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A Latin square $M$ is called Self-Orthogonal if $M$ and its transpose $M^{tr}$ form an orthogonal pair.

Which of the following statements are true about self-orthogonal Latin squares?

Select ALL that apply.

There is a $3\times 3$ self-orthogonal Latin square.

There is a $4\times 4$ self-orthogonal Latin square.

If $M=(m_{ij})$ is an $n\times n$ self-orthogonal Latin square, then the diagonal elements:

$$m_{ii}, 1\leq I \leq n$$

...must all be distinct.