# 5. Write the given table of numbers so that in each column and each row there is an arithmetic progression:

| $\mathbf{1}$ |  |  |  |
| :--- | :--- | :--- | :--- |
|  |  |  | $\mathbf{6}$ |
|  |  | $\mathbf{6}$ |  |
|  | $\mathbf{9}$ |  |  |
Let $a_{1}, a_{2}, a_{3}, a_{4}$ be the differences of the arithmetic progressions written in the rows of the table. Then its cells can be filled as shown in the figure

| 1 | $1+a_{1}$ | $1+2 a_{1}$ | $1+3 a_{1}$ |
| :---: | :---: | :---: | :---: |
| $6-3 a_{2}$ | $6-2 a_{2}$ | $6-a_{2}$ | 6 |
| $6-2 a_{3}$ | $6-a_{3}$ | 6 | $6+a_{3}$ |
| $9-a_{4}$ | 9 | $9+a_{4}$ | $9+2 a_{4}$ |

Using the property of an arithmetic progression, we can write the system of equations:

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ 2 ( 6 - 3 a _ { 2 } ) = 6 - 2 a _ { 3 } + 1 , } \\
{ 2 ( 6 - 2 a _ { 3 } ) = 6 - 3 a _ { 2 } + 9 - a _ { 4 } , } \\
{ 1 2 = 6 - a _ { 2 } + 9 + a _ { 4 } , } \\
{ 1 2 = 6 + a _ { 3 } + 1 + 3 a _ { 1 } ; }
\end{array} \Rightarrow \left\{\begin{array}{l}
12-6 a_{2}=7-2 a_{3}, \\
12-4 a_{3}=15-3 a_{2}-a_{4}, \\
a_{2}-a_{4}=3, \\
a_{3}+3 a_{1}=5 ;
\end{array} \Rightarrow\right.\right. \\
& \left\{\begin{array}{l}
6 a_{2}-2 a_{3}=5, \\
3 a_{2}-4 a_{3}+a_{4}=3, \\
a_{2}-a_{4}=3, \\
a_{3}+3 a_{1}=5
\end{array}\right.
\end{aligned}
$$

From the third equation, express $a_{4}=a_{2}-3$ and substitute into the second equation. We get:

$$
\left\{\begin{array} { l } 
{ 6 a _ { 2 } - 2 a _ { 3 } = 5 , } \\
{ 4 a _ { 2 } - 4 a _ { 3 } = 6 , } \\
{ 3 a _ { 1 } + a _ { 3 } = 5 ; }
\end{array} \quad \Rightarrow \quad \left\{\begin{array}{l}
6 a_{2}-2 a_{3}=5 \\
2 a_{2}-2 a_{3}=3 \\
3 a_{1}+a_{3}=5
\end{array}\right.\right.
$$

Subtract the second equation from the first, we get $4 a_{2}=2$; hence, $a_{2}=0.5$. Then $2 a_{3}=2 a_{2}-3$ $=1-3=-2$ and $a_{3}=-1 ; a_{4}=a_{2}-3=0.5-3=-2.5$ and $a_{4}=-2.5$. Therefore, $3 a_{1}-1=5$ and $a_{1}=2$.

Now fill in the table. Verification shows that the found numbers satisfy the condition of the problem.

| 1 | 3 | 5 | 7 |
| :---: | :---: | :---: | :---: |
| 4.5 | 5 | 5.5 | 6 |
| 8 | 7 | 6 | 5 |
| 11.5 | 9 | 6.5 | 4 |

The answer is shown in the figure.