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 |