Realizations with five subsquares

TL;DR

This paper completes the classification of when a Latin square of order n can contain five disjoint subsquares of specified sizes, extending previous results for up to four subsquares.

Contribution

It provides a complete solution for the existence of Latin squares with five disjoint subsquares of given sizes, filling a gap in the existing literature.

Findings

Solved the case for five disjoint subsquares in Latin squares

Extended previous results from k≤4 to k=5

Proved additional results for specific partitions with five subsquares

Abstract

Given an integer partition $(h_1,h_2,\dots,h_k)$ of $n$, is it possible to find an order $n$ latin square with $k$ disjoint subsquares of orders $h_1,\dots,h_k$? This question was posed by L.Fuchs and is only partially solved. Existence has been determined in general when $k\leq 4$, and in this paper we will complete the case when $k=5$. We also prove some less general results for partitions with $k=5$.