### Abstract

Fuchs asked: For which integer partitions g_{1} +· · · g_{n} = N does there exist a latin square of side N having n subsquares of sides g_{1}, …,g_{n} having no rows, columns, or symbols in common? Only when at most two distinct integer parts are used is the answer known completely; in general even the necessary conditions are elusive. Two conjectures giving plausible sufficient conditions are advanced. The first asserts than whenever the largest three integer parts are the same, such a latin square always exists. The second asserts that whenever the largest part is no larger than n − 2 times the smallest, such a latin square always exists. Partial results on both are established.

Original language | English (US) |
---|---|

Pages (from-to) | 501-536 |

Number of pages | 36 |

Journal | Australasian Journal of Combinatorics |

Volume | 71 |

Issue number | 3 |

State | Published - Jan 1 2018 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

## Fingerprint Dive into the research topics of 'On a latin square problem of fuchs'. Together they form a unique fingerprint.

## Cite this

*Australasian Journal of Combinatorics*,

*71*(3), 501-536.