Question 110
A Latin Square is an $n \times n$ board filled with $n$ distinct symbols so that there are no repeated symbols in any row or in any column. For example, the figure below shows an example of a Latin Square of dimensions $3 \times 3$. The name was inspired by works of the mathematician Leonhard Euler, who used Latin characters as symbols.
| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 1 | 2 |
| 2 | 3 | 1 |
We know that there are 576 distinct Latin Squares of dimensions $4 \times 4$. In how many ways can we complete the square below, which already has two filled cells, with the digits 1, 2, 3 and 4 so that each row and each column contains all four digits?
| 1 | 2 | 3 |
| :--- | :--- | :--- |
| 3 | 1 | 2 |
| 2 | 3 | 1 |
We know that there are 576 distinct Latin Squares of dimensions $4 \times 4$. In how many ways can we complete the square below, which already has two filled cells, with the digits 1, 2, 3 and 4 so that each row and each column contains all four digits?