Latin square puzzle vs Gemini thinking mode

Community
1

Hello all!

Ever since AI with reasoning capabilities came out, I kept coming back to this old puzzle relating to the Mensa logo I saw many years ago…
Neither I nor AI, not even Gemini 2.0 Thinking mode seems do deal with this… I really thought Gemini 2.0 Thinking mode could provide some helpful observations, but eh, it didn’t, and it didn’t even follow my suggestions to exclude trying some solutions :frowning:

So I was told the arrangement of colors in a Mensa logo was not random… I could translate these colors into a square which turned out to be a latin square, but I couldn’t figure out any pattern here… here’s the latin square with color indices:

0 1 2 3 4
1 0 3 4 2
3 4 1 2 0
2 3 4 0 1
4 2 0 1 3

I know this is not the simplest latin square possible, but I (just like Gemini and o1) fail to see why it isn’t just a random latin square.

I’m losing hope that AI could really deal with this, but I know that Google engineers that’d stumble upon this post could do it easily… perhaps you could help a bit? :slight_smile:
Or maybe I’m not giving Gemini the most optimal prompt for this task?

(yes I see the “2013” part, but these were colors so I’d be surprised to learn it was an encoded message that relied on color index representation)

2

Welcome to the forum, and thank you for the puzzle.

I am not affiliated with Google and not in Mensa, so no insider information here. The first thing to notice is, the Latin square is one permutation away from being reduced, if the middle and next-to-last rows are swapped, the first column is ordered in ascending order (same order as the first row).

Then, it becomes the interesting square that features in the Wikipedia article Small Latin squares and quasigroups - Wikipedia , the representative of the second isotopy class (the class that isn’t the boring cyclic group of order 5).

I don’t know for sure if that is the point of the logo; any other observations are very welcome.

3

Thanks for your observations!

Flipping these rows makes sense, given that they were inverted vertically in the original image and I’d already flipped those that were inverted horizontally.

This is the original logo (if I can upload it):

image
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