Go Zendo

I have a feeling some of the GCHQ puzzles might be easier https://www.gchq.gov.uk/information/stay-at-home-and-stay-busy-with-our-brainteasers

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Ok, I’m 99.9% sure now. I looked through all the boards again, and it also fits with this thing Martin said:

(notice that if we allowed the whole board to be filled with black stones, it would be green)

I have something special prepared for the next game so I’m gonna call dibs on hosting it, but maybe I should wait until tomorrow before starting it up? It feels like there’s not many people active right now, and it would be nice to get Martins confirmation that the rule was indeed correct before starting the next game.


I agree. That’s got to be it!


So far , all these squares (I know there are more) makes it green but two are red.

Congratulations @le_4TC, that is indeed my rule :slight_smile: Well done finding it out!


Game 8

For this game, valid koans are any positions on the 5x5-board (they do not have to be legal go positions).

The rule is invariant to rotating/mirroring the board, and also to switching black and white.

Now here’s the new part: You can check as many boards as you like on https://zendo.4tc.xyz/. I will not be answering questions about specific boards, or collect them in this post. I will still be here to give counterexamples to any claims about the rule, and also provide hints if necessary. I’ll make an empty wiki post below this one, where you can collect what you’ve learned so far if you want, but it’s up to you how you want to organize things.

I encourage you not to post every single board you’ve checked. Instead, check some boards of some specific form and try to learn something from that, then share your findings. For example, let’s say you test every board with just one stone and find that they are all green. Instead of posting images of all those boards, simply share the information that “Every board with one stone is green”.

If you think you found part of the rule (maybe you think that every board with two stones is also green, and haven’t found a counterexample yet, but you also haven’t checked every possibility), you can write

Conjecture: Every board with two stones is green.

and I will then either confirm the conjecture or provide a counterexample.

Good luck!


Confirmed properties

  • The rule is invariant to rotating or mirroring the board.
  • The rule is invariant to switching black and white.
  • If the position stays the same after rotating it 90 degrees, the board is green.
  • If no two stones lie on the same gridline, the board is green.

Let’s post the koans we checked in this post .(must be important)I checked like twenty and found this.

Like I said, please don’t post every koan you check, that will be way too many and will just make the whole thread harder to overlook. Just choose ones that you find interesting, and share those. A couple of boards at a time. And add some explanation with your thinking, don’t just post pictures of boards.

I do like the two boards you chose so far, those are good examples of green boards!

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Conjecture: Assuming there are at most two stones on the board, the rule is invariant to toroidal translation.

This is true, but I feel like I should warn that it feels a bit like it’s true “by accident”. The rule makes no mention of a torus, and in general it’s not invariant to toroidal translation (as you might’ve seen before restricting the conjecture to at most 2 stones).

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That was not every koan I checked.

Yes, I saw that after I wrote the first half of the message. Like I said, those are good choices! Just wanted to remind not to post too many :slightly_smiling_face:

I get that we don’t want to post everything, but heres a fun sequence where if you add one more black stone it always changes from Red to Green and vice versa. It’s a zig zag pattern with one black group.
image image image image

I was playing around with other group sizes and liberties, and not sure I understand the pattern yet.

It’s not purely about the group size since


I’ve played around a little bit, and this seems interesting to me:


Adding white stones in the middle of this diagram does not change the color (at least the white configurations I tried did not). Adding white stones to the corners turned the color to red unless the added white corner stones are symmetrical.

After evaluating a bunch of positions, I have a feeling the following might be true:

Conjecture: If the board is symmetrical with respect to the center point, then the Koan is green.

Edit: Nevermind, I forgot that I already know a counter-example:

Edit2: New conjecture: If the board position stays the same after rotating it 90 degree, then the Koan is green.


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But not all greens satisfy that I presume?


That’s mad though…

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Indeed, maybe I should emphasize this distinction since it might not be obvious to everyone. What Martin guessed and I confirmed was the implication

IF the position stays the same after rotating it 90 degrees THEN the board is green.

If it were also true that

IF the board is green THEN the position stays the same after rotating it 90 degrees.

this would mean that the rule is “the position stays the same after rotating it 90 degrees”. But since the above statement is not true (as illustrated by the above counterexample, a position which is green but doesn’t stay the same after rotating), this is not the rule.

Another way of looking at it is that Martin has found a subset of the green boards, but not all green boards.


Rotationally invariant boards are a strict subset.

However, rotational symmetry is not required.


And any mirror symmetry is not required either. Here is one that is toroidally one step to the right from a rotationally invariant board.


Also, I couldn’t resist checking the website source code. It looks like everything is computed client-side, but a reasonable amount of obfuscation was employed to make reverse-engineering the rule non-trivial.

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