1. Daan distributes the numbers 1 through 9 over the nine cells of a $3 \times 3$ table (each cell contains exactly one number). Then Daan circles the middle number in terms of size in each row. For example, if the numbers 8, 1, and 2 are in a row, he circles the number 2. He does this for each column and for each of the two diagonals as well. If a number is already circled, he does not circle it again.

| 8 | 1 | 2 |
| :---: | :---: | :---: |
| 7 | 6 | 3 |
| 9 | 5 | 4 |

The result of this process he calls a median table. Above you see an example of a median table where 5 numbers are circled.

(a) What is the smallest possible number of circled numbers in a median table? Prove that it cannot be less and provide an example where this minimal number is circled.

(b) What is the largest possible number of circled numbers in a median table? Prove that it cannot be more and provide an example where this maximal number is circled.
1. (a) The smallest possible number of circled numbers is 3. That it cannot be done with fewer follows from the fact that at least one number is circled in each row (and these are three different numbers).

A median table with only 3 numbers circled is shown alongside. In the rows, the numbers 7, 5, 3 are circled, in the columns the numbers 3, 5, 7, and on the diagonals the numbers 5 and 5. Together, these are three different numbers: 3, 5, and 7.

| 4 | 9 | 7 |
| :---: | :---: | :---: |
| 2 | 5 | 8 |
| 3 | 1 | 6 |

(b) The largest possible number of circled numbers is 7. That it cannot be more follows from the fact that the numbers 9 and 1 are never circled. Thus, no more than \(9 - 2 = 7\) numbers are ever circled.

A median table where actually 7 numbers are circled is shown alongside. In the rows, the numbers 2, 6, 8 are circled, in the columns the numbers 7, 5, 3, and on the diagonals 4 and 5. Together, these are the seven numbers 2, 3, 4, 5, 6, 7, 8.

| 4 | 1 | 2 |
| :--- | :--- | :--- |
| 7 | 5 | 6 |
| 8 | 9 | 3 |