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Looks nice! Your looping board seems to be what is known as Toroidal Go

I concacted him at OGS

I believe it is the user with the IP 2a03:7846:eb91:0101. I have unblocked him.

I am pleased to inform you that the PDF version of all 8 titles are now available on Patreon. The links can be found in the table on my Senseis page.

That happens to be Self-Atari.

No worries. :) If there is a definition of alive or pass-alive that uses construction rather than the inability to capture, then the community would have come out with one by now. Therefore, I suspect a definition by construction doesn't really exist.

However, I do admit that operationally, the resist capture definition is difficult to apply. Therefore people came out with various heuristics, of which many are of the form of one-sided implications (if certain properties are satisfied, then the group is alive). Examples include:

• If a chain has two real eyes, then it is pass-alive. (See recognizing an eye for a heuristic for determining whether an eye is real.)
  • If a group can obtain two real eyes with perfect play by the defender, then the group is alive. (The part on perfect play is typically omitted and implied in discussions.)
• If a chain is part of a seki, then it is alive but not pass-alive.

You can start by learning and applying such constructive heuristics first, then expanding your knowledge by figuring out how to make or destroy eyes (this is a gigantic topic), and learning exceptions such as false eye life. I would say, the ability to make two eyes or to make seki covers the vast majority of the cases, so gun for those first.

All the best! :)

My use of the phrase "directly or indirectly" is redundant now that I think of it. Connected is connected. Thanks for the feedback that really helps, the effort in trying to create a definition has helped and learning more why this is not straightforward is helping more. :) Also it is great to be reminded that go is always rewarding of more attention and analysis, and there is so much to discover from other peoples work on it. Not many games like that!

In the above, I should have used the term pass-alive rather than unconditional life because that would have been clearer.

Anyway deMangler, I saw your SL page and I wish you all the best in your journey! Don't worry at all about losing games in the beginning. You may have heard of proverbs like the best way to start at Go is to lose 100 games as quickly as possible.

You wrote in your SL page that you lose many games through not recognising false eyes. That's a common thing for beginners. There are some common simple situations that one can become familiar with quickly. I recommend having a look at Capture Three To Make An Eye - and I was thinking of a similar, related page Capture Two To Make A Half Eye? which doesn't exist at the moment - it may or may not exist at some point in the future.

I saw Xela's comments which are very sensible. I feel that for most beginners, trying to get precise / rigorous / formal definitions for some things is not be the best approach. Indeed, the New Zealand rules are short and precise - and they don't mention concepts such as life and death; they don't need to.

I'm sorry to tell you that what you're looking for simply doesn't exist! I looked for it too when I started learning go :-)

First you have the theory. Yes, it's true that if all stones are "connected" to two liberties, roughly speaking, then the group is alive. (Unkx80 and Malcolm have linked to more precise versions of this statement.) But this is not the only way to live! Exceptions in real games are rare, but if you want your definitions to be logically correct then you have to deal with things like moonshine life and hanezeki.

Then there's the practical side. It's common to end the game with some groups that surround large empty spaces but don't have two distinct eyes. Usually, two experienced players will agree that these groups are alive beyond all reasonable doubt, but won't require a mathematical or logical proof that they are 100% alive. (Unless the surrounded space is so huge that an invasion is possible, in which case one player won't pass but will have a go at invading.)

If you're playing with Chinese rules (or some other form of area scoring), you could of course fill in most of the space until you've created a pass-alive group, and there you have your mathematical proof. But this is time-consuming and annoying. If you make a habit of doing this past the beginner stage, people won't want to play with you. And under Japanese rules or territory scoring, you will also lose a lot of points by doing this. So at some point you have to get used to living with a tiny amount of ambiguity.

If you like simple logical, textual definitions, then you'll probably enjoy the New Zealand rules! They're not quite what you're looking for, but they are in a sense very beautiful.

The term live or alive can include seki which you missed. Further, alive can also include the cases where the defender's group cannot be captured regardless of how it is being attacked, provided that the defender mounts a perfect defense - see Alive - Introductory for an example. In this aspect, a group that is alive can become dead if the defender fails to defend correctly, or fails to respond to an attacking move at all (perhaps to resolve a ko, in which case the attacking move is a ko threat).

I guess what you are trying to define is pass-alive in a mathematically rigorous manner. However, there are at least two problems:

• You do not define the term indirectly.
• You seem to be describing properties of a pass-alive group, but due to the final or world in your sentence, merely satisfying these properties does not guarantee that a group is pass-alive. However, merely changing the final or to and appears to be insufficient (depending on what you mean by indirectly).