3. In the cells of a $3 \times 3$ square, the numbers $1,2,3, \ldots, 9$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is $18?$
Answer: 7.

Solution. Let's color the cells in a checkerboard pattern: suppose the corner and central cells are black, and the rest are white. From the condition, it follows that in cells of different colors, the numbers have different parities. Since there are five black cells and four white cells, we get that the odd numbers are in the black cells. Their total sum is $1+3+5+7+9=$ 25. Therefore, the number in the central cell is $7=25-18$.

Evaluation. 12 points for a correct solution. If an example of the arrangement of numbers satisfying the problem's condition is provided but the uniqueness of the answer is not proven, 6 points.