Sofia plays by writing all the 4-digit numbers with different digits that can be written with the digits 1, 2, 4, and 7. She adds 3 of these numbers - all different - and gets 13983. What are these 3 numbers?
Since 13983 ends in 3, the sum of the units digits of the 3 numbers must be 13, and for this we only have one option: $2+$ $4+7=13$.

|  |  |  |  | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  |  |  |  | 4 |
|  |  |  |  | 7 |
| 1 | 3 | 9 | 8 | 3 |

Now, the sum of the tens digits must be $8-1=7$, so it must be $1+2+4=7$. We complete the tens digits, being careful not to repeat the same digit in the same number. We have three options:

|  |  |  | 1 | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  |  |  | 2 | 4 |
|  |  |  | 4 | 7 |
| 1 | 3 | 9 | 8 | 3 |

![](https://cdn.mathpix.com/cropped/2024_05_01_dbce7bc2a200e234c3e2g-25.jpg?height=300&width=355&top_left_y=1910&top_left_x=793)

|  |  |  |  |  |  |  |  | 1 |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  | 4 | 2 |  |  |  |  |  |  |  |
|  |  |  | 1 | 4 |  |  |  |  |  |  |  |
|  |  |  | 2 | 7 |  |  |  |  |  |  |  |
| 1 | 3 | 9 | 8 | 3 |  |  |  |  |  |  |  |

The hundreds digits must sum to 9, so we have two options: $4+4+1$ and $1+1+7$. Since in the three previous possibilities the digit 4 appears in two of the three numbers, we choose the second option to avoid the digit 4 being repeated. We must be careful that 1 and 7 also do not appear repeated.

|  |  | 7 | 1 | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  |  | 1 | 2 | 4 |
|  |  | 1 | 4 | 7 |
| 1 | 3 | 9 | 8 | 3 |


|  |  | 1 | 4 | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  |  | 7 | 1 | 4 |
|  |  | 1 | 2 | 7 |
| 1 | 3 | 9 | 8 | 3 |

Finally, the thousands digits must sum to 13, now it is easy to choose them:

|  | 4 | 7 | 1 | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  | 7 | 1 | 2 | 4 |
|  | 2 | 1 | 4 | 7 |
| 1 | 3 | 9 | 8 | 3 |


|  | 7 | 1 | 4 | 2 |
| :--- | :--- | :--- | :--- | :--- |
|  | 2 | 7 | 1 | 4 |
|  | 4 | 1 | 2 | 7 |
| 1 | 3 | 9 | 8 | 3 |