# Mutually orthogonal latin squares of order $mn$ from order $m$ and order $n$

Given a pair of mutually orthogonal latin squares (MOLS) of order $m$ and a pair of MOLS of order $n$, how would we construct a pair of MOLS of order $mn$?

[EDIT: MOLS means Mutually Orthogonal Latin Squares]

-
@Gerry: Yes, I added the parenthetical "MOLS" right after the first use to clarify that (original post contained only "MOLS", no explanation) –  Arturo Magidin Jul 13 '12 at 3:35

Let $M_1$ and $M_2$ represent the pair of MOLS of order $m$. Let $N_1$ and $N_2$ represent the pair of MOLS of order $n$.Take the cross product of $M_1$ and $N_1$ to produce one of the MOLS of order $mn$ and then take the crossproduct of $M_2$ and $N_2$ to produce the $2$-nd MOL of order $mn$. These two will then give you a pair of MOLS of order $mn$.