orthogonal Latin squares

Given two Latin squaresMathworldPlanetmath L1=(A,B,C1,f1) and L2=(A,B,C2,f2) of the same order n, we can combine them coordinate-wise to form a single square, whose cells are ordered pairs of elements from C1 and C2 respectively. Formally, we can form a function f:A×BC1×C2 given by


This function f says that we have created a new square A×B, whose cell (i,j) contains the ordered pair of values, the first coordinate of which corresponds to the value in cell (i,j) of L1, and the second to the value in cell (i,j) of L2. We may write the combined square L1*L2.

For example,


In general, the combined square is not a Latin square unless the original two squares are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath: f1(i,j)=f1(k,) iff f2(i,j)=f2(k,). Nevertheless, the more interesting aspect of pairing up two Latin squares (of the same order) lies in the function f:

Definition. We say that two Latin squares are orthogonalPlanetmathPlanetmathPlanetmath if f is a bijectionMathworldPlanetmath.

Since there are n2 cells in the combined square, and |C1×C2|=n2, the function f is a bijection if it is either one-to-one or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.



  • 1 H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 1963
Title orthogonal Latin squares
Canonical name OrthogonalLatinSquares
Date of creation 2013-03-22 16:04:47
Last modified on 2013-03-22 16:04:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 62K10
Classification msc 05B15
Synonym mutually orthogonal Latin squares
Synonym MOLS
Synonym pairwise orthogonal Latin squares
Defines complete set of Latin squares