Recent results on Choi's 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 3ˆ3 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 n2 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 D8. 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.