## Latin Squares in Practice and in Theory II

**Feature Column Archive**

## 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*.