The $5 \times 5$ square grid shown in the figure can be filled with positive integers such that in each row and each column, consecutive terms of an arithmetic sequence are placed. Determine the number that will be placed in the cell marked with (*).

|  |  |  | * |  |
| :--- | :--- | :--- | :--- | :--- |
|  | 74 |  |  |  |
|  |  |  |  | 186 |
|  |  | 103 |  |  |
| 0 |  |  |  |  |
Let's start filling in the lower square of the first column of the table. Denoting the common difference of the arithmetic sequence by $d$, the terms are: $0, d, 2d, 3d, 4d$. Let the third element of the second row be denoted by $x$. Then $74=\frac{3d+x}{2}$, from which $x=148-3d$, which stands in the third column of the second row.

![](https://cdn.mathpix.com/cropped/2024_05_02_61689a540b9c4f2b9137g-1.jpg?height=474&width=525&top_left_y=332&top_left_x=781)

Figure 1

The third element of the third column:

$$
\frac{148-3d+103}{2}=\frac{251-3d}{2}
$$

This, as the third element of the third row, can be written as the arithmetic mean of the first and fifth elements symmetrically placed with it in the same row, that is,

$$
\frac{251-3d}{2}=\frac{2d+186}{2}.
$$

From the equation, $d=13$. Now we can start filling in the table with numbers. We only wrote down those numbers that were necessary to determine the number marked with *.

| 52 |  |  | $\underline{142} *$ |  |
| :--- | :--- | :--- | :--- | :--- |
| 39 | $\mathbf{74}$ | 109 | 144 |  |
|  |  | 106 | 146 | $\mathbf{186}$ |
|  |  | $\mathbf{103}$ |  |  |
| $\mathbf{0}$ |  |  |  |  |

Figure 2

The sought number: 142.

Remark. We could have started the solution elsewhere.