Basic endgame problem -- feedback requested

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Re: Basic endgame problem -- feedback requested

Post by Mnemonic »

:shock: Arrrg, what? I got slaughter!?! :sad:

Dan players just got a whole lot scarier :sad: Like you weren’t scary enough already :sad:

Answering jts
For your (i), how do you count your moves? Is first black, then white 1 or 2? If it's 2 then 8 seems reasonable, otherwise correct down to 3 :sad:
About your (ii), ddk are pretty good at life and death. I mean the reason we got to 10k is because our groups survived (That is was with 2 point in gote, completely enclosed is another topic, but that's what sdk is for :) )


And then, can someone please explain where I went wrong? :scratch:
:b6: at :wc:
Click Here To Show Diagram Code
[go]$$W copy of mw42's run
$$ ---------------------------
$$ | . . X O . . . O O . 0 . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 7 . 9 . O O O X O . |
$$ | . X X X X . X . X O 8 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 5 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 4 W O X O O . O 3 . |
$$ ---------------------------[/go]


Click Here To Show Diagram Code
[go]$$Wcm11 copy of mw42's run
$$ ---------------------------
$$ | . . X O . . . O O . X . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O O . O . O O O X O . |
$$ | 2 X X X X 3 X 1 X O X X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X 4 X . X O X . X . |
$$ | . . O X O X . X O X X X . |
$$ | . O O 5 O X . X O X O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . O . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O X X . X O O . O O . |
$$ ---------------------------[/go]


My :b2: puts the white stones in atari at :b14:. Now, you cannot capture the black stone, but it is sente (and you should play sente before gote, right?) or is it just a stylistic choice?

:b4: ???
:b6: ?????

The rest looks pretty much like what I had, but I really don't understand the bottom side. Isn't that just a 2 point gote move? (And then a 1 point gote move??)

Edit: @ Bill
Yes, that would have helped a lot. Although I would also like to know why :) At least I figured the 1,5 gote moves out.
2. Edit: The same for me. The only way I could read this was to actually write down what I was thinking and double checked everything. A few more moves weren't that much harder. (Like I said, there is no chance in Hell that this will ever play a role in any of my games)
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Re: Basic endgame problem -- feedback requested

Post by Phelan »

Loons wrote:I'm a little curious; ye fellow sdks who are saying "seems very hard" - could you go into more detail ? Which step(s) are ambiguous, for example. I know I find the e-1 affair a little niggle-y to consciously think about (a two point gote that makes a new two point gote if black plays it? Is that how you describe it?), but not really write-off material.


I missed this move, for instance:

Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . . . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O . . . . O O O X O . |
$$ | . X X X X . X . X O . X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O a O X . X O . O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . . . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O . O O X O O . O . . |
$$ ---------------------------[/go]

My guess for first move was 'a'. Other than that, it's hard for me to calculate the point values of gote moves and pick them in order. I think I do it better when the values are further apart, or there are less zones of contention. I'm often surprised by the score in my games.
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Re: Basic endgame problem -- feedback requested

Post by daal »

Bill Spight wrote:
My thought was to be realistic by adding some small, supposedly easy plays. What if I eliminated them?


Better? Worse? Meh?


Better. All of the plays are small and easy, but since the problem asks us to find the right order of plays, less choices would make it easier to handle. Fewer possibilities would also make counting and comparing less of a chore.
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Re: Basic endgame problem -- feedback requested

Post by mw42 »

Mnemonic wrote::shock: Arrrg, what? I got slaughter!?! :sad:

Dan players just got a whole lot scarier :sad: Like you weren’t scary enough already :sad:

Answering jts
For your (i), how do you count your moves? Is first black, then white 1 or 2? If it's 2 then 8 seems reasonable, otherwise correct down to 3 :sad:
About your (ii), ddk are pretty good at life and death. I mean the reason we got to 10k is because our groups survived (That is was with 2 point in gote, completely enclosed is another topic, but that's what sdk is for :) )


And then, can someone please explain where I went wrong? :scratch:
:b6: at :wc:
Click Here To Show Diagram Code
[go]$$W copy of mw42's run
$$ ---------------------------
$$ | . . X O . . . O O . 0 . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 7 . 9 . O O O X O . |
$$ | . X X X X . X . X O 8 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 5 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 4 W O X O O . O 3 . |
$$ ---------------------------[/go]


Click Here To Show Diagram Code
[go]$$Wcm11 copy of mw42's run
$$ ---------------------------
$$ | . . X O . . . O O . X . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O O . O . O O O X O . |
$$ | 2 X X X X 3 X 1 X O X X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X 4 X . X O X . X . |
$$ | . . O X O X . X O X X X . |
$$ | . O O 5 O X . X O X O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . O . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O X X . X O O . O O . |
$$ ---------------------------[/go]


My :b2: puts the white stones in atari at :b14:. Now, you cannot capture the black stone, but it is sente (and you should play sente before gote, right?) or is it just a stylistic choice?

:b4: ???
:b6: ?????

The rest looks pretty much like what I had, but I really don't understand the bottom side. Isn't that just a 2 point gote move? (And then a 1 point gote move??)

Edit: @ Bill
Yes, that would have helped a lot. Although I would also like to know why :) At least I figured the 1,5 gote moves out.
2. Edit: The same for me. The only way I could read this was to actually write down what I was thinking and double checked everything. A few more moves weren't that much harder. (Like I said, there is no chance in Hell that this will ever play a role in any of my games)


:b4: is a 2 point gote move (2 prisoners for black if he plays there, 0 points for white if she plays there). :b6: is also a 2 point gote move (1 point of territory for black if he plays there, 1 prisoner for white if she plays there). :w5: is also a 2 point gote move (1 point of territory for black if he plays there, 1 point of territory for white if she plays there).
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Re: Basic endgame problem -- feedback requested

Post by Mnemonic »

@ mw42
Ah, it's starting to make sense (not really :) ), so :w3: , :b4: , :w5: , :b6: and :w7: are all 2 point plays. Is there a reason why black takes the 2 stones at :b4: ? With that he adds an additional 2 point play, doesn't he? (Thereby make it an uneven number with white getting one more.) Isn’t that bad for black? What I'm asking is, is this also a correct sequence?

Click Here To Show Diagram Code
[go]$$Wc
$$ ---------------------------
$$ | . . X O . . . O O . . . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 8 . 9 . O O O X O . |
$$ | . X X X X . X . X O 0 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X 2 X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O 3 O X . X O 4 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 6 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 5 O O X O O . O 7 . |
$$ ---------------------------[/go]

Click Here To Show Diagram Code
[go]$$Wcm11 Also posible?
$$ ---------------------------
$$ | . . X O . . . O O . 2 . . |
$$ | X . X O O O O . O . X X . |
$$ | . X . X . O . O O O X O . |
$$ | 3 X X X X 1 X 4 X O X X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X X X . X O X . X . |
$$ | . . O X O X . X O X X X . |
$$ | . O O O O X . X O X O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . X . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O O O O X O O . O O . |
$$ ---------------------------[/go]


Edit
Seems like you answered my question already in the PM.
There is also a subtlety with D1 you might be missing.

Scenario 1: White plays D1 (+2), then black (-2), white (+2), black (-2) exchange two point moves. Therefore, white gains 0.

Scenario 2: White plays avoiding D1 (+2), black (-2), white (+2), black plays D1 (-2), white plays E1 (+2). Therefore, white gains 2, but at the loss of sente.

Many thanks :) (But is my :b2: in the above diagram right or wrong?)
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Re: Basic endgame problem -- feedback requested

Post by Dusk Eagle »

Endgame is definitely an area I urgently need to work on. The original problem overwhelmed me (though I'd like to come back to it from time to time to hone my skills). Anyway, my first thought in this simplified problem turned out wrong when I counted it, but I believe I have solved it.
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O O O X . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 4 . X . O O O X O . |
$$ | X X X X X . X O X O O X . |
$$ | O O O O X . X . X X X X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 3 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 6 W O X O O . O 5 . |
$$ ---------------------------[/go]

:w7: at :wc:. :b4: and :w5: are miai, and can be played before :w3: if desired.
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Re: Basic endgame problem -- feedback requested

Post by daniel_the_smith »

If 10 kyus are supposed to get these problems, I must be really really bad at endgame... Do you all have a table of values memorized or do you calculate them out each time?
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Re: Basic endgame problem -- feedback requested

Post by Mnemonic »

daniel_the_smith wrote:If 10 kyus are supposed to get these problems, I must be really really bad at endgame

Dusk Eagle wrote:The original problem overwhelmed me

Dan players starting to look cute again :mrgreen:
Anyway, now that I understood the theory behind the move I formally request that Bill (or anybody else) makes another problem. I would like to see if I could get the next one. :rambo:

And mad props to Bill Spight for the first one :tmbup:
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Re: Basic endgame problem -- feedback requested

Post by quantumf »

daniel_the_smith wrote:If 10 kyus are supposed to get these problems, I must be really really bad at endgame


I know what you mean...otoh, it seems that some basic assessment and counting could pay off significantly.
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Re: Basic endgame problem -- feedback requested

Post by Dusk Eagle »

For the record, I considered myself bad at endgame before this thread. Though I still think Bill said 10k as a practical joke.
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Re: Basic endgame problem -- feedback requested

Post by Violence »

Well, I'm a 10 kyu.

I don't think I solved this correctly, but then again, I'm bad at endgame. To make this like a real endgame situation, I like to try and solve them in under 10 minutes.

I don't really count the value of moves very precisely, when things get close, I just... read out the endgame board states.

So here's my unrefined, "blunt club to the face" answer to this Gordian knot.

To hell with fractional endgame values. We're doing this simple and dumb.

Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . x . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O x . x . O O O X O . |
$$ | x X X X X . X x X O x X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X x X . X O X X X . |
$$ | . O O . O X . X O x O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . x . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O x O O X O O . O x . |
$$ ---------------------------[/go]


First all, let's look at all the endgame plays. I will mark them with x's.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . c . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O b . c . O O O X O . |
$$ | c X X X X . X c X O c X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X a X . X O X X X . |
$$ | . O O . O X . X O b O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . b . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O b O O X O O . O b . |
$$ ---------------------------[/go]


Knowing just the basics of counting, I put all the endgame plays that are roughly the same value together. A is worth a whopping 4 points, b's are worth around 2-3, and c's are worth around 1. I doubt I've lost anyone at this point. Let's play A because it's the biggest, and move on.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . c . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O b . c . O O O X O . |
$$ | c X X X X . X c X O c X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O a O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . b . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O a O O X O O . O b . |
$$ ---------------------------[/go]


And then we come here, and the first decision has to be made.

I've remarked the diagram. the b's are exactly 2 points, no tricksies. The A's are where things get weird, right? If black captures two stones at the bottom, it creates another 2 point gote play.

What an odd situation! Gee, I sure wish we could see the end of the game at the end of each branch. Oh wait, we can read.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . x . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O x . x . O O O X O . |
$$ | x X X X X . X x X O x X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X x X . X O X X X . |
$$ | . O O . O X . X O x O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . x . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O x O O X O O . O x . |
$$ ---------------------------[/go]


I don't know what the hesitation with reading is here. I know for a fact that people can read sequences deeper than than number of moves left in this game, so if you aren't comfortable with memorizing value calculation, "When in doubt, read it out!"

Here's we'll see if black takes the 3 points raw, or the 2 points that generate another 2 point gote. First, we have him take 3 points raw. Naturally, white takes the other one. The final three 2 pointers are bigger than the rest, so we take them in turn.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . b . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 6 . a . O O O X O . |
$$ | b X X X X . X b X O a X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 4 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 3 O O X O O . O 5 . |
$$ ---------------------------[/go]


Now we come to the 1 pointers, and we see we have 3 b's and 2 a's. The 2 a's are miai, right? both are 1 pointers that have 1 points after you play them. The b's are just straight up 1 pointers. We play a's first.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . a . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 6 . 8 . O O O X O . |
$$ | a X X X X a X a X O 7 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 5 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 3 O O X O O . O 4 . |
$$ ---------------------------[/go]


4 1 pt plays, they become miai. Finish the game!

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . 9 . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 6 . 8 . O O O X O . |
$$ | B X X X X 0 X W X O 7 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 2 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 5 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 3 O O X O O . O 4 . |
$$ ---------------------------[/go]


Now count the score! Wait, is that too hard? Not a problem, let's just do a comparison! Let's go back and see what the other board looks like if black moves the other way.


[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . x . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O a . x . O O O X O . |
$$ | x X X X X . X x X O x X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 3 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . a . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 2 a . X O O . O a . |
$$ ---------------------------[/go]


Same value, 2 points a piece. Let's make it look as much like the other one as possible.

[Hide]
Click Here To Show Diagram Code
[go]$$W White to play. Japanese rules. 6.5 komi.
$$ ---------------------------
$$ | . . X O . . . O O . 0 . . |
$$ | X . X O O O O . O . X X . |
$$ | . X O 7 . 9 . O O O X O . |
$$ | B X X X X W X W X O 8 X . |
$$ | O O O O X . X . X X . X . |
$$ | . . O X . X . X O X . X . |
$$ | . . O X 1 X . X O X X X . |
$$ | . O O . O X . X O 3 O X . |
$$ | . X O O X X X O O X . X . |
$$ | . O . O O X X . O X X X O |
$$ | . . O . 5 . X O O O O X X |
$$ | . . O O X X X O . O O X . |
$$ | . . O 2 4 . X O O . O 6 . |
$$ ---------------------------[/go]


What's the difference between these two diagrams? In the previous one, Black just captured 1 stone and made 3 points on the right, and in this one, he captured two stones and made 3 points on the bottom. Looks pretty equivalent to me! Maybe our choice didn't make a difference after all.

Let's count, finally.

Black, 28 points.

White, 29.5 points.

White wins by 1.5.


And yes. Your reading will be wrong sometimes. Mine probably is right now. I might be off by... a whole point!

But the way I see it, while I respect the mathematics behind the counting methods, when you apply it to other aspects of the game, the ability to calculate fractional endgame becomes rather useless.

But the ability to visualize and analyze the board state 7-8 moves from now?
That's pretty useful.

This is how I usually recommend kyu players do endgame. I feel like too many people stop reading and start counting. Ideally, you should do both, but if given the choice between one or the other, I'll stick with my reading.
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daniel_the_smith
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Re: Basic endgame problem -- feedback requested

Post by daniel_the_smith »

Heh, that's what I try to do to solve a problem like this. For me it's different from normal reading because everything is scattered all over the board. (In a game I just play whatever looks biggest unless it's really close, then I try this and fail).

Maybe if I ever gain another couple stones I'll be able to do full-board reading at that level of detail... (actually the arrow of causality probably goes the other way.) As it is, comparing points from more than two variations rapidly becomes extremely difficult for me...
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Re: Basic endgame problem -- feedback requested

Post by Bill Spight »

Wow! The responses have been terrific! Many thanks! :)

(I have tried to thank everyone at least once. :))

John Fairbairn calls for a new approach to writing about yose. I am afraid that he might call my approach just another way of justifying the numerical approach. <sigh> But I do want to satisfy two audiences, regular go players and mathematically inclined players. (I do write papers for mathematicians and computer scientists, as well, but that's another matter. :))

Your feedback has been very helpful. :)

Now for breakfast and a write-up on the problem.

P. S. About aiming at 10 kyus: This is close to minimum complexity (I thought) for a full board problem. So it is definitely in the beginner range. ALso, it seems to me that there is a low correlation between yose skill (aside from tesuji) and go strength. I once made a problem that I thought was elementary, but to my dismay it stumped a dan player. Fortunately, a DDK solved it. ;)
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Re: Basic endgame problem -- feedback requested

Post by daniel_the_smith »

I'm still curious, for those who can do these problems: Do you all have a table of values memorized, do you calculate them out each time, or do you just read everything out and compare results?
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Re: Basic endgame problem -- feedback requested

Post by Dusk Eagle »

I imagine that with experience, the value of certain moves would just be memorized naturally over time. If you constantly refer back to something (like, to take a mathematical example, the value of sin (0)), it eventually just gets ingrained in your memory.

There was something confusing me about calculating the value of one of the positions on the board, but as I typed it out to ask I figured it out. I'll just leave it here in case I forget or if anyone's curious to see my thought process.
Click Here To Show Diagram Code
[go]$$B
$$ -------------
$$ . O X . . X .
$$ . O O X X X .
$$ . . . . . . .[/go]

If white plays first, he has one point. If black plays first, he has one point. So, if I'm not mistaken, the count here is 0 and a play gains 1 point.

Click Here To Show Diagram Code
[go]$$B
$$ -------------
$$ . O . O O X .
$$ . O O X X X .
$$ . . . . . . .[/go]

In this case, if white plays first, he has no points, and if black plays first, he has 2 (from captures) + 0 (from the resulting position) = 2 points. So black has one point in this position, and a play here seems like it gains one point.

So, if black were to capture, and then white recaptures, it seems like they both gain one point, which should leave the count unchanged. What I was missing originally is that the count in the second diagram is +1. I overlooked this and was thinking of it as zero, which made me question how miai counting could say that the score after black captures and white recaptures is zero when clearly black gained a point.
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