Conjecture: Starting with a red board and removing some of the black stones always keeps it red.
Equivalently: It’s impossible to create a red board from a green one by adding black stones.
Asking for confirmation or a counterexample, if the master deems us deserving 
Thanks 
I just did that size because that was what the website was set to.
Again both red:
Alright, so we already have a counterexample:
-> 
But I’ll give you another one using more stones:
-> 
This one’s also red:
This might be the kind of rule where many board positions are red 
Can we see a green board with more black stones than white stones?
I’m glad you asked! This is a beautiful example:
I’m stumped…
May we see something almost opposite: a red position with only white stones?
Actually you may not, because I’m unable to come up with such a position 
Do you mean that you are unable, or could you prove that anybody is unable?
Hehe, I guess there’s no denying it now: There is no red Koan without black stones.
I look forward to the occasion when someone makes a conjecture that the rulemaker can neither prove nor provide a counterexample to, then we’ve got ourselves another puzzle to look at together once we’ve found the rule 
So every board with only white stones is green. Is every board with only black stones (apart from the empty board) red?
Indeed. At this point I should point out that I’m only considering valid go positions, so there are always some empty intersections.
Ah, thank you for clarifying.
I guess that could arise in several ways:
In the latter case, we might discover it quite quickly when the rule master struggles with labelling grey boards. In the former case, maybe the conjecturer would have to simplify their statements first.
There are countless ways to embed difficult, arcane or computationally expensive problems into rule/conjecture statements.
One example would be to say that the valid koans are optimal solutions to a particular set of board coloring problems.
Not necessarily. A rule may be easier to check than to say that no counter example exist.
For example, the statement “every even integer is the sum of two primes” is easy enough to check for any example, but nobody has succeeded to find a counter example nor prove it. Easy rules may have counter examples to certain situations that are easy to recognise, but very hard to find.
There are even statements where neither the statement nor the negation can be proven. As an example I refer to the continuum hypothesis https://en.wikipedia.org/wiki/Continuum_hypothesis
Edit: Of course I strongly doubt that we will stumble upon such a statement 
