Latin Squares in Practice and in Theory I

Graeco-Latin squares are nicely illustrated in Rob Beezer's page (University of Puget Sound). Cut-the-knot.com has an applet that leads you through the construction of a latin square: any suitable beginning can always be completed. An experimental latin square design and analysis is given on the site of the Washington State University Tree Fruit Research and Extension Center. A contemporary agricultural experiment with latin square design is Effect of Magnesium and Sulfur Fertilization of Alfalfa by K. L. Wells and J. E. Dollarhide, University of Kentucky.

1. What is a Latin Square, and what is a Graeco-Latin Square?

An N x N Latin Square arrangement is a way of putting N copies of each of N things in an N x N array such that in each row, and in each column, all the elements are different.

Examples:

A B

B A
is a 2 x 2 latin square. The name comes from the elements usually being represented, as they are here, by letters of the Latin alphabet (this terminology goes back to Euler).
A B C

B C A

C A B
is a 3 x 3 latin square, and this pattern can be extended to any size.

An N x N Graeco-Latin Square involves two sets of N elements. The first are traditionally represented by letters of the Latin alphabet, and the second by letters of the Greek, hence the name. Here I will use capital letters for the Latin, and lower-case Latin letters for the Greek. The task now is to put one Latin and one Greek letter in each box, so that no two pairs are the same (equivalently, so that all N² possible pairs appear in the array), and such that in each row, and in each column, all the 2N elements are different.

A a B b C c

B c C a A b

C b A c B a
is a 3 x 3 example, but there is no example of size 2, as is easy to check.

Latin and graeco-latin squares have an important application to the statistical theory of the design of experiments.

They also have attracted the attention of mathematicians since Euler, who conjectured that there was no graeco-latin square of size 2 plus a multiple of 4. He was right for 2 and 6, but wrong otherwise.

This is the first of two columns on this topic. Here we will look at the application of latin squares to statistics. The next column will look at some aspects of their mathematical study.

--Tony Phillips
Stony Brook