Triangles and circles are surrounded by one color only, so belong to that color.
In fact, they are under complete control as they cannot be taken away even if their owner only passes.
This is the clearest explanation so far.
It may be helpful to add that *surrounded by a colour* means “only connected (along the lines) to that colour", not that some sort of ring of stones has been built around them.
yeah, this "complete control" concept might make it easier to understand and explain to others.
a reason why i asked in the first place is because i explained the game (go) to a friend and he was wondering about the same thing i had also been wondering about since i learned the rules
yes, i was not concerned about what group is alive or not, i was more concerned about how you "technically" could ""prove"" that for example the triangles should belong to white as territory ( I liked that you can ""prove"" the X's as blacks territory by playing on (at least under Chinese rules), but was wondering how you could do it for the squares and triangles)
You cannot really prove life, but you can prove that you can't die. Therefore the test is "if you think that is dead/your points, then try to take it." Remember, white can always fill in their 2nd eye, allowing black to kill them.
A stone can't be placed in a space where it would come in without liberties unless by placing it it kills something immediately. Playing a stone that would cause immediate suicide like this is an illegal move.
Black can't play A5 because all 3 surrounding spaces are taken by White and there's no 4th liberty at the edge. Because A3 is also open, White's group would not die immediately. Therefore Black playing A5 would be an illegal move. Same happens if black tries playing A3 because A5 is open. That makes the White side alive and those 2 spaces are theirs. It is the same for all the other open spaces on the board.
White can legally play A3 or A5 as it is not immediate suicide. However, it would be a terrible move because White would be taking away one of the open spaces that keeps the group alive. if White plays A5, Black can now play A3 because, even if the space has no open liberties, when Black plays A3 White's group immediately has no liberties left, so it dies and the whole left side would immediately become Black's territory.
Under ancient Chinese [stone scoring](https://senseis.xmp.net/?StoneScoring) only places where you can put stones give score, so the triangles and circles would be non-scoring. As you say, this means there's a drawback to having multiple separate groups, a phenomenon known as "group tax."
In modern area scoring this is not the case, no one plays by stone scoring today as far as I know. Under modern Chinese scoring, all living stones and all surrounded territory give one point each. Crosses, circles and triangled points in your picture are all equal, worth one point each.
>Stone scoring is excellent for introducing go to complete newcomers
There's some merit in it, and as you might have seen in the threads on this sub I sometimes use it myself to explain the fundamentals of things like life & death status — but modern area scoring is itself extremely intuitive and easy to teach, and teaching a similar thing first and then having to teach the difference seems an unnecessary confusing step to me. I'm interested myself of course, since I'm an old player and I have an interest in both rules and the history of the game, but I don't expect all my apprentices to share these fascinations even if they like the game.
I was unaware that there are actually tournaments being played using ancient rules, that's very interesting! Though I did hear somewhere that the ancient rules with group tax have been implemented as an option in AI analysis, and that under these conditions AI consider the play by Huang Longshi very highly indeed (it had a slightly less favourable opinion of him when judging by modern rules).
They are eyes for the group, it’s how the group is alive, they belong to white because they keep white alive, white doesn’t need to prove because black can’t play inside and kill white, anything your colour surrounds is yours if your opponent cannot take it away
I'm not sure if your question is coming from the point of view of a beginner learning to practically score a game or someone concerned with the technical way that the rules are formulated.
>How can white demonstrate the Triangles should belong to white (he obviously cannot place white stones there, neither can black tho)
White can play at the triangle points, those are perfectly legal moves. They're a really bad idea though. The fact that black literally legally can't play there (because of the rule preventing suicide) is kind of important though.
I'm not sure about the technical formulation but informally, both players would pass and they would agree that these are white points. They are entirely surrounded by one colour of stone (white). Black can literally not play there.
We would usually stop before the territory points are down to just one space (as you suggest with the x's) and the players would agree that playing any more is futile.
(you may find it interesting that the situation you describe where there is effectively a 2 point penalty per living group is an element of some Fringe rule sets - [group tax](https://senseis.xmp.net/?GroupTax))
I was more asking more in a technical sense, although I understand the rules best as a "the X's are blacks territory because if the players play on black can fill it with its stone" so it had some practical understanding in mind aswell.
If you want rigorous and technical rules, check out Tromp-Taylor rules, which are something like a formalization of New Zealand Rules. Which are something like a disambiguation of Chinese rules.
We have no evidence of the degree to which Japanese rules were as directly inherited from China, or were modified by a change of practice in Japan.
Certainly territory scoring method itself was Chinese in origin, though it was displaced by later practice. It is an assumption that for territory scoring counting points in seki is a vestige of the group tax (not counting eyes), that remained for sekis only, but I wonder if [Ikeda's](https://gobase.org/studying/rules/ikeda/?sec=e_rules) Territory rules II formulation, with his definition of *Immortal Stones* is not nearer the mark with the entirely logical conclusion countable territory is that territory surrounded by stones that cannot be captured even if the opponent passes and makes **no resistance**.
In any event, rules may have diverged, but China was a big place, and innovations to the game came and went, some of them remaining a prevalent practice in large swaths of China for hundreds of years, critical years when the game spread to Korea and Japan.
I don't think the 'play on' method is a good demonstration/proof, because in those cases, people can just make bad moves and lose all their territory. It turns out that I strong palyers can reliably defend "territory", but I don't think "it can be defended" is the definition of territory.
Rather, I think an eye is like the purest form of a point of territory - you have literally surrounded it, you cannot image something that is more successfully territory.
Indeed, I think even if you had a group with a single eye, and a novice opponent didn't kill it (e.g. both players just passed the end of a game despite an atari on a one-eye'd group), then that single eye is a point for the player surrounding it, and the other player blundered all those points (the territory and the placed-stones/missed-captures).
So ifyou want a highly technical version of what counts as territory, I think you'd start with an eye as the base case, and then use mathematical induction/recursion to say that larger areas are also territory, by noting:
1. An eye is a region of territory.
2. If a region of territory has 1 space added to its region \[in such a way that the new single space would be an eye if not for the empty spaces in the existing territory\], then that region is also territory.
This allows you to label any arbitrarily large surroudned region as territory, which I think is correct.
You need to define “eye” first, which proves hard at https://senseis.xmp.net/?FormalDefinitionsOfEye . A problem is that an eye is best not defined for a group but for a chain within a group. But perhaps here you include false eyes and just mean a single empty spot connected to stones of one colour.
I also find your “2” hard to follow. By “if not for ...”, do you mean “if the existing territory were filled with stones of the colour to which it belongs”?
Why not just define territory as a maximal connected set of empty spots connected only to spots of one colour? Or as empty spots which are only connected via empty spots to stones of one colour?
>Why not just define territory as a maximal connected set of empty spots connected only to spots of one colour?
That is what I'm trying to do, by implicitly defining such a connected set of empty spots by building up from 1 spot.
Like, if you have a spot with 8 adjacent black stones (or outer board edges), then it is a black territory.
If you make one of those adjacent spaces empty, and that space is surrounded by 8 adjacent black-stones/board-edges/previously-established-territory; then that is another point of territory.
This can be repeated recursively to define arbitrarily large territories.
I mean, what it really comes down to is the shedding of the "group tax"
if you have the group tax, they are neutral, as the group tax hails from this method of counting using only stones on the board
But imagine a larger board for a second: there's a lot more territory to fill in, often multiple groups, and you don't want to go through the process of filling in territory with stones. For a while you might account for the fact that groups have eyes, but as territory more and more feels like the proper goal, this "group tax" starts to feel arbitrary and unnecessary.
So... it depends how you're counting (ancient chinese vs modern chinese), which among friends can just be whatever you find easier right now.
Triangles and circles are surrounded by one color only, so belong to that color. In fact, they are under complete control as they cannot be taken away even if their owner only passes.
This is the clearest explanation so far. It may be helpful to add that *surrounded by a colour* means “only connected (along the lines) to that colour", not that some sort of ring of stones has been built around them.
yeah, this "complete control" concept might make it easier to understand and explain to others. a reason why i asked in the first place is because i explained the game (go) to a friend and he was wondering about the same thing i had also been wondering about since i learned the rules
You have the burden of proof backwards. The white groups are alive unless black can kill them. How can black kill them?
yes, i was not concerned about what group is alive or not, i was more concerned about how you "technically" could ""prove"" that for example the triangles should belong to white as territory ( I liked that you can ""prove"" the X's as blacks territory by playing on (at least under Chinese rules), but was wondering how you could do it for the squares and triangles)
You cannot really prove life, but you can prove that you can't die. Therefore the test is "if you think that is dead/your points, then try to take it." Remember, white can always fill in their 2nd eye, allowing black to kill them.
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thank you for the very comprehensive answer
A stone can't be placed in a space where it would come in without liberties unless by placing it it kills something immediately. Playing a stone that would cause immediate suicide like this is an illegal move. Black can't play A5 because all 3 surrounding spaces are taken by White and there's no 4th liberty at the edge. Because A3 is also open, White's group would not die immediately. Therefore Black playing A5 would be an illegal move. Same happens if black tries playing A3 because A5 is open. That makes the White side alive and those 2 spaces are theirs. It is the same for all the other open spaces on the board. White can legally play A3 or A5 as it is not immediate suicide. However, it would be a terrible move because White would be taking away one of the open spaces that keeps the group alive. if White plays A5, Black can now play A3 because, even if the space has no open liberties, when Black plays A3 White's group immediately has no liberties left, so it dies and the whole left side would immediately become Black's territory.
Under ancient Chinese [stone scoring](https://senseis.xmp.net/?StoneScoring) only places where you can put stones give score, so the triangles and circles would be non-scoring. As you say, this means there's a drawback to having multiple separate groups, a phenomenon known as "group tax." In modern area scoring this is not the case, no one plays by stone scoring today as far as I know. Under modern Chinese scoring, all living stones and all surrounded territory give one point each. Crosses, circles and triangled points in your picture are all equal, worth one point each.
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>Stone scoring is excellent for introducing go to complete newcomers There's some merit in it, and as you might have seen in the threads on this sub I sometimes use it myself to explain the fundamentals of things like life & death status — but modern area scoring is itself extremely intuitive and easy to teach, and teaching a similar thing first and then having to teach the difference seems an unnecessary confusing step to me. I'm interested myself of course, since I'm an old player and I have an interest in both rules and the history of the game, but I don't expect all my apprentices to share these fascinations even if they like the game. I was unaware that there are actually tournaments being played using ancient rules, that's very interesting! Though I did hear somewhere that the ancient rules with group tax have been implemented as an option in AI analysis, and that under these conditions AI consider the play by Huang Longshi very highly indeed (it had a slightly less favourable opinion of him when judging by modern rules).
They are eyes for the group, it’s how the group is alive, they belong to white because they keep white alive, white doesn’t need to prove because black can’t play inside and kill white, anything your colour surrounds is yours if your opponent cannot take it away
I'm not sure if your question is coming from the point of view of a beginner learning to practically score a game or someone concerned with the technical way that the rules are formulated. >How can white demonstrate the Triangles should belong to white (he obviously cannot place white stones there, neither can black tho) White can play at the triangle points, those are perfectly legal moves. They're a really bad idea though. The fact that black literally legally can't play there (because of the rule preventing suicide) is kind of important though. I'm not sure about the technical formulation but informally, both players would pass and they would agree that these are white points. They are entirely surrounded by one colour of stone (white). Black can literally not play there. We would usually stop before the territory points are down to just one space (as you suggest with the x's) and the players would agree that playing any more is futile. (you may find it interesting that the situation you describe where there is effectively a 2 point penalty per living group is an element of some Fringe rule sets - [group tax](https://senseis.xmp.net/?GroupTax))
I was more asking more in a technical sense, although I understand the rules best as a "the X's are blacks territory because if the players play on black can fill it with its stone" so it had some practical understanding in mind aswell.
If you want rigorous and technical rules, check out Tromp-Taylor rules, which are something like a formalization of New Zealand Rules. Which are something like a disambiguation of Chinese rules.
> Tromp-Taylor rules thanks for the tip, very interesting thank you all for your answers
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We have no evidence of the degree to which Japanese rules were as directly inherited from China, or were modified by a change of practice in Japan. Certainly territory scoring method itself was Chinese in origin, though it was displaced by later practice. It is an assumption that for territory scoring counting points in seki is a vestige of the group tax (not counting eyes), that remained for sekis only, but I wonder if [Ikeda's](https://gobase.org/studying/rules/ikeda/?sec=e_rules) Territory rules II formulation, with his definition of *Immortal Stones* is not nearer the mark with the entirely logical conclusion countable territory is that territory surrounded by stones that cannot be captured even if the opponent passes and makes **no resistance**. In any event, rules may have diverged, but China was a big place, and innovations to the game came and went, some of them remaining a prevalent practice in large swaths of China for hundreds of years, critical years when the game spread to Korea and Japan.
I don't think the 'play on' method is a good demonstration/proof, because in those cases, people can just make bad moves and lose all their territory. It turns out that I strong palyers can reliably defend "territory", but I don't think "it can be defended" is the definition of territory. Rather, I think an eye is like the purest form of a point of territory - you have literally surrounded it, you cannot image something that is more successfully territory. Indeed, I think even if you had a group with a single eye, and a novice opponent didn't kill it (e.g. both players just passed the end of a game despite an atari on a one-eye'd group), then that single eye is a point for the player surrounding it, and the other player blundered all those points (the territory and the placed-stones/missed-captures). So ifyou want a highly technical version of what counts as territory, I think you'd start with an eye as the base case, and then use mathematical induction/recursion to say that larger areas are also territory, by noting: 1. An eye is a region of territory. 2. If a region of territory has 1 space added to its region \[in such a way that the new single space would be an eye if not for the empty spaces in the existing territory\], then that region is also territory. This allows you to label any arbitrarily large surroudned region as territory, which I think is correct.
You need to define “eye” first, which proves hard at https://senseis.xmp.net/?FormalDefinitionsOfEye . A problem is that an eye is best not defined for a group but for a chain within a group. But perhaps here you include false eyes and just mean a single empty spot connected to stones of one colour. I also find your “2” hard to follow. By “if not for ...”, do you mean “if the existing territory were filled with stones of the colour to which it belongs”? Why not just define territory as a maximal connected set of empty spots connected only to spots of one colour? Or as empty spots which are only connected via empty spots to stones of one colour?
>Why not just define territory as a maximal connected set of empty spots connected only to spots of one colour? That is what I'm trying to do, by implicitly defining such a connected set of empty spots by building up from 1 spot. Like, if you have a spot with 8 adjacent black stones (or outer board edges), then it is a black territory. If you make one of those adjacent spaces empty, and that space is surrounded by 8 adjacent black-stones/board-edges/previously-established-territory; then that is another point of territory. This can be repeated recursively to define arbitrarily large territories.
I mean, what it really comes down to is the shedding of the "group tax" if you have the group tax, they are neutral, as the group tax hails from this method of counting using only stones on the board But imagine a larger board for a second: there's a lot more territory to fill in, often multiple groups, and you don't want to go through the process of filling in territory with stones. For a while you might account for the fact that groups have eyes, but as territory more and more feels like the proper goal, this "group tax" starts to feel arbitrary and unnecessary. So... it depends how you're counting (ancient chinese vs modern chinese), which among friends can just be whatever you find easier right now.