If the mode of the observations 4, 5, 6, 6, 5, 4, 4, 5, x, 6, 6, 5, 4, 6, 5, `gamma` is 6 and `x ne gamma,` then find the values of x and`gamma.`

Answer

To solve the problem, we need to find the values of \( x \) and \( \gamma \) given that the mode of the observations is 6 and \( x \neq \gamma \).

### Step-by-Step Solution:

1. **List the Observations**:
The observations given are:
\[ 4, 5, 6, 6, 5, 4, 4, 5, x, 6, 6, 5, 4, 6, 5, \gamma \]

2. **Determine the Frequency of Each Number**:
We will count how many times each number appears in the list, including \( x \) and \( \gamma \).
- Frequency of 4: 4 times (4, 4, 4, 4)
- Frequency of 5: 5 times (5, 5, 5, 5, 5)
- Frequency of 6: 5 times (6, 6, 6, 6, 6)

So far, we have:
- 4 appears 4 times
- 5 appears 5 times
- 6 appears 5 times

3. **Understanding the Mode**:
The mode is the number that appears most frequently. We know that the mode is 6, which means it must appear more times than any other number. Currently, 4 appears 4 times, and both 5 and 6 appear 5 times.

4. **Adding \( x \) and \( \gamma \)**:
To make 6 the mode, we need to increase its frequency. Since \( x \) and \( \gamma \) are variables, we can assign values to them.

5. **Setting \( x = 6 \)**:
If we set \( x = 6 \), the frequency of 6 becomes:
\[ 5 + 1 = 6 \text{ times} \]
Now, we have:
- 4 appears 4 times
- 5 appears 5 times
- 6 appears 6 times (which is now the highest frequency)

6. **Determining \( \gamma \)**:
Since \( x \) is now 6, \( \gamma \) cannot be 6 (as given \( x \neq \gamma \)). The only remaining option is to set \( \gamma = 4 \) or \( \gamma = 5 \).

- If \( \gamma = 5 \), then the frequency of 5 becomes:
\[ 5 + 1 = 6 \text{ times} \]
This would make the frequency of 5 equal to that of 6, which contradicts our mode condition.

- If \( \gamma = 4 \), then the frequency of 4 becomes:
\[ 4 + 1 = 5 \text{ times} \]
This keeps the frequency of 4 at 5, while 6 remains the mode with 6 occurrences.

7. **Final Values**:
Thus, we conclude:
\[ x = 6 \quad \text{and} \quad \gamma = 4 \]

### Final Answer:
\[ x = 6, \quad \gamma = 4 \]