Greco-Latin square

Size 5 Greco-Latin square

A Greco-Latin square (GLQ) or Euler's square of size n is a square scheme with n  rows and n  columns, in which one character from a set G and one from another set  L is entered in each of the fields  . It is also called the orthogonal Latin square. ${\displaystyle n\cdot n}$

Each element from G and also each element from L must appear exactly once in each row and also in each column , and each tuple must appear exactly once in the entire square. ${\displaystyle (g,l)\in G\times L}$

A GLQ is a generalization of the so-called Latin square . While the Latin square is about a quantity, the GLQ is about two quantities. The concept was developed by Leonhard Euler introduced which for the quantity G letters of the Greek and L letters of the Latin alphabet used. This is where the name came from.

In the 1780s Euler found methods for constructing GLQ with the odd or divisible quantity n . However, he did not succeed in finding solutions. The case has become known as the problem of the 36 officers : six regiments each provide six officers with six different ranks, and they are supposed to line up in a 6 × 6 square so that in each row and in each column each regiment and each rank once occurs. ${\displaystyle n\equiv 2\;({\bmod {\,}}4)}$ ${\displaystyle n=6}$

Euler accordingly suspected that there is a GLQ if and only if . That there is no solution was shown by Gaston Tarry in 1901 , but in 1959 RC Bose and SS Shrikhande constructed counterexamples with and ET Parker with . Parker, Bose and Shrikhande proved that there is a GLQ for all sizes except and . ${\displaystyle n\not \equiv 2\;({\bmod {\,}}4)}$ ${\displaystyle n=6}$ ${\displaystyle n=22}$ ${\displaystyle n=10}$ ${\displaystyle n=2}$ ${\displaystyle n=6}$

Statistical design of experiments

An agronomist wants to find out which fertilizer concentration and amount of irrigation will maximize the yield of his crops. For this he divided his field in time individual areas. In each of the regions is one of the mineral concentrations , , or with and one of the irrigation quantities , , or with used. The growing conditions can differ depending on the position in the field, which is why block factors are necessary. Together with the two factors of interest, fertilizer concentration and amount of irrigation, this results in factors with each factor level. A randomized Latin square as a statistical test plan can be generated in R with the function from the package agricolae. Deviating from the definition above, the levels of the second factor are indicated in lowercase Latin letters in the following design. ${\displaystyle 4}$ ${\displaystyle 4}$ ${\displaystyle 16}$ ${\displaystyle 4}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle D-C=C-B=B-A>0}$ ${\displaystyle 4}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle d}$ ${\displaystyle d-c=c-b=b-a>0}$ ${\displaystyle (x,y)}$ ${\displaystyle 2}$ ${\displaystyle 4}$ ${\displaystyle 4}$design.graeco

A a	B b	C c	D d
B c	A d	There	C b
C d	D c	A b	B a
D b	C a	B d	A c

In this test plan, each factor level combination of the two factors of interest fertilizer concentration and amount of irrigation occurs exactly once, so that only the main effects can be estimated. If one is also interested in interactions, each factor level combination should be carried out several times in the experiment.

literature

• Victor Bryant: Aspects of Combinatorics: A Wide-ranging Introduction. Cambridge University Press, 1993, ISBN 0-521-42997-8 .

Web links

• Eric W. Weisstein : Euler Square . In: MathWorld (English).
• Lars Dovling Andersen: The History of Latin Squares . (PDF; 1.20 MB) Aalborg University
• Christoph Pöppe: Noble magic squares. (No longer available online.) In: Spektrum der Wissenschaft 1. 1996, p. 14 , formerly in the original ; Retrieved February 28, 2012 .  ( Page no longer available , search in web archives )  
• Spektrum.de
• free web application and source code for Euler Square
  • Analysis of Euler's squares / Graeco-Latin squares of sizes 1 × 1 to 10 × 10 for Firefox compatible browsers and HTML5 mobile devices
  • It includes more background information about Graeco-Latin squares in English

Individual evidence

1. ↑ Felipe de Mendiburu: agricolae: Statistical Procedures for Agricultural Research. June 12, 2016. Retrieved March 9, 2017 .