orthogonal Latin squares

Orthogonal Latin squares are a concept in combinatorics. A Latin square is a grid where each row and column contains distinct symbols. Two Latin squares are orthogonal if, when superimposed, each symbol from the first square appears exactly once with each symbol from the second square.1

A set of mutually orthogonal Latin squares means any two squares in the set are orthogonal. There can be at most k-1 mutually orthogonal Latin squares of order k. Constructing these squares is important for experimental designs. Orthogonal arrays are a generalization of orthogonal Latin squares, with applications in factorial designs and communication theory. The existence of orthogonal Latin squares has been a subject of mathematical inquiry and conjecture, with significant developments in disproving earlier conjectures.12