Weird and wonderful consequences of simple rules

antonTobi · August 14, 2021, 1:23pm

In this thread I want to share some interesting things that can happen under simple rulesets such as Tromp-Taylor. Sometimes I may consider different variations of rules, and often the exact details won’t matter much, but unless otherwise stated we’re assuming:

Positional superko (passes do not lift ko-bans).
Suicide is legal (this very rarely matters).
Two consecutive passes ends the game, and all stones on the board are assumed alive.

This last point may seem extreme if you’re not familiar with Tromp-Taylor rules, but all it means is that dead stones need to be “manually” removed from the board before passing. Even if you introduce a dead stone agreement phase as a practical shortcut, this is what would happen if both players stubbornly insist that all their groups are alive, requiring the other player to capture to prove them dead.

Index of main posts in this thread

Bent four in the corner (see below)
Frozen life (aka 1-eye-flaw)
Zombie groups (dead groups coming back to life)
Cycle tampering (starting a cycle and then tricking the rules that it was the opponent who did it)
Long cycles of the eternal flavor (eternal life, eternal ko, triple/quadruple hot stones)
Unremovable ko for both sides (what happens if neither side can finish the ko?)
Molasses ko (ko beast that can slow the game down to 20% playing speed)
No-pass go (ruleset without passing and scoring)

Index of smaller posts

Let’s start with a well-known example before I get into the more controversial stuff

Bent four in the corner

This is the go-to example to highlight the difference between Chinese and Japanese rules, so a lot of people will already be familiar with it, and I’ll only go over the basic shape quickly. If you want a better introduction, the sensei’s page has some more variations.

In this shape, white has no continuation, since both A2 and D1 are self-atari. The best white can hope for is seki.

But black can at any time start a ko to kill the white group, and black will take the ko first (5 at 1):

This means that as long as the surrounding black stones are alive, the white group will usually die. Black can leave it until no other point-gaining moves are available, then remove all ko threats, and only then start and win the ko.

Some people might even accept the above as a proof that Bent four in the corner is always dead. But there is an important flaw in the “proof”, it’s not always possible to remove all ko threats!

A relatively common example of an unremovable ko-threat is a seki which one side can afford to sacrifice, something like this:

White can use B2 as a ko-threat. It loses 7 points, but can easily be worth it to win a big ko. And there is no way for black to remove the ko threat!

So let’s put these two situations on the same board and see what happens:

If both sides pass here, the score on the board is B+3. Can black do better by starting the ko?

It’s a one-way street up to here, where white is forced to use the point-losing threat:

Now black has two options: respond to the threat or finish the ko.

If black responds to the threat, white will take back the ko, and there are no more threats:

Black will pass, white will play C1, and the final score is B+2, one point worse than before!
Black gained 7 points in the upper right, but lost 8 points in the lower left

Then how about black ignoring the threat?

Black kills lower left, white kills upper right. The final score is W+2, even worse for black!

In summary: Black could kill the lower left in this game, but will lose points in the process. Black is better off leaving it as the seki in the first diagram, so this is a case where Bent four in the corner is not dead.

If you’re familiar with Japanese rules, you may note that they allow black to claim the benefits of killing the lower left group without paying the price of giving up the upper right group! For fans of simple rules this seems really unfair.

But this is a matter of subjective opinion, and if you prefer the Japanese result that’s perfectly fine!
In particular, I really hope this thread won’t devolve into heated debates over which rules are better.
I just want to share and discuss some interesting positions since I know there is a small group of people out there which enjoy them as much as I do

The fact that we used a point-losing unremovable ko threat to construct the above position made it quite a close call whether black should start the ko or not. As it turns out, there are several ways to have unremovable ko threats which are not point-losing, but I’m saving those for later installments in this series where they are actually needed!

yebellz · August 14, 2021, 2:40pm

antonTobi · August 14, 2021, 2:49pm

Thank you @yebellz, I had forgotten about that post of yours! This thread could perhaps be thought of as a sister-thread to Odd Cases 🤔 in the Japanese Rules, which is one of my favorite threads on the forums

I wanted to include a quick discussion about bent four here for completeness, but I promise that there are much stranger things coming soon

yebellz · August 14, 2021, 2:51pm

At first, I thought you were going focus on this interesting aspect of the strict Tromp Taylor rules (without life and death confirmation)

Here’s an interesting example from Sensei’s Library

White just played at 1 (capturing a black stone) and wins!

antonTobi · August 14, 2021, 2:52pm

Indeed, coming soon!

yebellz · August 14, 2021, 3:02pm

Looking forward to your exposition of superko “anomalies”. I believe that we might agree with the subjective view of them simply being acceptable consequences of the rules, rather than something that would motivate more complex rules in order to avoid them.

antonTobi · August 14, 2021, 3:58pm

The established name for this shape is 1-eye-flaw, but I dislike this name, so instead I’ll call it…

Frozen life

White has just captured at A1 - black cannot recapture at A2 due to ko.
The other moves are suicide/self-atari for black, so passing is the best option.
Similarly for white: filling at A2 would be suicidal, so white passes.

Thus the game ends here: the one-eyed white group is alive with 7 points!

I think “Frozen life” is descriptive, because the group is only alive as long as the game remains frozen in this state: if it was possible to unfreeze the game by making a non-passing move elsewhere on the board, white would die.

It may seem like the reason for this strange result is only the fact that the game immediately ends after two passes, but even if black were allowed to dispute the result after two passes, superko would still forbid him from capturing at A2, so there really is no way to kill the group.

Here I should mention that in some rulesets, passing lifts some or all ko-bans. This is not the case in Tromp-Taylor rules, and I believe not in Chinese, AGA/BGA or NZ rules either. So under all these rulesets, the white group lives.

(NZ rules use situational superko, but the distinction does not matter in this case)
(also Chinese rules in practice are different from Chinese rules in theory, which is why Tromp-Taylor rules is my go-to ruleset for this thread)

Ok, so with all of that boring rules stuff out of the way, what fun things can we do with this weird and wonderful way to live?

Consider this position, with a frozen life in the top left:

If the game ends here, black is losing by 1 point. Can black do better?

Yes! Black should fill an eye with F2 (or G1, it doesn’t matter), to unfreeze the game:

Black voluntarily dies in gote with the lower right group! Regardless of whether white passes or captures on the next move, the position has changed, so black can now freely capture at C7 and then at B6 on the move after that.

Since the upper left group is slightly bigger than the lower right, this exchange swings the game in black’s favor: the final result is B+1.

Next consider this position:

Here the top left is smaller than the bottom right, so it’s not in white’s interest to die in gote at F2.

Instead, white should suicide at A1! After that the upper left group can be killed without problems.

If suicide is not legal, then the upper left group and white’s chain of four stones are both alive!

Black never wants to capture the four stones, since then he loses the frozen group.

In summary, not only can one-eyed groups live, unconditionally pass-alive groups can die! And if suicide is illegal, we can get big chains in atari, surrounded by solidly alive groups, but which are nonetheless alive!

Personally I think this is lovely, but if you find it ugly, don’t worry - the chances of a frozen life happening in a real game on a 19x19 board are practically 0

shinuito · August 14, 2021, 4:49pm

Weird things happen in 9x9 all the time Especially with low time controls

~~I definitely find it interesting but it makes the two-passes ends the game combined with positional superko a very unattractive idea to me~~ Actually once I start thinking about it I don’t know what I want in a set of go rules really tbh

In practice, I think a tromp-Taylor rule set like that would cause awful headaches on OGS It would absolutely demand a non-adjustable accurate scoring tool. Maybe it wouldn’t be too hard to make, since one just assumed anything left on the board is alive, and counts empty points that only reach one color.

It could be interesting to add that I think on OGS passing (twice?) lifts these? At least on demo boards.

I vaguely remember some examples though on the forums of games with resumptions, superko etc which were disputed that could be looked up to see how the games behave in practice. Of course one could set up some practice games either.

antonTobi · August 17, 2021, 4:24pm

Today I want to share two examples of groups that look dead but aren’t. For dramatic effect, let’s name them:

Zombie groups

Let’s start with this position, which we’ve discussed here on the forums before:

@yebellz did an excellent analysis here, so I’ll just redirect you there rather than redoing the work myself!

The quick summary is that although the white group on the left is locally dead, it’s not in black’s interest to kill it, because doing so would give white a ko threat, enabling her to start and win the E4 ko. Therefore the white group is globally alive in seki.

Such small-board examples are often completely constructed and would never happen with reasonable play from an empty board. But this position actually comes from near-perfect play! Only 5 and 7 are not optimal in this sequence:

Source: Solving Go for Rectangular Boards (section 5.2)

A few weeks ago, this Lifein19x19 post by Gérard Taille brought to my attention an even more shocking version of this phenomenon.

Surely the top left group is dead! It only has one eye, and it’s surrounded by an unconditionally alive group. Even with arbitrarily many moves in a row, black can’t live.

But, just like in the previous example, an issue arises when white actually tries to take the group of the board. After white has played two more moves inside, black can capture as a ko threat! Therefore black will choose that exact time to start the ko in the upper right:

When black makes the ko threat at 6, white is forced to choose between killing the left and staying alive on the right. The right group is bigger, so correct play is to finish the ko with 7, and let black live on the left:

Now here is where this example differs a bit from the previous one: white should play out this sequence at the end of the game! If she instead leaves the original position as seki (just adding a stone at B6 for an extra point) the score is W+3. But in the final diagram above, the score is W+6.

If you’d like to make the sequence above necessary to win, just imagine that C1 and D1 are black stones - I’m too lazy to change the diagrams

In conclusion, with correct play the obviously dead black group comes back to life!

Note that it’s not too far-fetched for this to happen in a real game. Unlike the frozen life in my previous post, this has nothing to do with super-ko, and it doesn’t matter if both sides have lots of territory-filling moves available. We just need these ingredients:

A black dead group where white capturing it will create a ko-threat. This is true of many nakade-shapes.
Some shape where black can start a ko at any time, and white can’t remove the ko. The upper right corner from above is not a common shape, but a standard Ten Thousand Year Ko has similar properties and could have the same effect under the right circumstances.
Black must not have been able to start the ko earlier in the game for a larger profit.

On one hand, it seems likely that black could get some nice profit by starting the ko earlier, since it’s so heavy for white. On the other hand, living with a dead group like we saw above may be even bigger! So if black has such a dead group, and is aware of the implications, he may choose to leave the ko on the board until the final moments of the game.

Glunkolin · August 17, 2021, 4:26pm

death of 23 eyes

elsantodel90 · August 18, 2021, 10:24pm

Awesome examples!

I had never seen it shown so clearly with a dead nakade example, but I think that the 10 thousand year ko principle here is well known and probably occurs in pro games (maybe less spectacularly :P).

The principle is that 10 thousand year ko fights revolve not so much around “using” the available ko threats, as normal kos do, but they are intrinsically about “Creating and destroying ko threats”, as is said in TenThousandYearKo/Fighting the ko at Sensei's Library

Indeed, since whoever starts the ko is at a disadvantage, and each player prefers the opponent to be the one that starts a 10 thousand year ko, when ko-threats are more or less balanced so that “right now” no player can start the ko and win, each player would like to get to a better situation “kothreat-wise”, in order to have enough surplus ko threats to start the ko and win. This can be done by playing moves that create large ko threats, or moves that destroy large ko threats of the opponent. Thus a “pre - ko fight” so to speak occurs before the “normal ko fight”, where players fight to create and destroy ko threats to get to a position where they can afford to take the ko, starting the “normal ko fight”, and win it.

The novelty to me here is the observation that, under rulesets where stones must be removed to be counted as dead, the act of playing them out at the very end of the game very often requires the creation of ko-threats for the opponent, and thus has the potential to affect a ten thousand year ko. It would be awesome to know if this has ever been used and would be accepted under actual Chinese Rules on the board (as used by Chinese professionals, as opposed to the “online servers Chinese Rules”).

antonTobi · August 20, 2021, 5:36pm

Cycle tampering

Let’s look at a few smaller variations of the wonderful beast shared by @Glunkolin above. The basic building block here is a seki where one side can repeatedly sacrifice two stones:

Without superko, black could infinitely repeat the sequence [A3 A1 A2 Pass]. This does not pose a problem for Japanese rules, because white gains one prisoner each cycle. However, in area scoring rules, we really need some rule such as positional superko to prevent this repetition.

There are multiple variations on generalized ko-rules, but the two most common are:

Positional superko (PSK) forbids any move that would recreate an earlier board position.

Situational superko (SSK) forbids any move that would recreate an earlier board position with the same player to move.

This means that in the sequence [A3 A1 A2] above, the final black move would not be legal under PSK (because the position is repeated), but it would be legal under SSK (because in the repeated position, the player to move has changed from black to white).

Note though that passing is always legal - superko restrictions only apply to board plays. So under SSK the sequence [A3 A2 A1 Pass] is totally fine, even though white’s final pass recreates the initial position with the same player to move. However, black would not then be able to continue the sequence with a new play at A3.

My personal preference is for PSK - it’s slightly cleaner to state, and it seems easier to keep track of. All analysis in this thread assumes PSK unless otherwise explicitly stated.

Normally, superko means that a position like the above would stay as seki. But in exceptional cases, strange things can happen:

At first glance, both white groups look alive. But black has this sneaky sequence:

3 cannot be at F5 because it would repeat the initial position.
5 cannot be at F5 because then white kills with D1.

After 7, all white moves are self-atari, so she passes. Then black plays 9, probably accompanied by an evil laugh:

White is forbidden from capturing the two stones, because that would repeat the position after move 6 in the sequence! As a result, all the white stones will die.

One way to make sense of what just happened is that normally, PSK prevents black from doing anything nasty in the corner. But by playing some forcing moves in the middle of the “cycle”, black turned PSK to his favor: suddenly white is the one who is forbidden from repeating, even though black was the one that initiated the cycle. Quite an evil trick indeed!

Seeing this threat in advance, white would do better by playing 4 at F5, sacrificing the bottom left group to save the upper group.

This particular example seems to be due to John Tromp himself, more discussion about it here.

Two crucial and somewhat “unrealistic” elements of the position above are:

The ko threat at B2, which white could easily remove if she had the chance.
The lack of non-passing moves for white after move 7.

Here is an alternate version with an unremovable ko threat and some black territory:

Black can at any time exchange A5-A7 as a ko threat, but white cannot move first in the seki without killing herself.
However, the black territory around G4 means that black can’t start the J7 sequence immediately, because white has several non-passing moves available.
Correct play is for black to first play inside his own territory until it consists entirely of 1-point eyes (white cannot prevent this), and only after that start the J7 sequence.
White will die somewhere, but she can choose quite freely where she will die. She may ignore the ko threat on the left and die there, or she may play an eye-filling move on the bottom instead of passing, or she may go along with the entire sequence and allow black to kill the top right.

What happens if we give white a third eye?

Now killing is no longer possible for black, white has a move available at G1, and there’s no way for black to take that away.

With all this in my mind, you may be able to solve the following problem!

How should black play from here, and what is the result when both players play optimally?
As usual: Area scoring, no komi, positional superko.

Click the diagram above if you want to play out sequences! This is especially helpful because OGS will prevent you from making moves that violate PSK. You may also download the sgf and open it in CGoban or some other program which may be more helpful with scoring than OGS.

antonTobi · August 22, 2021, 2:42pm

Solution to the problem above

If black plays the natural move at 1, white makes a third eye with 2:

This position is identical to the second to last diagram in the above post - because white has a guaranteed move at G1, black will never be able to successfully start the J7 sequence. All stones on the board are alive, and white wins by 1 point.

The correct move for black is to deny white the third eye! White makes an extra reduction with 2-3 before coming back to 4:

(If white instead tries 2 at J7, black J9 kills)

Now we have reached an extremely delicate position!

White has a non-passing move at G1, which black cannot remove (note that black suiciding at G1 doesn’t help). This makes it impossible for black to kill the upper right.
However, this means that white must not play G1 as long as the threat of the J7 sequence remains on the board.
Thus the single black stone is alive! (and G1 is dame, not a point for white)
Black must take care not to play J9 or exchange A5-A7, because doing so would remove the threat of killing the upper right, allowing white to safely capture H1.
Similarly, black must not start the J7 sequence, because it accomplishes nothing except allowing white to capture H1 in the process.

Both players should pass and end the game here, the final score is a tie.
To see exactly where black’s 1 point gain came from, note that:

The 2-3 exchange is a 2 point swing in white’s favor (black lost 1 point, white gained 1 point).
The H1 invasion is a 3 point swing in black’s favor (black gained 1 point, white lost 2 points).

This problem is my own creation, so it’s entirely possible that I’ve made some oversight! Please view the analysis with a critical eye and let me know if you find a mistake.

elsantodel90 · September 7, 2021, 8:45pm

Does the 23 eye group die here? This examples illustrates the principle by which, if black has many ko-threats and white’s only “safe moves” are eye-filling in the lower-left group, then for just enough black ko-threats, white will prefer to fill all her eyes to avoid black killing all the rest of the board with the superko trick.

The thing is, I do not think that “eye filling in the lower-left” is the only option of safe-moves for white in this example. Maybe I am horribly misreading, but isn’t any move on the right safe for white? That is, joining one of the bamboo joints seems to me like a perfectly fine waiting-move, and also destroys one black ko-threat so it is definitely better than eye-filling.

The “23 eyes death” would indeed be inevitable if the position were modified so that these 23 black ko threats were “unremovable by white moves”. In that case, white has indeed no other safe “waiting move” than playing in the lower-left. This is what happens in the nice example given by le_4TC (for only one eye instead of 23, but you could copy his left-side seki 23 times to create 23 unavoidable black ko-threats, showing that in principle for large boards and artificial positions, any number of eyes can be made to die as a consequence of the superko rule).

antonTobi · September 7, 2021, 8:54pm

Joining one of the bamboo-joints actually loses the seki for white! It may be easier to see for a smaller version like this one:

The principle is the same in the large version, if white plays any move black will push through all the joints and then eventually capture. So those ko-threats are indeed unremovable, and the example works

elsantodel90 · September 7, 2021, 8:59pm

Oh yes, you are totally right. The fact that those are sente for black creates an asymmetry, since once white plays black can “play sente everywhere else” and we imagine that only two liberties were “actually” remaining, and white just filled one so she loses the fight.

I was horribly misreading once again!

shinuito · September 8, 2021, 12:18am

I feel like this is a good a reason as any to have passing lift the superko restriction no? In that case black would just run out of ko threats?

It seems quite silly (as opposed to wonderful) to have to fill in ones own eyespace in order to avoid repetition.

antonTobi · September 8, 2021, 1:16am

This is of course a subjective issue, and there are plenty of rulesets which agree with your opinion.

For me, what’s so nice about the superko rule is that it’s the most obvious and natural way to remove all cycles from the game and make it provably finite. If you allow passes to lift ko-bans, cycles are once again possible, and you will need some additional rules to handle these cases. This is less elegant in my eyes.

Also, why is it silly to have to fill in your own eye-space? There are so many strange things in go which we wouldn’t dream of trying to “patch out”. For a total beginner, the idea that stones can be alive without two eyes may seem like a flaw! Or perhaps the fact that an obviously alive group can die as the result of a ko fight? We’ve learned to accept these strange things only because we’ve seen them many times and they have become normal, but I think in some sense they are just as weird as the stuff in this thread.

And even if you find the “23 eyes can die”-example appalling, there is no need to worry, since it will not happen in an actual game within our lifetimes

Some of the other “strangeness” might happen a few times in recorded go history, but probably never in your own games (see for instance eternal life). Is it really worth it to make the rules more complicated and harder to understand just to avoid such a rare issue?

To me the much more common confusions that arise from the Japanese rules is a far larger problem. My local tournaments use Japanese rules, but of course nobody present actually knows the rules! And I’m not sure anybody can be expected to - it seems that even experts on these rules can have lengthy debates on how they should be interpreted and applied. (see recent threads on lifein19x19 )

Of course Tromp-Taylor and Japanese are not the only two options - there are rulesets where passes lift ko-bans which are still relatively simple. But I think the horrificness of the Japanese rules is a good cautionary tale for what happens when you try to make the rules conform to some preconceived notions of how the game “should” be, rather than adapting simple rules and accepting (or in my case, enjoying ) the rare oddity as a natural consequence.

yebellz · September 8, 2021, 1:53am

Another example is (I think) the Ing ko rules. I don’t fully understand the Ing ko rules (so, I qualified my previous sentence with an “I think”, and if someone wants to try to explain them, please go ahead!), but I think the general motivation was to patch out a superko “anomaly” that Ing did not like, so he wound up with ko rules that mostly behave like superko, except that some exceedingly rare life/death situation is resolved as he wished (rather than how they would be resolved under typical superko rules). However, the drawback is that virtually no one understands what the Ing ko rules actually are.

Feijoa · September 8, 2021, 7:02am

I wonder what would happen to “sending two, returning one” under the rule I suggested here which you might call delayed-capture go or sanitary go.

After 2, the captured black stones stay on the board until Black’s turn:

So Black needs to play a ko threat elsewhere (3, 4) and remove his stones before returning to capture:

White can play elsewhere or just pass, removing the stone at A1, and we return to the original corner position. White has lost nothing while Black wasted a ko threat, right?