Task 1.1. A $4 \times 4$ table is divided into four $2 \times 2$ squares.

Vika wrote 4 ones, 4 twos, 4 threes, and 4 fours in the cells of the table such that in each column, each row, and each $2 \times 2$ square, all numbers are different.

The hooligan Andrey erased some of the numbers. Help Vika restore which cells contained fours.

|  | 1 | 2 | 3 | 4 |  |
| :--- | :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  | 3 |  |  |
|  |  |  |  | 3 |  |
|  | 2 |  |  | 1 |  |
|  | 1 |  |  | 1 |  |

Construct the correspondence

- In row $A$, the four is in
- In row $B$, the four is in
- In row $C$, the four is in
- In row $D$, the four is in
- column 1.
- column 2.
- column 3.
- column 4.
Answer: $A-4, B-1, C-3, D-2$.

Solution.

In the fourth column, the number 1 is already present, so in the upper right $2 \times 2$ square, the number 1 will be in cell $A 3$. By similar reasoning, the number 3 will be in cell $D 4$.

|  | 1 | 2 | 3 | 4 |
| :---: | :---: | :---: | :---: | :---: |
| $A$ |  |  | 1 |  |
| $B$ |  |  | 3 |  |
| c | 2 |  |  | 1 |
|  | 1 |  |  | 3 |

In columns 1, 3, 4 and rows $A, C, D$, there are three number 1s. Therefore, the number 1 is also in cell $B 2$.

The number 2 cannot be in cell $B 1$, so it must be in cell $B 4$. The number 4 must be in cell $B 1$ because it is the only number missing in that row.

In the fourth column, the number 4 can only be in cell $A 4$. In the lower right $2 \times 2$ square, the number 2 is missing: it cannot be in row $C$, so it must be in cell $D 3$.

In the third column, the number 4 can only be in cell $C 3$. Considering rows $C$ and $D$, we find that the number 3 is in cell $C 2$, and the number 4 is in cell $D 2$.

Considering the first two columns, we understand that the number 3 is in cell $A 1$, and the number 2 is in cell $A 2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2a01f5e31e9fc3823579g-03.jpg?height=1116&width=380&top_left_y=96&top_left_x=952)

![](https://cdn.mathpix.com/cropped/2024_05_06_2a01f5e31e9fc3823579g-03.jpg?height=354&width=375&top_left_y=1240&top_left_x=953)

Considering the first two columns, we understand that the number 3 is in cell $A 1$, and the number 2 is in cell $A 2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2a01f5e31e9fc3823579g-04.jpg?height=353&width=375&top_left_y=94&top_left_x=953)