# Problem 4. Maximum 15 points

A trader sells stocks. It is known that stocks are sold once a month except during the summer, when the trader is on vacation. After economic analysis and entering the results of the past year into a table by months, it was found that the number of stocks sold corresponds to a $3 * 3$ magic square. The numbers in the square are unique and form a sequence with a difference of one. One of the numbers is 31, and the sum of the numbers in each row, column, and both diagonals of the square is 102.

| September | December | March |
| :---: | :---: | :---: |
| October | January | April |
| November | February | May |

Find the number of stocks sold each month and fill in the cells of the given square with numbers.
# Solution:

Let's sum up the three rows in the square, then this sum is obviously equal to 306. On the other hand, since the numbers in the square form a sequence with a difference of one, we can use an arithmetic progression, where the first term is the minimum number in the square (let's call it a), and the maximum element in the square is then a + 8.

$$
\frac{a+(a+8)}{2} \cdot 9=306
$$

Then \( \mathrm{a}=30 \), the elements of the square range from 30 to 38, which can be arranged, for example, as follows:

373233

303438

353631

## Criteria:

Finding all numbers: 5 points

An example of the correct arrangement in the square: 10 points

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