$:$ Folkiore

The numbers $1,2,3, \ldots, 25$ are arranged in a $5 \times 5$ table such that the numbers in each row are in increasing order.

What is the greatest and the least value that the sum of the numbers in the third column can have?
Evaluation. The sum of the numbers in the first three columns is not less than the sum of the first 15 natural numbers, which is not less than 120.

If we add 2 to each number in the first column and 1 to each number in the second column, the numbers in each row will remain in non-decreasing order. Therefore, the sum of the numbers in the third column will be no less than the sum in each of the first two columns, which is no less than $(120+5+10): 3=45$.

Example of number placement where the sum of the middle column is 45:

| 1 | 2 | 3 | 16 | 17 |
| :---: | :---: | :---: | :---: | :---: |
| 4 | 5 | 6 | 18 | 19 |
| 7 | 8 | 9 | 20 | 21 |
| 10 | 11 | 12 | 22 | 23 |
| 13 | 14 | 15 | 24 | 25 |

Similar reasoning gives the maximum sum of 85 (considering the largest possible sum of the numbers in the last three columns).

## Answer

The smallest is 45, the largest is 85.