# Latin Squares in Practice and in Theory II

## 2. Orthogonal Latin Squares

A *latin square* of size `n` is an array of `n`
copies each of `n` different objects
(typically the latin letters A, B, C, ...)
so that all the objects in any row, and all the objects in any
column, are different. Two size `n` latin squares,
one with objects A, B, C, ...,
one with objects a, b, c, are *orthogonal* if superimposing
them
leads to a square array containing all `n`^{2}
possibile pairs (A,a), (A,b), ... , (B,a), (B,b), ..., ... .
For example, the two `5 x 5` latin squares

are orthogonal: they can be superimposed to give every possible
combination of rank and color

If we could find two orthogonal latin squares of size 6, they
would combine to give a solution to Euler's problem of the 36
officers. So an equivalent statement to the impossibility of
solving that problem is: *There are no two orthogonal latin
squares of size 6*.