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Can we find two mutually orthogonal diagonal latin squares of orders $4$ and $8$? A diagonal Latin square is a Latin square of order $n$ where the symbols $1$ thru $n$ fil both the forward diagonal and the back diagonal? It'd be great if you could show me the actual MOLS as well as the methodology.

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    I added an 8x8 example to my answer.2012-07-27

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