Problem 4.6. Some digits in the rectangle are already placed. Place the remaining digits so that:

- the sum of the digits in each column is the same;
- the sum of the digits in each row is the same;
- the sum of the digits in the red cells is equal to the sum of the digits in any row. Enter the three-digit number $A B C$ as your answer.

| 4 |  | 3 |
| :--- | :--- | :--- |
|  | $A$ | 1 |
| 1 |  | 6 |
| $B$ | 2 | $C$ |
Answer: 652.

Solution. If we add 3 to the sum of the two upper red numbers, we get the same number as if we added the sum of the two lower red numbers. Therefore, the unknown lower red number is 2. Then, in each row, the sum is $1+2+6=$ 9. This means that the unknown upper red number complements $4+3$ to 9, i.e., it is also 2. We have found that the sum of the digits in each row is 9. This means that the sum of the numbers in the entire table is 36, and in each column, it is 12. Now, it is not difficult, by sequentially considering columns and rows with one unknown number, to fill in the entire table (for example, we can first determine that $C=2$ and $A=6$, and then $B=5$):

| 4 | 2 | 3 |
| :--- | :--- | :--- |
| 2 | 6 | 1 |
| 1 | 2 | 6 |
| 5 | 2 | 2 |

We get that the number $A B C$ is 652.