If you have at least 2 eyes then you’re safe. Here both white and black have so many eyes, you can stop counting eyes. But if you really want to count, then you can add A3 and B1 for white, which I see no reasons not to count as eyes, and G9, G7, H8 and J9 for black, which I see no reason not to count as eyes.
Note that the goal of the game is to have the most territory+prisoners.
Here, white has 19 prisoners, plus 12 territory: A8, A6, A3, A1, B2, B1, D4, D1, E3, G1, H3, J2 are all white territory. We also give 5.5 extra points to white to compensate for black’s first-player-advantage.
Black has 4 prisoners plus 6 territory: E9, E6, G9, G7, H8, J9 are all black territory.
Every intersection which is not covered by a stone is going to be a point of territory for someone, with the exception of C7 which is a neutral point. I suggest playing neutral points at the end of the game, to make it more visually clear that they’re no one’s territory.
So, Black has 6+4 = 10 points, White has 12+19+5.5=36.5 points, White wins by 26.5 points.
Note that the main goal is typically said to be to have the most territory. Prisoners tend to be a smaller fraction of the score. In this case, White’s score appears to come mostly from prisoners, but that’s only because Black played many aimless moves in White’s territory, so White filled their own territory capturing all the black stones, which doesn’t change the score.
If you had ended the game a bit sooner, then it would have been more clear that White won because of a large territory, not because of prisoners:
I think @Revolutionizer99 might be trying to clarify in their head what counts as an eye and what doesn’t.
But this definition is not correct:
the point at A, for example, is surrounded on all four sides by Black stones, but is not a real eye, as it can be played by White, atari’ing Black in the process. The Black group is dead, unless White somehow lets its surrounding stones be captured. The point at A is what’s called a “false eye”.
Basically the question you need to answer, to know if a hole is a “real eye”, is:
Can the opponent play enough moves inside the hole to capture my group? Or is that illegal?
A hole that is securely and completely surrounded by one color can also be too big to be an eye. One of the smallest examples is this:
It looks like Black has two eyes, and virtually it does. Normally you wouldn’t expect this group to be captured in a game. But it can technically be captured if Black doesn’t react:
Now Black is in atari, and if they don’t react, the group can be captured on the next move by White.
Now in this case Black can react when atari’ed:
But in other cases the hole might be too big to even do that:
Here, if Black thinks “whatever, I’m alive, I have two eyes”, it’s technically possible for White to not only atari from the inside, but even make life inside!
in this case, if Black doesn’t prevent this from happening, it’s not even possible for Black to survive after being atari’ed.
So, again, operative rule:
If the opponent can play enough legal moves inside one of your “eyes” to put your group in atari, then it’s not a real eye.
In practice, in intermediate and higher level games, most eyes never become “real eyes” in this strict sense, and saying that a group is “alive” usually means that it is virtually alive, as in, a good enough player has the ability to defend it no matter what.
But I think it’s good to be aware of these subtleties, and as a beginner, to start working your way up from these very concrete foundations.
A more practical and useful guideline for a double digit kyu might be:
If the opponent is as good as me, can they capture this group?
If the answer is “no”, it’s probably good enough for now. As you face opponents with different experiences and stronger, and as you practice tsumegos harder and harder, you will learn that some of your ideas were wrong, and your understanding of when a group is alive will gradually expand and improve.
But then again, I’m barely an intermediate player myself, so someone more experienced might chime in and disagree on what I said!
That’s not clear, since we don’t use any definition of how many eyes might be
possessed by a group consisting of more than one chain. Chain at Sensei's Library
(Once a set of stones is immortal, it wouldn’t matter whether it has just 2 eyes or instead 100 eyes.)
Also, the logic your calculation is based upon gives sufficient conditions, but not necessary conditions.
For example, see the Second,Third,Fourth corner group from Smallest Group with Two Eyes at Sensei's Library : White can never play inside them.
Note that the difference between true eye and fake eye can be slightly more subtle in some edge cases.
Really, a fake eye is something that looks like an eye, but where eventually your opponent can capture some of your stones, or force you to fill your own eye to connect those stones.
In many cases, this definition coincides with the definition you give, but there can be edge-cases. For instance, here are a few examples where black has either a true eye or fake eye at F1. For every example, can you tell whether F1 is a true eye or a fake eye?
1., 2. and 4. are correct. 3. is correct but I would prefer a more complete answer.
For 3., I would prefer a slightly more subtle answer: F1 will become a true eye or a fake eye, depending on who plays first. If black plays G2, we get the situation of 1., where F1 is a true eye and black is alive. But if white plays G2, then we get the situation of 2., where F1 is a fake eye and black is dead.
Note that in 4., black has two chains of stones: the chain E1-E2-F2, and the chain G3-H3-H2-H1-G1. The eye at F1 is a true eye, even though it doesn’t completely fit the definition your gave earlier.
We use the word “chain” for the concept defined by the rules, of stones which are truly connected by being directly adjacent.
“Group” is a concept which is completely useless to determine whether an eye is a true eye or a false eye.
For instance, in diagram 3, we’ll say that all black stones are part of the same group, but there are two chains of stones in the group; the stone at G1 is not perfectly connected to the chain of the other stones.