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We are given a list of numbers: \( 5, 3, n, 9, 3, 7, 9, 6, 6 \), and we need to determine which value of \( n \) makes the mean, median, and mode of the set equal.

### Steps to solve:

1. **Mean**:
   The mean is the average of the nine numbers. The formula for the mean is:
   \[
   \text{Mean} = \frac{\text{Sum of all numbers}}{9}
   \]
   So, sum the given numbers:
   \[
   5 + 3 + n + 9 + 3 + 7 + 9 + 6 + 6 = 48 + n
   \]
   Thus, the mean is:
   \[
   \frac{48 + n}{9}
   \]

2. **Mode**:
   The mode is the number that appears most frequently. In the list \( 5, 3, n, 9, 3, 7, 9, 6, 6 \), without knowing \( n \), the current modes are 3, 9, and 6, which each appear twice.

   - If \( n = 6 \), 6 becomes the mode (appearing 3 times).
   - If \( n = 3 \), 3 becomes the mode (appearing 3 times).
   - If \( n = 9 \), 9 becomes the mode (appearing 3 times).
   - If \( n = 8 \), there is no clear mode, as no number appears more than twice.

3. **Median**:
   The median is the middle value when the numbers are ordered. With 9 values, the median will be the 5th number in the sorted list.

   Let's consider different values of \( n \) and find the sorted order of the numbers:
   - **If \( n = 3 \)**: The sorted list is \( 3, 3, 3, 5, 6, 6, 7, 9, 9 \). The median is 6.
   - **If \( n = 6 \)**: The sorted list is \( 3, 3, 5, 6, 6, 6, 7, 9, 9 \). The median is 6.
   - **If \( n = 8 \)**: The sorted list is \( 3, 3, 5, 6, 6, 7, 8, 9, 9 \). The median is 6.
   - **If \( n = 9 \)**: The sorted list is \( 3, 3, 5, 6, 6, 7, 9, 9, 9 \). The median is 6.

4. **Finding the correct value of \( n \)**:
   The mean, median, and mode must all be equal to 6. From the previous steps, when \( n = 6 \):
   - The sum of the numbers becomes \( 48 + 6 = 54 \), so the mean is:
     \[
     \frac{54}{9} = 6
     \]
   - The mode is 6 (since 6 appears 3 times).
   - The median is also 6.

Thus, the value of \( n \) that satisfies all conditions is \( \boxed{6} \).

Would you like more details or further clarification?

Here are 5 related questions to further your understanding:
1. What happens if the value of \( n \) was 9? Would the mean, mode, or median still be equal?
2. How would the mean change if you added another number to the list?
3. Can a set of numbers have two different modes?
4. What is the difference between mean, median, and mode conceptually?
5. Why is the median a better measure of central tendency in some cases than the mean?

**Tip**: When comparing mean, median, and mode, always sort the data first to make analyzing easier!

The mean, median, and mode of the nine numbers above are equal. Which of the following is a possible value of n?

Solve for n in the sequence where the mean, median, and mode of nine numbers are equal. Understand how to compute the mean, determine the median by sorting, and identify the mode. This problem is ideal for students in Grades 7-9.

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