TITLE: Are there complete Graeco-Latin squares?
QUESTION [21 upvotes]: Are there two orthogonal complete Latin squares of any order greater than 1? If so, what is the smallest order for which they exist?
(A Latin square of order $n$ is an $n\times n$ array of symbols $\{ s_1\ldots s_n \}$ such that each of the symbols appears exactly once in each row and in each column.  Two Latin squares $L_{ij}$ and $G_{ij}$ are orthogonal if each of the $n^2$ pairs $(L_{ij}, G_{ij})$ is distinct; such a pair together form a Graeco-Latin square.  A Latin square  $L_{ij}$ is complete if each of the $n\cdot(n-1)$ pairs $(L_{ij}, L_{i+1,j})$ is distinct and if each of the $n\cdot(n-1)$ pairs $(L_{ij}, L_{i,j+1})$ is distinct.)

REPLY [51 votes]: The problem is open no longer. It turns out that there are many pairs
of orthogonal complete Latin squares of order 12.
Here is one example. I also know how to build some bigger ones.
\begin{pmatrix}
  1 & 2 & 5 & 7 & 4 & 8 & 9 &11 & 6 &12 &10 & 3\\
  7 &12 & 9 & 3 &10 & 2 & 1 & 5 & 8 & 4 & 6 &11\\
  2 & 3 & 6 & 8 & 5 & 9 &10 &12 & 1 & 7 &11 & 4\\
  8 & 7 &10 & 4 &11 & 3 & 2 & 6 & 9 & 5 & 1 &12\\
 10 & 9 &12 & 6 & 7 & 5 & 4 & 2 &11 & 1 & 3 & 8\\
  3 & 4 & 1 & 9 & 6 &10 &11 & 7 & 2 & 8 &12 & 5\\
  9 & 8 &11 & 5 &12 & 4 & 3 & 1 &10 & 6 & 2 & 7\\
 12 &11 & 8 & 2 & 9 & 1 & 6 & 4 & 7 & 3 & 5 &10\\
 11 &10 & 7 & 1 & 8 & 6 & 5 & 3 &12 & 2 & 4 & 9\\
  6 & 1 & 4 &12 & 3 & 7 & 8 &10 & 5 &11 & 9 & 2\\
  4 & 5 & 2 &10 & 1 &11 &12 & 8 & 3 & 9 & 7 & 6\\
  5 & 6 & 3 &11 & 2 &12 & 7 & 9 & 4 &10 & 8 & 1
\end{pmatrix}
\begin{pmatrix}
  1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 &11 &12\\
  6 & 4 &11 & 2 & 8 & 1 & 9 & 5 & 7 &12 & 3 &10\\
  4 &10 & 7 & 1 &11 & 9 & 3 &12 & 6 & 2 & 5 & 8\\
  9 & 1 & 5 &10 &12 & 4 & 6 &11 & 3 & 8 & 7 & 2\\
  2 &12 & 9 & 6 & 3 & 7 &11 &10 & 1 & 4 & 8 & 5\\
  3 & 9 &12 & 8 & 2 &10 & 4 & 7 & 5 &11 & 6 & 1\\
 10 & 8 & 6 & 9 & 7 & 3 & 5 & 2 & 4 & 1 &12 &11\\
  5 & 3 & 2 &11 & 1 & 8 &10 & 6 &12 & 7 & 4 & 9\\
 11 & 7 &10 & 5 & 4 &12 & 2 & 9 & 8 & 3 & 1 & 6\\
  8 &11 & 4 & 3 & 6 & 5 &12 & 1 &10 & 9 & 2 & 7\\
  7 & 6 & 8 &12 &10 & 2 & 1 & 3 &11 & 5 & 9 & 4\\
 12 & 5 & 1 & 7 & 9 &11 & 8 & 4 & 2 & 6 &10 & 3
\end{pmatrix}