To find the smallest and largest values that the average $m$ of the medians $A, B, C, D, E$ can take, let's break down the problem.

Step 1: Understanding the Problem

• The natural numbers from 1 to 25 are divided into 5 groups of 5 numbers each.
• For each group, the median is the middle number when the group is arranged in increasing order.
• $m$ is the average of these 5 medians.

Step 2: Properties of the Groups

To minimize or maximize $m$, we need to understand how the numbers can be arranged to affect the medians.

• If the groups are arranged with lower numbers in one group and higher numbers in another, it affects the median values.

Step 3: Minimizing $m$

To minimize $m$, we want the medians to be as small as possible. This happens when the smallest numbers are grouped together:

• Group 1: $\{1, 2, 3, 4, 5\}$ with median $3$
• Group 2: $\{6, 7, 8, 9, 10\}$ with median $8$
• Group 3: $\{11, 12, 13, 14, 15\}$ with median $13$
• Group 4: $\{16, 17, 18, 19, 20\}$ with median $18$
• Group 5: $\{21, 22, 23, 24, 25\}$ with median $23$

So, the smallest possible value of $m$ is: $m_{\text{min}} = \frac{3 + 8 + 13 + 18 + 23}{5} = \frac{65}{5} = 13$

Step 4: Maximizing $m$

To maximize $m$, we want the medians to be as large as possible. This happens when the highest numbers are grouped together:

• Group 1: $\{1, 2, 23, 24, 25\}$ with median $23$
• Group 2: $\{3, 4, 19, 20, 22\}$ with median $19$
• Group 3: $\{5, 6, 7, 8, 21\}$ with median $7$
• Group 4: $\{9, 10, 11, 12, 18\}$ with median $11$
• Group 5: $\{13, 14, 15, 16, 17\}$ with median $15$

So, the largest possible value of $m$ is: $m_{\text{max}} = \frac{23 + 19 + 7 + 11 + 15}{5} = \frac{75}{5} = 15$

Final Answer:

The smallest value of $m$ is 13, and the largest value of $m$ is 15.

Would you like to explore further details or have any questions? Here are some related questions:

1. How does the median behave under different groupings of data?
2. What are other ways to arrange groups to get specific median values?
3. How would this approach change if the range of numbers was different?
4. Can this method be generalized to more groups or different group sizes?
5. How do extreme values in data affect the overall median or average?

Tip: In problems involving medians, always consider both extreme and central arrangements to understand the range of possible outcomes.