Interesting, so black and white are not the same!
Rule: Koans need to have an even number of black stones
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Edit: This was supposed to be a reply to @Vsotvep
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I wonder if black stones can only be on the board if white stones are also on the board in a particular arrangement in relation to those black stones.
Guess: Either there are no black stones, or some must be adjacent to white stones.
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Rule: there is a corner of the board, such that the centre of each black stone is contained in the convex hull of the centres of the white stones and said corner.
Looks promising!
Rule guess: The set of white stones can be translated and rotated (all together as one rigid shape) to completely cover all the black stones.
Edit: Ehm, whoops. This holds for all boards so far, both red and green… never mind me.
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Ha, my rule still works 
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Took me a while to think about your rule. I hope these are good counterexamples:
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Let’s do some tests on missing black / white stones:
I’m suspecting that every green koan must have at least as many white stones as black stones, but it’s clearly not the only thing that is at play…
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Guess: every black stone must be adjacent to at least two other stones (and one of those needs to be white) or on the edge of the board.
Oh… Congrats again to martin for spotting these rules. The game is really interesting, I just have no idea how to get better at spotting these patterns!
They’re called vacuously true statements or properties. Like the empty set {} has the property that all of it’s elements are multiples of two.
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Unfortunately the board size of your submitted Koan is bigger than the board size I decided on for this game. I’ve taken the liberty to adjust the pattern to the 5x5 size and this is the result:
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