---
tag: H6TN6BADVY
kind: concept
name: Mutually orthogonal Latin squares
msc: [05B15]
sources:
  - url: https://en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares
same_as:
  wikidata: Q615342
  wikipedia: Mutually orthogonal Latin squares
---

## Summary

Two [[Latin squares|CDVRMVED2V]] of the same order are orthogonal if, when one is laid over the other, every ordered pair of symbols appears exactly once.
A family of Latin squares is mutually orthogonal if every two of them are orthogonal.
For $n \ge 2$, there are at most $n - 1$ mutually orthogonal Latin squares of order $n$.
A family of $n - 1$ of them exists exactly when a [[finite projective plane|2HWB7P1DZN]] of order $n$ exists.
The node [[mutually orthogonal Latin squares|KHD2QMNR7R]] gives the precise definition.

## Explanations

### L1 human

Imagine a $3 \times 3$ field planted with three crops and treated with three fertilisers.
Each crop should appear once per row and column, and so should each fertiliser.
If in addition every crop meets every fertiliser exactly once, the two Latin squares are orthogonal.
Such a pair exists for every order except $2$ and $6$.