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Suppose $A$ and $B$ are Latin squares of order $n$. And suppose any column of $A$ and any column of $B$ have common entry only once. Then are $A$ and $B$ mutually orthogonal?

I know the converse is true, but I don't know whether this is true, and how to prove this.

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    $\begingroup$ What do you mean by the converse and by mutual orthogonality? There is an example on Wikipedia of mutually orthogonal Latin squares of order 3, and the first columns are identical: en.wikipedia.org/wiki/Mutually_orthogonal_Latin_squares $\endgroup$
    – 1001
    Commented May 20, 2023 at 19:46
  • $\begingroup$ I forgot to mention this. They are Mutually orthogonal latin squares of order n and their first row is fixed by (1,2,...,n). Then I think converse is true. $\endgroup$
    – Lim do
    Commented May 21, 2023 at 2:33
  • $\begingroup$ Do you also require the fixed first row property for the original question? $\endgroup$
    – 1001
    Commented May 21, 2023 at 17:52
  • $\begingroup$ Yes. A and B should have same first row. (Both original question and converse) $\endgroup$
    – Lim do
    Commented May 23, 2023 at 2:02
  • $\begingroup$ Please edit the clarification into the original question (it makes it more visible, and save readers the time of having to go through the comments). $\endgroup$
    – Willie Wong
    Commented Jun 1, 2023 at 15:00

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