---
tag: CDVRMVED2V
kind: concept
name: Latin square
msc: [05B15]
specializes: [PAJTMQ604P]
sources:
  - url: https://en.wikipedia.org/wiki/Latin_square
same_as:
  wikidata: Q679367
  wikipedia: Latin square
---

## Summary

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols.
Each symbol occurs exactly once in each row and exactly once in each column.
Latin squares are the multiplication tables of quasigroups, and the Cayley tables of finite groups are examples.
They are used in the design of experiments and in coding theory, and orthogonal families of them are tied to finite projective planes.
The node [[Latin square|44HTG620GB]] gives the precise definition.

## Explanations

### L1 human

A completed Sudoku grid is a Latin square of order $9$ with extra conditions on the boxes.
Without the boxes, the rule is only: no symbol repeats in a row or in a column.
For order $3$, the rows $1\,2\,3$, then $2\,3\,1$, then $3\,1\,2$ form a Latin square.
Each new row is the previous row shifted by one place.