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\homework{15: Finite Fields and Latin Squares}{Due Monday, Week 9}

Solve \textbf{one} of the following \textbf{three} problems.    As always, prove your claims/have fun!  Also, this set is \textbf{extra-credit}, because of the holiday!  Take care, and I'll see you all in a week!
--Paddy


\begin{enumerate}
\item Given a Latin square of even order $2n$, must it have an orthogonal mate?  In other words, for any even number $2n$, can you find a Latin square $L$ that is not orthogonal to any other $2n \times 2n$ Latin square?   

(For $n = 2, 6$, this is trivially true because there are no pairs of MOLS of order 2 or 6.  What about other values of $2n$?)

\item Given a pair of mutually orthogonal Latin squares $(A,B)$ of order $m$ and another pair of mutually orthogonal Latin squares $(C,D)$ of order $n$, create a pair of mutually orthogonal Latin squares $(X, Y)$ of order $mn$.
\item Write a computer program that, given inputs $n, k$, will try to find $k$ mutually orthogonal Latin squares of order $n$.  For what values of $n, k$ does your program finish running in (say) under half a hour?
\end{enumerate}

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