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Jul 24, 2023 at 17:28 comment added David E Speyer Remark: I wrote this answer using the standard description of mutually orthogonal Latin squares, but the definition I prefer is that having $d-2$ MOLS's is the same as having an $N^2$ element subset of $[N]^d$ which projects bijectively onto every $2$-dimensional coordinate plane. From this perspective, the subset is given by, for each person, listing the sequence of $d$ tables at which they have sat.
Jul 24, 2023 at 17:23 history edited David E Speyer CC BY-SA 4.0
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Jul 24, 2023 at 16:03 history answered David E Speyer CC BY-SA 4.0