Perfect Latin Squares

Question

The following Latin Square has an interesting property: there are 6*5=30 possible “ordered dominos” containing distinct digits, each occurring exactly once horizontally and exactly once vertically. For instance 36 (63) occurs horizontally in the top (bottom) row and no other. Also 36 (63) appears vertically in column Two (Five) and no other etc. Let us say that such a Latin Square is perfect.
I can’t find any perfect Latin Squares of odd order except for the trivial case of 1x1. Similarly, I can’t find any perfect non-symmetrical Latin Squares of even order. Can you find an example of either of the above, or conversely prove that none exists?
Special thanks to my friend @DmitryKamenetsky for a similar problem about painting a rectangle with K different colours.
I don’t know the answer to this puzzle, so this is a chance to prove you are one of the Awesome People ?

Text version of image:
1   4   3   6   5   2
6   1   5   4   2   3
5   3   1   2   6   4
4   6   2   1   3   5
3   2   4   5   1   6
2   5   6   3   4   1

Paul Panzer · Answer

Here are perfect latin squares of sizes 8,10,12,14 and 16 all as far as I can tell without obvious symmetries. Sorry about the formatting, at least it's copy-n-paste friendly (you wouldn't want to check them by visual inspection anyway, I assume).

How they were constructed and why this method doesn't work for odd sizes:

AxiomaticSystem · Answer

I don't know how to enumerate all the "perfect Latin squares", so I started off by enumerating all the possible templates: sets of rows satisfying the domino and Latin criteria. A template in itself creates a perfect Latin square if it is symmetric around either the center or the main diagonal, but it does not rule out the possibility that some permutation of rows, columns, and/or numbers creates another possibly-asymmetric perfect square, with row set and column set possibly belonging to different templates.

Answered by AxiomaticSystem on September 4, 2021

Vassilis Parassidis · Answer

In the Latin square below you read the numbers always in pairs. When you read horizontally you start from left to right, when you read vertically you start from top to bottom.
524163
156234
645312
213546
432651
361425
I hope this satisfies your conditions.
Here is another square in response to your comment:
362154
135642
523416
614325
246531
451263

Perfect Latin Squares

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