choose the color for the center doorway at the back.

3. **Coloring the remaining doorways:**
   - The remaining doorways must be colored such that any two adjacent doorways have different colors.
   - We need to consider the doorways on the left and right sides separately.

4. **Coloring the doorways on the left side:**
   - There are 2 doorways on the left side.
   - The first doorway on the left side must be a different color from the adjacent front doorway.
   - The second doorway on the left side must be a different color from both the first doorway on the left side and the center doorway at the back.
   - There are \(2\) choices for the first doorway on the left side and \(2\) choices for the second doorway on the left side.
   - Therefore, there are \(2 \times 2 = 4\) ways to color the doorways on the left side.

5. **Coloring the doorways on the right side:**
   - Similarly, there are \(2 \times 2 = 4\) ways to color the doorways on the right side.

6. **Combining all choices:**
   - The total number of ways to paint the first level is:
     \[
     3 \text{ (front)} \times 2 \text{ (back)} \times 4 \text{ (left)} \times 4 \text{ (right)} = 3 \times 2 \times 4 \times 4 = 96
     \]

### Part (b): In how many ways can the whole tower model be painted?

1. **Painting the top level:**
   - The top level is painted with only one color.
   - There are \(3\) ways to choose the color for the top level.

2. **Painting the second level:**
   - The second level follows the same rules as the first level.
   - From part (a), we know there are \(96\) ways to paint the second level.

3. **Painting the first level:**
   - The first level follows the same rules as the second level.
   - Therefore, there are \(96\) ways to paint the first level.

4. **Combining all choices:**
   - The total number of ways to paint the entire Tháp Rùa tower is:
     \[
     3 \text{ (top level)} \times 96 \text{ (second level)} \times 96 \text{ (first level)} = 3 \times 96 \times 96 = 27648
     \]

The final answer is \(\boxed{27648}\)