In the board below, in each row and in each column, exactly one number from the set $\{1,2,3,4\}$ is written. Which number is in the square with the symbol $\star$?

| 1 | 2 |  |  |
| :--- | :--- | :--- | :--- |
|  | 1 | 2 |  |
| 2 |  | $\star$ | 1 |
| 3 |  | 1 |  |
Solution

Consider the positions indicated by the letters: $A, B, C$ and $D$. Whenever a row or column already has three numbers written, its fourth element is completely determined and is equal to the number that has not yet appeared in it. Thus, $C=4$. Analyzing the numbers in the second row, we have $D=3$. The cell in the top right corner cannot be any number in the set $\{1,2,3\}$, as these have already appeared in the first row and last column. Therefore, $B=4$. Analyzing the numbers in the first row, we can conclude that $A=3$. Finally, observing the third column, we can conclude that $\star=4$.

| 1 | 2 | $A$ | $B$ |
| :---: | :---: | :---: | :---: |
| $C$ | 1 | 2 | $D$ |
| 2 |  | $\star$ | 1 |
| 3 |  | 1 |  |