Question 21
Consider a rectangular array of single digits $d_{i, j}$ with 10 rows and 7 columns, such that $d_{i+1, j}-d_{i, j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_{i}$ be the median of $d_{i, 1}, \ldots, d_{i, 7}$. Determine the least and greatest possible values of the mean of $m_{1}, m_{2}, \ldots, m_{10}$.
Example:
| | $d_{i, 1}$ | $d_{i, 2}$ | $d_{i, 3}$ | $d_{i, 4}$ | $d_{i, 5}$ | $d_{i, 6}$ | $d_{i, 7}$ | $m_{i}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $i=1$ | 2 | 7 | 5 | 9 | 5 | 8 | 6 | median is 6 |
| $i=2$ | 3 | 8 | 6 | 0 | 6 | 9 | 7 | median is 6 |
| $i=3$ | 4 | 9 | 7 | 1 | 7 | 0 | 8 | median is 7 |
| $i=4$ | 5 | 0 | 8 | 2 | 8 | 1 | 9 | median is 5 |
| $i=5$ | 6 | 1 | 9 | 3 | 9 | 2 | 0 | median is 3 |
| $i=6$ | 7 | 2 | 0 | 4 | 0 | 3 | 1 | median is 2 |
| $i=7$ | 8 | 3 | 1 | 5 | 1 | 4 | 2 | median is 3 |
| $i=8$ | 9 | 4 | 2 | 6 | 2 | 5 | 3 | median is 4 |
| $i=9$ | 0 | 5 | 3 | 7 | 3 | 6 | 4 | median is 4 |
| $i=10$ | 1 | 6 | 4 | 8 | 4 | 7 | 5 | median is 5 |
Example:
| | $d_{i, 1}$ | $d_{i, 2}$ | $d_{i, 3}$ | $d_{i, 4}$ | $d_{i, 5}$ | $d_{i, 6}$ | $d_{i, 7}$ | $m_{i}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $i=1$ | 2 | 7 | 5 | 9 | 5 | 8 | 6 | median is 6 |
| $i=2$ | 3 | 8 | 6 | 0 | 6 | 9 | 7 | median is 6 |
| $i=3$ | 4 | 9 | 7 | 1 | 7 | 0 | 8 | median is 7 |
| $i=4$ | 5 | 0 | 8 | 2 | 8 | 1 | 9 | median is 5 |
| $i=5$ | 6 | 1 | 9 | 3 | 9 | 2 | 0 | median is 3 |
| $i=6$ | 7 | 2 | 0 | 4 | 0 | 3 | 1 | median is 2 |
| $i=7$ | 8 | 3 | 1 | 5 | 1 | 4 | 2 | median is 3 |
| $i=8$ | 9 | 4 | 2 | 6 | 2 | 5 | 3 | median is 4 |
| $i=9$ | 0 | 5 | 3 | 7 | 3 | 6 | 4 | median is 4 |
| $i=10$ | 1 | 6 | 4 | 8 | 4 | 7 | 5 | median is 5 |