I still think the number of liberties is the key.
(Is it ok to submit two koans at a time?)
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I hope you donāt mind, I put all koans on the same line and made them all a bit smaller, so that I can fit them on the same screen, instead of having to scroll.
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Iāve looked at
- number of black stones vs number of white stones
- number of black chains vs number of white chains
- number of blackās liberties vs number of whiteās liberties (=empty intersections next to white stones)
but I didnāt find anything interesting.
data
c Green Koans
bs 2, bc 2, bl 8, ws 2, wc 2, wl 8
bs 4, bc 1, bl 10, ws 4, wc 1, wl 10
bs 2, bc 2, bl 8, ws 2, wc 2, wl 8
bs 3, bc 1, bl 8, ws 3, wc 1, wl 8
bs 2, bc 2, bl 4, ws 2, wc 2, wl 4
bs 3, bc 3+, bl 9, ws 3, wc 2, wl 9
bs 4, bc 1-, bl 8, ws 4, wc 3, wl 8
bs 2+, bc 2+, bl 6+, ws 1, wc 1, wl 2
bs 3, bc, 1, bl 5, ws 3, wc 1, wl 5
bs 1, bc 1, bl 3, ws 1, wc 1, wl 3
bs 3, bc 1, bl 5, ws 3, wc 1, wl 5
bs 32+, bc 1, bl 20, ws 16, wc 1, wl 20
bs 6, bc 2, bl 10, ws 6, wc 2, wl 10
bs 4, bc 3, bl 13+, ws 4, wc 3, wl 12
Red Koans
bs 4, bc 3+, bl 8, ws 4, wc 1, wl 8
bs 2+, bc 2+, bl 3, ws 1, wc 1, wl 3
bs 4-, bc 4, bl 11-, ws 6, wc 4, wl 14
bs 28+, bc 4+, bl 24, ws 16, wc 1, wl 24
bs 4, bc 3+, bl 8, ws 4, wc 1, wl 8
bs 7, bc 4+, bl 13, ws 7, wc 2, wl 13
bs 8, bc 6+, bl 16, ws 8, wc 2, wl 16
bs 2-, bc 2+, bl 4-, ws 7, wc 1, wl 14
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I was quite excited with āWhenever a white group touches a black group, the number of black stones should be larger than or equal to the number of white stonesā
Until I discovered that I had made a mistake with 25, which doesnāt follow that ruleā¦
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Number 25 is extremely upsetting to me, and I refuse to accept it as canon 
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25 does seem like the exception here. Why is it red?
Black is in the majority or equal in a number of different counts in green koans, while white is in the majority in red koans, except for that one.
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A few more ideas that almost work:
- Capturing a black stone takes at least as many moves as capturing a white stone.
(27 is an exception)EDIT oh, maybe this works? - Capturing the largest black group takes at least as many moves as capturing the largest white group. (17 is an exception)
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For your first rule, 27 is not an exception?
Black can be captured in 3 moves, but it takes 4 for white.
I donāt see any exception to that rule yet.
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If this were the case, shouldnāt this koan be undecideable:
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Right, itās not well-defined when there are no capturable groups. Maybe they would count as equal infinities? Or maybe the white one is ā+1 and black is ā+9? But even before getting to that, we could test whether itās about capturing or liberties with something like this:
(Whiteās group with 6 liberties canāt be captured in 6 moves.)
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Sure why not 
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The minimum of blackās liberties per chain is greater than or equal to the minimum of whiteās liberties per chain?
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You got it! 
My formulation:
There is a white chain with the minimum number of liberties among all chains.
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hmm, those two rules are not exactly alike.
how would you rule this?
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Green:
True, the difference hinges on how the minimum of an empty set is defined (relevant for @Jon_Ko s rule). But I thought of this as an āedge caseā that could be understood one way or another, and that @Jon_Ko s answer was good enough for me
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Almost always the most useful definition is to make the minimum of the empty set to be larger than any possible value, and the maximum of the empty set to be smaller than any possible value. That is, this would indeed be green.
I guess, technically, one should call it the infimum and the supremum, though, in those cases. Minima and maxima are expected to be contained in the set.
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Can I do the next one?
Green Koans
Red Koans
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This is counterintuitive at first glance but on reflectionā¦
Regardless, yes it is an edge case.
On to the next one! Maybe we can crack it in less than a month this time.
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