We have written the first nine positive integers in a $3 \times 3$ table, each one appearing once. Suppose that the three-digit numbers that can be read from left to right in each of the three rows, from top to bottom in each of the three columns, and from the top-left corner along the main diagonal are all divisible by 11. What can be the value of the three-digit number that can be read along the diagonal starting from the top-right corner?
We denote the nine numbers with the lowercase letters of the alphabet from $a$ to $i$ and write them into the table (figure). According to the condition, $11 \mid \overline{a b c}$, which means $11 \mid a-b+c$, and similarly,

$$
11 \mid d-e+f, \quad 11 \mid g-h+i, \quad 11 \mid b-e+h
$$

From the last one, it follows that

$$
11 \mid 2 b-2 e+2 h
$$

By adding the first three signed sums on the right side and using (1) and the fact that the sum of the first nine positive integers is 45, we get that

$$
11 \mid a+b+c+d+e+f+g+h+i-4 e=45-4 e
$$

From this, it follows that the number in the center of the table is $e=3$, since only then will 11 be a divisor of $(45-4 e)$.

Next, let's see which numbers can stand in the middle column of the table, in the row, and on the diagonal starting from the upper left corner.

A three-digit number is divisible by 11 if the difference between the sum of its outer digits and the middle one is either 0 or 11. Now we are only looking for numbers different from 3 and from each other.

If $x-3+y=0$, then $x+y=3$, from which $x=1$ and $y=2$ or $x=2$ and $y=1$.

If $x-3+y=11$, then $x+y=14$. The following pairs of numbers satisfy this:

$$
(5,9) ; \quad(6,8) ; \quad(8,6) \quad \text { and } \quad(9,5) .
$$

Thus, the numbers $1,2,5,6,8,9$ stand in the six circled places of the table in some order. Among these, 4 and 7 do not appear, so the other diagonal has 437 or 734.

The numbers can be arranged in several ways, and we present two of them.

| $a$ | $b$ | $c$ |
| :---: | :---: | :---: |
| $d$ | $e$ | $f$ |
| $g$ | $h$ | $i$ |


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


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