# Recent results on Choi's orthogonal Latin squares

## TL;DR

This paper generalizes Choi's orthogonal Latin squares of order 9 to larger sizes using Kronecker products, explores their geometric structure with dihedral groups, and links them to magic squares.

## Contribution

It introduces a new generalization method for Choi's Latin squares of order 9 to size n^2, incorporating Lih's construction and geometric analysis.

## Key findings

- Generalization of Choi's Latin squares to size n^2 using Kronecker product
- Geometric description of Choi's Latin squares via dihedral group D_8
- New construction method for magic squares from orthogonal Latin squares

## Abstract

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.