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.

Latin Square - dCode

Tag(s) : Number Games

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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 |

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**.

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.

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

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

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