We are given six boxes and a set of numbers (0-9) to fill them. We must ensure that the following conditions are met:

• Median: 6
• Mean: 5
• Range: 8
• Mode: 6

Let's break down each condition and try to find a solution.

1. Median of 6:

Since we have 6 numbers, the median will be the average of the 3rd and 4th numbers when arranged in order. So, the 3rd and 4th numbers must average to 6. To make this simpler, let's assume that both the 3rd and 4th numbers are 6. This ensures the median is exactly 6.

2. Mean of 5:

The mean is calculated by adding all 6 numbers and dividing by 6. Since the mean must be 5, the sum of the 6 numbers should be: $\text{Sum of numbers} = 5 \times 6 = 30$

3. Range of 8:

The range is the difference between the highest and lowest numbers. For a range of 8, if the smallest number is $x$, the largest number should be $x + 8$. Since we are working with numbers from 0 to 9, the largest possible number is 9. Therefore, the smallest number should be: $\text{Smallest number} = 9 - 8 = 1$ Thus, the smallest number is 1 and the largest number is 9.

4. Mode of 6:

The mode is the number that appears most frequently. For the mode to be 6, the number 6 must appear more frequently than any other number. Since we are using six boxes, it makes sense to use the number 6 at least twice (and possibly more) to satisfy the mode condition.

Putting it all together:

We need to select six numbers that satisfy all the conditions. Here's one possible solution:

• Numbers: $1, 6, 6, 6, 9, 2$

Now, let's verify each condition:

1. Median: The sorted list is $1, 2, 6, 6, 6, 9$. The 3rd and 4th numbers are both 6, so the median is 6.
2. Mean: The sum of the numbers is: $1 + 2 + 6 + 6 + 6 + 9 = 30$ The mean is $30 \div 6 = 5$, so the mean condition is satisfied.
3. Range: The range is $9 - 1 = 8$, which satisfies the range condition.
4. Mode: The number 6 appears three times, more than any other number, so the mode is 6.

Thus, the set $1, 6, 6, 6, 9, 2$ satisfies all the conditions: a median of 6, a mean of 5, a range of 8, and a mode of 6.

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Would you like further clarification on any part of the solution? Here are some additional questions you might find useful:

1. How do we handle ties in the median calculation when the number of boxes is even?
2. What happens if we change the range requirement to 9 instead of 8?
3. Can you think of another set of numbers that satisfies all the conditions?
4. How does the mode affect the flexibility of the solution?
5. What if the mean had to be 6 instead of 5?

Tip: When solving a problem with multiple constraints, try breaking it down step by step and ensuring each condition is satisfied as you add elements to the solution.