The sums of the three columns and the three rows of the table are equal.

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

What is the minimum number of cells in the table that need to be changed so that all the new six sums are different from each other?
Solution

If three or fewer cells are changed, there will either be two rows without any changed cells or one cell is the only one changed in its row and column. In the first case, these two rows without any changed cells have the same sum. In the second case, if only one cell is the only one changed in its row and column, they also have the same sum. The table below represents an example with only 4 changes and all 6 row and column sums are distinct.

| $\mathbf{5}$ | $\mathbf{9}$ | $\mathbf{7}$ |
| :--- | :--- | :--- |
| 8 | $\mathbf{3}$ | $\mathbf{9}$ |
| 3 | 5 | 7 |

Therefore, the minimum number of changes is 4.