A Graeco-Latin square design is a design of experiment in which the experimental units are grouped in three different ways. It is obtained by superposing two Latin squares of the same size. If every Latin letter coincides exactly once with a Greek letter, the two Latin square designs are orthogonal.
How do you make a Graeco-Latin square?
To create a Graeco-Latin square, we add a second dimension, superimposing a square with Greek letters over the Latin square. The two squares must be mutually orthogonal, meaning that each Greek-Latin pair occurs only once.
How many Latin squares are there?
This example illustrates one of the 576 possible Latin squares for a 4-by-4 layout; larger squares have many orders of magnitude more combinations (e.g., 161,280 for a 5-by-5 layout).
How many mutually orthogonal Latin squares are there?
From the Graeco-Latin square construction, there must be at least two and from the non-existence of a projective plane of order 10, there are fewer than nine. However, no set of three MOLS(10) has ever been found even though many researchers have attempted to discover such a set.
What are the advantages of a Latin square design?
The advantage of the Latin square design is to control the variation from different labels and different experimental runs. The Latin square also provides better efficiency than the RCBD [5].
Why we use Graeco-Latin square design?
Graeco-Latin squares, as described on the previous page, are efficient designs to study the effect of one treatment factor in the presence of 3 nuisance factors. They are restricted, however, to the case in which all the factors have the same number of levels.
What is split plot design?
The split-plot design is an experimental design that is used when a factorial treatment structure has two levels of experimental units. The whole plot is split into subplots, and the second level of randomization is used to assign the subplot experimental units to levels of treatment factor B.
What does mutually orthogonal mean?
“Mutually orthogonal” means that the dot product of any pair of distinct vectors in the set is 0.
What is meant by orthogonal array?
In mathematics, an orthogonal array is a “table” (array) whose entries come from a fixed finite set of symbols (typically, {1,2,…,n}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row …
What was the purpose of the Graeco Latin squares?
Graeco-Latin squares are a fascinating example of something that developed first as a puzzle, then as a mathematical curiosity with no practical purpose, and ultimately ended up being very useful for real-world problems. As early as 1725, Graeco-Latin squares existed as a puzzle with playing cards.
Can you make a Latin square with Greek letters?
You can see that Latin squares are not difficult to create, and the number of possible permutations increases with the size of the square. To create a Graeco-Latin square, we add a second dimension, superimposing a square with Greek letters over the Latin square.
How many different types of Latin squares are there?
The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflexion, and/or permutation of the symbols) for n = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively (sequence A264603 in the OEIS) . We give one example of a Latin square from each main class up to order 5.
When do you call a set of Latin squares a complete set?
When it does have that many, it is also called a complete set of mutually orthogonal Latin squares. Graeco-Latin squares have an interesting link to finite projective planes: a Hyper-Graeco-Latin square of order n can have n-1 dimensions if and only if a finite projective plane of order n exists.
