Indeed, if no black and white stones are in the same row/column, and the black multiset is the same for rows and columns, and the white multiset is the same for rows and columns, then the board is green. However, you can’t count black and white in the same multiset, even when they are “separated”, as this board demonstrates:
Inspired by @shinuito 's multiset theory, I would like to make a rule guess:
For a row / column we write down the order of colors in the row / column and call it the reduced row / column. For example the row e / b / b / e / w becomes b / w. (here b is black stone, w is white stone, e is empty intersection.)
My guess is the rule compares the multiset of reduced rows and the multiset of reduced columns. Then the Koan is green if and only if they are equal.
So the reduction works by removing all empty intersections, and merging adjacent colors together if they are equal? But b / w / b would stay b / w / b?
Yes. This seemed to work for me for a lot of Koans, but I think I found a counterexample already.
Edit: Specifically: 
There are some more subtleties on how to define your proposed rule (is b / w the same as w / b? if it’s not, then it’s unclear how to compare rows and columns), but I’ll say that you are somewhat on the right track. In case you want another counterexample, one can be built from a board you posted early on:
Conjecture: Adding a stone at a position where stones exist to the right, left, above and below (i.e. the shortest paths from the position to the four edges of the board are blocked) never changes the color of the Koan.
True!
Does the rule count the number of white stones that are visible from the edges along the rows and compares this number to the number of visible white stones along the columns? And then the same for black stones. The numbers have to match both times for the rule to be satisfied.
Yes! The same number of black and white stones has to “visible” from each of the four edges.
Nice! With the tool you coded and being able to check a lot of Koans, it feels almost as if this was an entirely different game.
Being able to ask about confirmation on conjectures is very potent. The last Conjecture confirmed that the rule depends only on the outmost stones in every row and column, for example.
So how does that rule work. Is it like if I look at the board side on, I count the number of white and black stones I see. Like if eye level was where the stones are placed on a real board. Then I do that for every direction?
That’s a cool rule.
I think the combination of being able to check lots of cases quickly with the updated tool, and being able to check conjectures was fun.
I know what you mean it does kind of feel like a different game. This game I was actually able to guess part of a rule though 
I couldn’t really formulate many ideas matching the boards in some of the other games.
Yes, exactly!
I’m glad you enjoyed the new variation on the game. If someone else wants to host a game like this (which I think you should, because it’s more fun to host in this way in my opinion, less tedious stuff and more interesting stuff), I would be happy to help with implementing an automatic check for the rule (of course then I won’t be able to participate in that particular game, but that’s fine 
).
I too enjoyed the new variation on the game. I would host another game, and ideally I would like you to be able to play as well, but there’s some hurdles;
I can think of another rule, and I can think of an algorithm to check that rule. However I wouldn’t know what programming language to use or how to make the program accessible on the internet, like you did.
I really don’t mind not taking part as a player in the next game, and I enjoy the little programming exercise of implementing the rule. Also there’s some advantages to having 2 people know the rule, there’s less chance of accidentally confirming a false conjecture, for instance. And I’m sure there will be more chances for me to play later 
Alternatively, if you have some experience with any programming language, I’m sure you could quickly learn to implement your rule in JavaScript (which is what I’m using). But it might be easiest if we implement the rule you’re thinking of now together, and then you can try to implement one by yourself later.
I have experience with some programming languages, but it’s been a while. I’ll have to learn the syntax and so on, but I have thought of a rule that could be interesting. If I should learn JavaScript, it would be very helpful if I could look at some code and learn from that.
Here’s the unobfuscated version of the code from before: https://zendo.4tc.xyz/graphics.js
I threw everything together in one file before obfuscating it, but the check()-function is the part that actually implements the rule. For each orientation, it counts how many black and white stones can be “seen”, and afterwards checks if it ever got two different counts from two different directions.
The underlying Board class only stores the color of each intersection (-1 for white, 0 for empty, 1 for black) and the connections between intersections (the concept of chains and liberties is not pre-implemented right now, but it’s not too hard to do).
Thank you! This helps a lot. I really want to do this now, but on second thoughts, I still have work to do until wednesday. After that I would love to implement my rule in this way 
Great! Just don’t learn too much from that code, it’s not very good 
In the meantime, anyone else is also welcome to submit a rule idea to me if they would like my help implementing it. Of course several of the participants in this thread wouldn’t really need my help ^^
You don’t want to waste time lol.
I am very good with scratch.