in each row and column. In 1782 Euler posed the same problem for ‘36 officers’.
In essence he was looking for two orthogonal squares of order 6. He couldn’t
find them and conjectured there were no pairs of orthogonal Latin squares of
orders 6, 10, 14, 18, 22 ... Could this be proved?
Along came Gaston Tarry, an amateur mathematician who worked as a civil
servant in Algeria. He scrutinized examples and by 1900 had verified Euler’s
conjecture in one case: there is no pair of orthogonal Latin squares of order 6.
Mathematicians naturally assumed Euler was correct in the other cases 10, 14,
18, 22 ...
In 1960, the combined efforts of three mathematicians stunned the
mathematical world by proving Euler wrong in all the other cases. Raj Bose,
Ernest Parker and Sharadchandra Shrikhande proved there were indeed pairs of
orthogonal Latin squares of orders 10, 14, 18, 22, ... The only case where Latin
squares do not exist (apart from trivial ones of orders 1 and 2) is order 6.
We’ve seen that there are two mutually orthogonal order 3 Latin squares. For
order 4 we can produce three squares which are mutually orthogonal to each
other. It can be shown that there are never more than n − 1 mutually orthogonal
Latin squares of order n, so for n = 10, for example, there cannot be more than
nine mutually orthogonal squares. But finding them is a different story. To date,
no one has been able to even produce three Latin squares of order 10 that are
mutually orthogonal to each other.
Are Latin squares useful?
R.A. Fisher, an eminent statistician, saw the practical use of Latin squares. He
used them to revolutionize agricultural methods during his time at Rothamsted
Research Station in Hertfordshire, UK.
Fisher’s objective was to investigate the effectiveness of fertilizers on crop
yield. Ideally we would want to plant crops in identical soil conditions so that soil
quality wasn’t an unwanted factor influencing crop yield. We could then apply the
different fertilizers safe in the knowledge that the ‘nuisance’ of soil quality was
eliminated. The only way of ensuring identical soil conditions would be to use the
same soil – but it is impractical to keep digging up and replanting crops. Even if
this were possible different weather conditions could become a new nuisance.
A way round this is to use Latin squares. Let’s look at testing four treatments.