Question 90
Exercise 9. Throughout the exercise, we are interested in tables $5 \times 5$, whose rows (resp. columns) are denoted $L_{0}, L_{1}, L_{2}, L_{3}, L_{4}$ from bottom to top (resp. $C_{0}, C_{1}, C_{2}, C_{3}, C_{4}$, from left to right). In each of the 25 boxes of such a table, a number is written. The number that is in the box located on the row $L_{i}$ and column $C_{j}$ is noted $f(i, j)$. An array is said to be consistent when, for all $i$ and $j$: the number $f(i, j)$ is the number of $i$ written in the row boxes $L_{j}$. Determine all consistent tables of size $5 \times 5$.