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I cannot find a proof for the following theorem anywhere: A latin square has an orthogonal mate iff it can be decomposed into disjoint transversals.

Could you perhaps link me to one?

Also, how can we derive from this whether we can, for latin square of any order $n$, find a latin square that has no orthogonal pairing?

I think the result might involve the use of finite fields.

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See the first paragraph of this paper for the first question. The second question is also discussed in that paper. For $n$ even, the Cayley table of the integers modulo $n$ has no transversals. For odd $n$ it's a conjecture of Ryser that there is always a transversal.

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