For example, Mary from Team One, has a meeting with Tom from Team Two
on Monday (the intersection of the M row with the T column is a = Monday).
The Latin square arrangement ensures a meeting takes place between each pair
of team members and there is no clash of dates.
This is not the only possible 3×3 Latin square. If we interpret A, B and C as
topics discussed at the meetings between Team One and Team Two, we can
produce a Latin square which ensures each person discusses a different topic with
a member of the other team.
So Mary from Team One discusses topic C with Ross, topic A with Sophie and
topic B with Tom.
But when should the discussions take place, between who, and on what topic?
What would be the schedule for this complex organization? Fortunately the two
Latin squares can be combined symbol by symbol to produce a composite Latin
Square in which each of the possible nine pairs of days and topics occurs in
exactly one position.
Another interpretation for the square is the historical ‘nine officers problem’ in
which nine officers belonging to three regiments a, b and c and of three ranks A,
B and C are placed on the parade ground so that each row and column contains
an officer of each regiment and rank. Latin squares which combine in this way
are called ‘orthogonal’. The 3×3 case is straightforward but finding pairs of
orthogonal Latin squares for some larger ones is far from easy. This is something
Euler discovered.
In the case of a 4×4 Latin Square, a ‘16 officers problem’ would be to arrange
the 16 court cards in a pack of cards in a square in such a way that there is one
rank (Ace, King, Queen or Jack) and one suit (spades, clubs, hearts or diamonds)