Orthogonal Latin Squares of Order 6 do not Exist

Theorem

Two orthogonal Latin squares of order $6$ do not exist.

Proof

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Also see

• Euler's Conjecture on Orthogonal Latin Squares

Historical Note

This problem was posed by Leonhard Paul Euler, who couched it as follows:

Place $36$ officers,

comprising a colonel, lieutenant-colonel, major, captain, lieutentant and sub-lieutenant

from each of $6$ regiments,

in a square array

so that no rank or regiment will be repeated in any row or column.

This turns out to be impossible to do.

This was not proved until Gaston Tarry achieved it in $1901$.

Sources

• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The Thirty-six Officers Problem