Latin Square

A Latin square is a square of size $ n \times n $ filled with exactly $ n $ distinct symbols (letters, numbers, numbers) so that each symbol appears only once per row and per column.

There are several methods for generating Latin squares. Here is the shortest to explain:

1- Take an empty NxN grid,

2- Fill the first row with N symbols

3- Fill the next row by shifting all the symbols by 1 box to the right (and return to the beginning of the row if necessary)

4- Repeat step 3 until you reach the last row of the grid.

And voilà! The square is thus valid but always identical, to obtain others:

5- Randomly swap 2 lines

6- Randomly swap 2 columns

7- Repeat steps 5 and 6 as many times as desired, the result is always a Latin square.

The resolution algorithm consists in noting, for each unfilled cell, the list of possible symbols respecting the rules (prohibition of 2 identical symbols on the same row or the same column), if only one symbol among the N is possible then fill in the cell with this symbol.

The Latin square name comes from Leonhard Euler who studied these grids and who used Latin letters as symbols.

The best known variant is sudoku, which uses the same bases, but adds a constraint on blocks of 3x3 (and sometimes other constraints for irregular sudoku).

Ken-ken (kendoku) is also a Latin square with constraints of mathematical calculations.

The magic square is a distant mathematical variant which takes up the fact that the sum of the rows and the columns is always identical, but it is not limited to N distinct symbols.

The number of possible Latin squares increases rapidly with the size of $ N $. For $ N = 3 $ there are 12 different Latin squares, for $ N = 4 $ there are 576, then 161280 for 5, 812851200 for 6 etc. for larger sizes this number becomes exponential, see here

No direct formula to find these numbers is known to date.

The first written traces of work on these grids date from 1700, but the principle itself was known long before, perhaps even from antiquity.