I’m still trying to find the significance of this:
Based on this comment I’m assuming empty intersections play some sort of role in the rule:
But I’m stuck on that, so let’s see what happens when we modify that last hint instead:
All green!
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I think that white move makes one less libertie and one more stone , but it might ruin two squares and in the last green example it also had to squares
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I think the rule is that there will be two squares someway.But there are some counter examples but there will be a basic shape(or whatever it is called)in all of the koans.
What about this? The difference is that the black stones are just one chain.
How about these?
I think this is green still.
I think this is all green
Green again 
Once again there are two squares so I think the rule includes something that makes two squares green.
I didn’t read all post concerning this puzzle. Is the board really a quadratical 5x5 board with an edge and angles of 90°?
These are different:
Ah, there is touching involved!
Ok so I think the rule about touching is above the squares green thing.
The board is a 5x5 board as you can see. How the rule works on the board is a different matter though.
However I will say that only valid board positions are Koans, i.e. all chains have at least one liberty.
I think we never asked for confirmation on this, but it seems like we should indeed be looking at chains and liberties in some way.
(… one shouldn’t play Go with mathematicians…)
What about this:
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Maybe the touching made that group of white stone lose one liberty which will make in way different.
This one’s red:
