Euler's Conjecture on Orthogonal Latin Squares

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Conjecture

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Two orthogonal Latin squares do not exist for order $4 n + 2$.


Refutation

While the conjecture is true for order $6$, it fails for order $10, 14, \ldots$.

The following is an example of two orthogonal Latin squares presented in the same array.

The elements of the first Latin square are shown in normal type, while those of the second Latin square are shown in italic type:

$\begin{array} {|c|c|c|c|c|c|c|c|c|c|} \hline 4 \mathit 6 & 5 \mathit 7 & 6 \mathit 8 & 7 \mathit 0 & 8 \mathit 1 & 0 \mathit 2 & 1 \mathit 3 & 2 \mathit 4 & 3 \mathit 5 & 9 \mathit 9 \\ \hline 7 \mathit 1 & 9 \mathit 4 & 3 \mathit 7 & 6 \mathit 5 & 1 \mathit 2 & 4 \mathit 0 & 2 \mathit 9 & 0 \mathit 6 & 8 \mathit 8 & 5 \mathit 3 \\ \hline 9 \mathit 3 & 2 \mathit 6 & 5 \mathit 4 & 0 \mathit 1 & 3 \mathit 8 & 1 \mathit 9 & 8 \mathit 5 & 7 \mathit 7 & 6 \mathit 0 & 4 \mathit 2 \\ \hline 1 \mathit 5 & 4 \mathit 3 & 8 \mathit 0 & 2 \mathit 7 & 0 \mathit 9 & 7 \mathit 4 & 6 \mathit 6 & 5 \mathit 8 & 9 \mathit 2 & 3 \mathit 1 \\ \hline 3 \mathit 2 & 7 \mathit 8 & 1 \mathit 6 & 8 \mathit 9 & 6 \mathit 3 & 5 \mathit 5 & 4 \mathit 7 & 9 \mathit 1 & 0 \mathit 4 & 2 \mathit 0 \\ \hline 6 \mathit 7 & 0 \mathit 5 & 7 \mathit 9 & 5 \mathit 2 & 4 \mathit 4 & 3 \mathit 6 & 9 \mathit 0 & 8 \mathit 3 & 2 \mathit 1 & 1 \mathit 8 \\ \hline 8 \mathit 4 & 6 \mathit 9 & 4 \mathit 1 & 3 \mathit 3 & 2 \mathit 5 & 9 \mathit 8 & 7 \mathit 2 & 1 \mathit 0 & 5 \mathit 6 & 0 \mathit 7 \\ \hline 5 \mathit 9 & 3 \mathit 0 & 2 \mathit 2 & 1 \mathit 4 & 9 \mathit 7 & 6 \mathit 1 & 0 \mathit 8 & 4 \mathit 5 & 7 \mathit 3 & 8 \mathit 6 \\ \hline 2 \mathit 8 & 1 \mathit 1 & 0 \mathit 3 & 9 \mathit 6 & 5 \mathit 0 & 8 \mathit 7 & 3 \mathit 4 & 6 \mathit 2 & 4 \mathit 9 & 7 \mathit 5 \\ \hline 0 \mathit 0 & 8 \mathit 2 & 9 \mathit 5 & 4 \mathit 8 & 7 \mathit 6 & 2 \mathit 3 & 5 \mathit 1 & 3 \mathit 9 & 1 \mathit 7 & 6 \mathit 4 \\ \hline \end{array}$

Note that all integers from $0 \mathit 0$ to $9 \mathit 9$ appear exactly once each.

$\blacksquare$


Source of Name

This entry was named for Leonhard Paul Euler.


Historical Note

Euler made this conjecture in $1782$.

It was refuted $1959$ by Raj Chandra Bose, Sharadchandra Shankar Shrikhande and Ernest Tilden Parker.

This discovery was significant enough to be reported in the mainstream news of the time. As a result, Bose, Shrikhande and Parker were given the nickname Euler's spoilers.


Sources

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