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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

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.

### 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.

## Source code

dCode retains ownership of the source code of the script Latin Square online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Latin Square script for offline use on PC, iPhone or Android, ask for price quote on contact page !

## Need Help ?

Please, check our community Discord for help requests!

Thanks to your feedback and relevant comments, dCode has developped the best Latin Square tool, so feel free to write! Thank you !

Source : https://www.dcode.fr/latin-square
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