Adam wrote the numbers from 1 to 9 into a $3 \times 3$ table as shown in the image:

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

For this filling, the sum of the numbers in the three cells along each side is always the same. Adam found that the numbers can be filled into the table in other ways without breaking the property of having the same sums along the sides.

What is the smallest value that this sum can have? Provide an example of a table with the smallest sum along the sides and explain why it cannot be smaller.

(J. Tkadlec)
Given that each corner cell appears in two sums, we try to place the smallest possible numbers in these cells and somehow fill in the rest. After some trial and error, the following filling can be revealed, where the sum of the numbers along each side is 12:

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

Filling with smaller sums is not possible, because it is not feasible. The smallest possible sum along a side with the addend 9 is $1+2+9=12$. Thus, the number 9 would have to be in the center of the table, and the remaining numbers would be along the sides. The smallest possible sum along a side with the addend 8 is $1+2+8=11$. Thus, a smaller sum cannot be achieved, and we consider filling the table such that the numbers 1, 2, and 8 are along one side. Along the opposite side, there would be three of the remaining five numbers. The smallest possible numbers are 3, 4, and 5, whose sum is $3+4+5=12$, not 11.

The smallest possible value of the sum in Adam's table is 12.

Note. Each of the four corner cells contributes to two sums, and each of the remaining four cells along the sides contributes to one sum. Therefore, the sum of all four sums along the sides is at least

$$
2 \cdot(1+2+3+4)+(5+6+7+8)=46.
$$

The requirement for the sums along the sides to be equal means that the previous sum must be divisible by four. The nearest larger number divisible by four is 48. Thus, the smallest possible value of the sum in Adam's table is $48: 4=12$. The table above shows that such a filling is possible.