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Latin Square

Tool to create or solve Latin squares, a square grid of NxN with N distinct symbols distributed without repetition in each row and column, ideal for lovers of logic and puzzles similar to sudoku.

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Latin Square -

Tag(s) : Number Games

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Latin Square

Latin Square Solver

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See also: Sudoku Solver

Latin Square Generator




See also: Sudoku Solver

Answers to Questions (FAQ)

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 row and per column.

Example: A 3x3 Latin square:

123
312
231

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 row with N symbols

Example:

ABC
---
---

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)

Example:

ABC
CAB
---

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

Example:

ABC
CAB
BCA

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.

There are variants of Latin squares with additional constraints, such as diagonal Latin squares where each symbol must also be unique on both long diagonals of the grid.

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 row 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.

How many different Latin squares can be generated for a given size?

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.

When was the Latin square invented?

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

Source code

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Cite as source (bibliography):
Latin Square on dCode.fr [online website], retrieved on 2024-12-03, https://www.dcode.fr/latin-square

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