A star square is a square table of numbers for which the sums of the numbers in individual rows and columns are always the same. In the image, there is a remnant of a star square, in which numbers in one row and one column have been erased.

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

Fill in the missing numbers so that all are integers and exactly four are negative. Determine all possibilities.

(E. Semerádová)

Hint. What are the sums of the remaining numbers in the rows and columns?
If the sum of the numbers in each row and column is denoted by $s$, then the missing numbers in the magic square are

| 1 | 2 | 3 | $s-6$ |
| :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | $s-15$ |
| 7 | 8 | 9 | $s-24$ |
| $s-12$ | $s-15$ | $s-18$ | $-2 s+45$ |

The number $s-6$ is negative for $s<6$. Exactly four negative integers are obtained for $s=12, 13$ or 14:

| 1 | 2 | 3 | 6 |
| :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | -3 |
| 7 | 8 | 9 | -12 |
| 0 | -3 | -6 | 21 |


| 1 | 2 | 3 | 7 |
| :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | -2 |
| 7 | 8 | 9 | -11 |
| 1 | -2 | -5 | 19 |


| 1 | 2 | 3 | 8 |
| :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | -1 |
| 7 | 8 | 9 | -10 |
| 2 | -1 | -4 | 17 |

Note. Instead of using general expressions with the unknown $s$, one can start by filling in specific values to form a magic square. By gradually increasing or decreasing the partial sums, the above completions can be derived, in which exactly four numbers are negative.