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Latin Square

Tool to solve or generate Latin squares (generalization of sudoku). The Latin square is a square grid of size NxN comprising N distinct symbols distributed such that each row and column also have N distinct symbols.

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Latin Square -

Tag(s) : Number Games

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# Latin Square

## Latin Square Generator

### What is a latin square? (Definition)

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 line and per column.

Example: A 3x3 Latin square:

 1 2 3 3 1 2 2 3 1

### How to generate a latin square?

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

1- Take an empty NxN grid,

Example:

 - - - - - - - - -

2- Fill the first line with N symbols

Example:

 A B C - - - - - -

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

Example:

 A B C C A B - - -

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

Example:

 A B C C A B B C A

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.

### How to solve 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 line or the same column), if only one symbol among the N is possible then fill in the cell with this symbol.

### Why is a Latin square so called? (Origin of the name)

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

### What are the variants of the latin square?

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.

### When latin square have been invented ?

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

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