4.4. In a $3 \times 4$ rectangle, natural numbers $1,2,3, \ldots, 12$ were written, each exactly once. The table had the property that in each column, the sum of the top two numbers was twice the bottom number. Over time, some numbers were erased. Find all possible numbers that could have been written in place of $\star$.

| 6 |  |  |  |
| :--- | :--- | :--- | :--- |
| 4 |  |  |  |
|  | 8 | 11 | $\star$ |

Answer, variant 1. 11.

Answer, variant 2. 8.

Answer, variant 3. 11.

Answer, variant 4. 2.
Solution 1. According to the condition, the number 2 is written in the bottom-left cell. The sum of the two unknown numbers in the second column is 10, and in the third column, it is 16. Therefore, the sum of the numbers in the first three columns is $6+15+24$, and the sum of all numbers in the table is $1+2+\ldots+12=78$. Thus, the sum of the numbers in the fourth column is $78-6-15-24=33$. From the condition, it follows that the number at $\star$ is one-third of 33, which is 11.

Remark. The arrangement of numbers described in the condition is indeed possible. For example, such an arrangement:

| 1 | 6 | 9 | 10 |
| :--- | :--- | :--- | :--- |
| 3 | 4 | 7 | 12 |
| 2 | 5 | 8 | 11 |