### Abstract

Suppose that n≡±1 mod 6 and n≥7. We construct a Latin square Ln of order n with the following properties: Ln has no proper subsquares of order 3 or more. Ln has exactly one intercalate (subsquare of order 2). When the intercalate is replaced by the other possible subsquare on the same symbols, the resulting Latin square is in the same species as Ln. Hence Ln generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, Ln is what is known as a self-switching Latin square and possesses a near-autoparatopism.

Original language | English |
---|---|

Pages (from-to) | 279-293 |

Number of pages | 15 |

Journal | Journal of Combinatorial Designs |

Volume | 24 |

Issue number | 6 |

DOIs | |

Publication status | Published - 1 Jun 2016 |

### Keywords

- autoparatopism
- intercalate
- Latin square
- self-switching
- subsquare

## Cite this

Mendis, M. J. L., & Wanless, I. M. (2016). Latin squares with a unique intercalate.

*Journal of Combinatorial Designs*,*24*(6), 279-293. https://doi.org/10.1002/jcd.21430