Your third image is already red I think.
this but rotated. The others are interesting 
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Correct, the boards are, among other things, rotation and reflection invariant.
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Just summarizing some observations:
There are cases where adding a single white stone makes a red board green.
There are cases where adding a single black stone makes a red board green.
There are cases where adding a single white stone makes a green board red.
I can’t find an example yet where adding a single black stone makes a green board red. (I bet there is one though)
Maybe this could be helpful in excluding some possible rules (not that we have an abundance of possible rules to choose from).
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I feel fairly confident about part of the answer being
“Black has at least as many groups as white…” but/except a few cases or rules.
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Wait so if a rotation it wrong the other one is too.But there is a example which if we rotate something it is ok but it is also not ok.
Such as?
I’m guessing that the tendency of black having at least as many groups as white falls out in some way out of the more general rule (but since we have no idea what that is right now, I don’t have any better plan than trying to come up with a rule of your form).
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Red?
Green?
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Correct
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Both green
(you don’t have to ping me, I’ve already got the thread on “watching” and get automatically pinged with each message)
I’ve added one extra:
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Both green 
You just did the first one, but the second one is red indeed.
You’re probably right.
These guys make it seem like green if black has more groups than white. Less or the same could have some rule attached.
Maybe it does come out simpler from some more general rule.
Oh wait nevermind (again) - number of groups is out.
Wait I forgot to save the other one!
I guess I have to know about this one
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Red
But:
