Given two Latin squares $L_1=(A,B,C_1,f_1)$ and $L_2=(A,B,C_2,f_2)$ of the same order $n$ we can combine them coordinate-wise to form a single square, whose cells are ordered pairs of elements from $C_1$ and $C_2$ respectively. Formally, we can form a function $f:A\times B\to C_1\times C_2$ given by $$f(i,j)=(f_1(i,j),f_2(i,j)).$$ This function $f$ says that we have created a new square $A\times B$ whose cell $(i,j)$ contains the ordered pair of values, the first coordinate of which corresponds to the value in cell $(i,j)$ of $L_1$ and the second to the value in cell $(i,j)$ of $L_2$ We may write the combined square $L_1*L_2$

For example,

\begin{equation*} \left(\begin{array}{cccc} a & b & c & d\\ c & d & a &b\\ d & c & b & a\\ b & a & d & c \end{array}\right) * \left(\begin{array}{cccc} 1 & 2& 3& 4\\ 4 & 3 & 2 & 1\\ 2 & 1 & 4 & 3\\ 3 & 4 & 1 & 2 \end{array}\right) = \left(\begin{array}{cccc} (a,1) & (b,2) & (c,3) & (d,4)\\ (c,4) & (d,3) & (a,2) & (b,1)\\ (d,2) & (c,1) & (b,4) & (a,3)\\ (b,3) & (a,4) & (d,1) & (c,2) \end{array}\right) \end{equation*} In general, the combined square is not a Latin square unless the original two squares are equivalent: $f_1(i,j)=f_1(k,\ell)$ iff $f_2(i,j)=f_2(k,\ell)$ Nevertheless, the more interesting aspect of pairing up two Latin squares (of the same order) lies in the function $f$

Definition. We say that two Latin squares are orthogonal if $f$ is a bijection.

Since there are $n^2$ cells in the combined square, and $| C_1\times C_2| = n^2$ the function $f$ is a bijection if it is either one-to-one or onto. It is therefore easy to see that the two Latin squares in the example above are orthogonal.

Remarks.

• The combined square is usually known as a Graeco-Latin square, originated from statisticians Fischer and Yates.
• It can be shown that if $L_1,\ldots, L_m$ are Latin squares of order $n\ge 3$ such that each pair of them are orthogonal, then $m\le n-1$ If the equality occurs, then the set of Latin squares are said to be complete.
• (Bose) If $n\ge 3$ then $L_1,\ldots,L_m$ form a complete set of pairwise orthogonal Latin squares of order $n$ iff there exists a finite projective plane of order $n$

Bibliography

1

H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, 1963