Latin-Graeco Squares contain
combinations of two sets of n symbols
combined in such a way that neither pair
of symbols is repeated in the square and
that neither symbol is repeated in the
same row or column.
Aα
Bγ
Cβ
Bβ
Cα
Aγ
Cγ
Aβ
Bα
In a 1782 paper Euler considers the 36 officers problem. This
problem states that there are 6 different ranks and 6 different
regiments. The problem then poses the question whether the
officers can be arranged in a 6×6 grid with no two officers from
the same regiment or of the same rank in the same row or
column. Euler denoted the regiments of the officers with a Latin
letter and the rank with a Greek letter and began by arranging the
officers in regiment to create a Latin square. The characteristics
of the officer would be made up by a combination of the two
letters creating a Latin-Graeco square. However the creation of
an order 6 Latin-Graeco square proved to be impossible…
This lead Euler to suggest that Latin-Graeco squares did not
exist for squares of the order n where:
n = 4k + 2
Therefore Latin-Graeco squares could not be created of the
order n = 2,6,10,…
This conjecture had however not been proved leading to
many attempts to prove the conjecture true or false. It was
known that squares to the order 2 and 6 could not be
constructed and therefore assumed by many that the theory
was correct.
However in 1959 Bose, Shrikhande and Parker found a
square of order 10 and several other counter examples of
higher numbers which fitted the pattern of n = 4k +2.
Therefore proving Euler’s conjecture wrong.
It has since been found that all orders greater than 10 can
form a Graeco-Latin square. Therefore showing that
squares can be formed for all orders n ≥ 3 (n ≠ 6)
Latin-Graeco squares are constructed by combining two orthogonal Latin
squares. Orthogonal implies that the values in the squares are perpendicular
with only one square of each colour being in the same place. This can be
seen best in the creation of a square of the order 3:
The main use of Graeco-Latin squares is in the design of experiments
where there are four variables to take into consideration. As there are
three ways of combining variables. The variables can be combined by
the columns, rows and the combination of the two letters.
In testing the economy of cars there can be four factors which account for the rate of fuel
consumption:
Fuel Type

Car

Weather

Tyre Type

These four factors can be tested in an experiment using a Graeco-Latin square to
design the series of tests.
In this grid
Columns = Different Cars

Rows = Different Weather

Latin Letters = Different Tyres

Greek Letters = Different Octane Levels of Petrol

By carrying out the series of tests in the order
outlined in the table each of the four factors can be
tested against each of the other factors without
repetition.
Car 1 Car 2
Car 3
Day 1
Aα
Bγ
Cβ
Day 2
Bβ
Cα
Aγ
Day 3
Cγ
Aβ
Bα
By increasing the order of the square the number of samples
which are being tested can be increased for example:
A square of order four can be used to test four samples

A square of order five can be used to test five samples
