Orthogonal Latin Square

Question

Find a Latin square orthogonal to the following Latin square:

    0 2 1 3
    2 0 3 1
    3 1 2 0
    1 3 0 2

I have done this by using trial and error. But my lecturer stated that if the Latin square is larger, it is quite difficult to use trial and error. And she gave me the hint that it can be solved by using the relation of Latin square and magic square.

Answer 1

It's true it can be quite difficult to use trial and error to find orthogonal mates for Latin square. However, that doesn't mean there's a better method.

It can even be impossible to find a mate: bachelor Latin squares (those without orthogonal mates) exist for all orders except $1$ and $3$. For example, the Cayley table of $\mathbb{Z}_{2n}$ has no orthogonal mate, for all $n \geq 1$.

I suspect there is some misinterpretation as to what the lecturer was talking about here.

One relationship between orthogonal Latin squares and magic squares is that if $L$ and $M$ are orthogonal Latin squares of order $n$, then $nL+M$ is a magic square.