Problem 2. In a $3 \times 3$ table, natural numbers (not necessarily distinct) are placed such that the sums in all rows and columns are different. What is the minimum value that the sum of the numbers in the table can take?
Answer: 17.

Solution. Note that in each row and column, the sum of the numbers is no less than 3. Then, the doubled sum of all numbers in the table, which is equal to the sum of the sums of the numbers in the rows and columns, is no less than \(3+4+\ldots+8=33\), so the simple sum of the numbers in the table is no less than 17.

Example of a table with a sum of 17:

| 1 | 1 | 1 |
| :--- | :--- | :--- |
| 1 | 2 | 2 |
| 2 | 3 | 4 |

## Criteria

The largest suitable criterion is used:

7 p. Any complete solution to the problem.

4 p. It is proven that the sum of the numbers in the table is no less than 17.

3 p. An example of a table with a sum of 17 is provided.