Recent results on Choi's orthogonal Latin squares

Jon-Lark Ki̇m,Jae Woo Park,Doo Young Park,Dong Eun Ohk

doi:10.13069/jacodesmath.1056511

Jon-Lark Ki̇m, Jae Woo Park + Show 2 more

Open Access

https://doi.org/10.13069/jacodesmath.1056511

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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.

Highlights

A Latin square of order n is an n × n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book
We find a geometric description of Choi’s orthogonal Latin squares of order 9 using the dihedral group D8

Summary

Introduction

A Latin square of order n is an n × n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column This Latin square is one of the most interesting mathematical objects. 9(1) (2022) 17–27 pair of two orthogonal Latin squares of order 9 was introduced in Koo-Soo-Ryak (or Gusuryak) written by Choi Seok-Jeong [2]. We observe that the Latin squares have self-repeating patterns This simple structure of Choi’s Latin squares motivates some generalization of his idea. We give a new generalization of Choi’s orthogonal Latin squares of order 9 to orthogonal Latin squares of size n2 using the Kronecker product including Lih’s construction [9]. We 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

A generalization of Choi’s orthogonal Latin squares

Latin squares acted by the dihedral group D8

Magic squares from Latin squares