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+--- Thema: Does this work as a puzzle? Graeco-Latin squares (/showthread.php?tid=2238)
Does this work as a puzzle? Graeco-Latin squares - Screature - 23.02.2025
I'm terrible at explaining things as I tend to waffle, so this is going to be difficult, apologies in advance. Please see attachments.
I've got a generator that creates graeco-latin squares (or rather, as the depth is greater than 2, the correct term would be mutually orthogonal Latin squares MOLS)
Using the image on the left which is a 5x5 MOLS square of depth 4, I translated it into the grid in the middle. Each 2x2 box is a representation of each cell of the square, with 1 being cyan, 2 is purple, 3 is red, 4 is green, 5 is orange. The topleft 2x2 of the grid is the topleft cell of the image, so 1 5 4 4 is cyan orange green green. Compare that to the image on the left to see how it's been translated. The next 2x2 is 2 3 2 3 which is purple red purple red, then the next is 3 1 1 1 which is red cyan cyan cyan
What makes a MOLS square is that every pair of orthogonal grids is fully unique. What this means for the grid in the middle is that you can pick any pair (of the 6 pair combinations) in each 2x2 box and it'll be unique compared to the likewise pairs in any other 2x2 box. Also the topleft digits in each 2x2 box together form a latin square (e.g. digit 1 appears once in the topleft box in every row and column of 2x2 boxes). Same for topright, bottomleft, bottomright.
Another way of explaining it, is every 1 in the top left of a 2x2 will have digits 1 to 5 to the right of it exactly once throughout the grid. Every 3 in the top left of a 2x2 will have digits 1 to 5 beneath it exactly once throughout the grid. I've highlighted those examples in yellow and green, but that applies to all 12 likewises pairs across all 2x2 boxes (all four digits in a 2x2 box has three other digits to pair with, hence 12 directional pairs)
Knowing how the grids came to be (if you understand so far, well done!), if you were given the grid on the right *on it's own* without the other parts of the image for solution/context, just the rules, would you be able to fill in the missing cells? Is there enough information there to solve it?