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[go]$$B
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 5 2 X O . .
$$ | . . 9 . 7 6 4 8 . .
$$ +---------------------[/go]

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[go]$$B
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 5 2 X O . .
$$ | . . . 7 6 . 4 . . .
$$ +---------------------[/go]

1. I knew it by memorization
2. was clearly the best try, so I just assumed it was right
3. was clearly the best try, and I could tell at a glance that White's clamp at doesn't work
4. was clearly the best try, and I had to verify that doesn't work for White, which was clear after reading out to move N (please specifiy N
5. Other?

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[go]$$B What I saw
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X . . . X O . .
$$ | . . . . 1 . . . . .
$$ +---------------------[/go]

BTW, this is the move that would have leaped out at me when I was a raw beginner. It is not actually so bad.

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[go]$$B First attempt, seki
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 5 X O . .
$$ | . O X 2 6 4 X O . .
$$ | . . . 3 1 . 7 . . .
$$ +---------------------[/go]

is kind of stupid. at 5 is much more sensible.

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[go]$$B First attempt, death
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 4 X O . .
$$ | . O X 2 . 5 X O . .
$$ | . . . 3 1 . 6 . . .
$$ +---------------------[/go]

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[go]$$B Second try
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X 1 . 3 X O . .
$$ | . . . . 2 . . . . .
$$ +---------------------[/go]

Same points, but switching the order of play.

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[go]$$B Variation
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 . 2 X O . .
$$ | . . 7 . 5 6 4 . . .
$$ +---------------------[/go]

Bingo!

Note: Not switching and in the previous diagram, but preventing the connection at 3 first. Zwischenzug.

----
Analysis (with thanks to Znosko-Borovsky):

General comment: Life and death problems are mainly about single point eyes and damezumari. Why single point eyes? Because space is constrained.

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[go]$$B Candidate eyes
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X C X . X O . .
$$ | . O X . C . X O . .
$$ | . . . C . C . . . .
$$ +---------------------[/go]

The first thing to do is to identify the likely one point eyes. This position has four of them.

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[go]$$B First seen
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X C X . X O . .
$$ | . O X . C . X O . .
$$ | . . . C 1 C . . . .
$$ +---------------------[/go]

The move that leapt out at me is not too bad, because it potentially helps to form three one point eyes.

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[go]$$B Solution, first move
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X S X . X O . .
$$ | . O X 1 C . X O . .
$$ | . . . C . C . . . .
$$ +---------------------[/go]

But why isn't obvious? It actually forms a one point eye, and in addition potentially helps to form two more.

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[go]$$B Solution, var. 1
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X 1 C 3 X O . .
$$ | . . . C 2 C . . . .
$$ +---------------------[/go]

destroys potential three one point eyes, because if any of them becomes an eye, it will have to include . Unfortunately for White, encloses for the second eye.

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[go]$$B Solution, var. 2
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 a 2 X O . .
$$ | . . 7 . 5 6 4 . . .
$$ +---------------------[/go]

destroys two potential one point eyes and manages to escape, but and form a second one point eye. Because of damezumari White cannot now play at "a".

Again, why isn't obvious? It forms a one point eye.

1. Increase/decrease the eyespace for living/killing
2. Usually this will reveal the vital point, when one color decisively lives or kills by playing there

In your example:

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 1 X O . .
$$ | . O X 4 a . X O . .
$$ | . . 2 3 . . . . . .
$$ +---------------------[/go]

B1 increases, W2 decreases, B3 increases, W4 cuts (and next A will create a killing shape)
We already see W4 as a likely candidate

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[go]$$B Vital point at 3?
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 1 X O . .
$$ | . O X 3 . . X O . .
$$ | . . 2 5 4 a 6 . . .
$$ +---------------------[/go]

So What if B3 plays there? Then Bill's point comes to the surface: W4. After W6 Black can not throw in at A because this will create shortage of liberties for himself

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[go]$$B Different way of increasing
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 2 X O . .
$$ | . O X . 4 3 X O . .
$$ | . . 1 . . . . . .
$$ +---------------------[/go]

In previous diagrams, the hane was lethal, so what if Black increases his eyespace there? White will break through at W2 and next W4 is atari.

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X a X O . .
$$ | . O X 1 . . X O . .
$$ | . . . . . . . . . .
$$ +---------------------[/go]

Back to our suspect: B1 as the vital point. As we can see, it also protects against the cut at A. So this is our solution

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 2 X O . .
$$ | . O X 1 . 3 X O . .
$$ | . . . . . . . . . .
$$ +---------------------[/go]

As said, B3 is the obvious answer, if W2 again decreases the space.

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X 1 . 3 X O . .
$$ | . . . . . . 2 . . .
$$ +---------------------[/go]

The same point is the answer if W2 decreases the eyespace from the other side.

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 . 2 X O . .
$$ | . . . . . . 4 . . .
$$ +---------------------[/go]

This makes W2 a tempting point for White to try and kill the black group. After connecting with W4, she seems to succeed.

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[go]$$B Increasing and decreasing eye space
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 . 2 X O . .
$$ | . . b . 5 a 4 C . .
$$ +---------------------[/go]

This is where the stylish but not so obvious B5 comes into play: it makes miai of a and b to live, using the shortage of liberties for White due to the circled point being empty and White's stones not being firmly connected.

Indeed, your first instinct was the same as mine. It follows the "split the eyespace in two" principle. I think one reason this problem took me a little while is that I spent a little too long trying to make that move work.

"Making the one point eye" (another good principle, as you point out) is what eventually motivates my discovery of the correct solution. I think one reason it takes me a little while to find is that my instinct is that White will clamp in response, and there goes my second eye. One reason I am asking people here is that I'm curious how clear it is to people right away that White's clamp won't work after .

Do you instantly see that White's clamp won't work?

Your systematic approach is very much like what In-Seong Hwang recommends (unsurprisingly); instead of being tricky right away, try the simplest move first, and if it doesn't work, refute your opponent's refutation (often by playing in the same spot).

In this case it turns out that the refutation to the simple move (your second diagram) is actually pretty tricky itself, involving shortage of liberties many moves in. So here it is nice that your instinct actually leads you to the "trickier" move first.

Do you instantly see that White's clamp won't work?

It's hard to say: in the meantime I have analyzed the position and I'm inclined to say, "no but the sequence is a one way street"
But in days when I'm much involved with Go, the shape will be so familiar that I will know the clamp doesn't work, without necessarily going through the whole sequence.

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[go]$$B Hitachi number 1
$$----------------------------
$$| . 2 1 a . . . . . . . . .
$$| C . O X X b . . . . . . .
$$| . . O O O X . . . . . . .
$$| O . O . . X . . . . . . .
$$| . O X . X . . . . . . . .
$$| X O X . . . . . . . . . .
$$| . X . . . . . . . . . . .
$$| . . X . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .[/go]

The decrease/increase exchange of 1-2 instantly reveals the marked vital point. We also see that White A provokes Black B

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[go]$$B Hitachi number 1
$$----------------------------
$$| . 2 1 4 . . . . . . . . .
$$| 6 3 O X X 5 . . . . . . .
$$| . . O O O X . . . . . . .
$$| O . O . . X . . . . . . .
$$| . O X . X . . . . . . . .
$$| X O X . . . . . . . . . .
$$| . X . . . . . . . . . . .
$$| . . X . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .[/go]

For example, Black has no time to cut, because of the danger 4 represents. Next White plays the vital point.

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[go]$$B Hitachi number 1
$$----------------------------
$$| . 2 1 4 . . . . . . . . .
$$| 3 a O X X 5 . . . . . . .
$$| b c O O O X . . . . . . .
$$| O . O . . X . . . . . . .
$$| . O X . X . . . . . . . .
$$| X O X . . . . . . . . . .
$$| C X . . . . . . . . . . .
$$| . . X . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .[/go]

Let's play B3 and remove the complexity of 4-5
This reduces the problem to its canonical form: can White live? Candidates are a/b/c but in all cases, the surprisingly effective move Black has at the marked spot, does the trick.

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[go]$$B Hitachi number 1
$$----------------------------
$$| 8 . a 2 . . . . . . . . .
$$| 1 4 O X X 3 . . . . . . .
$$| 6 . O O O X . . . . . . .
$$| O 7 O . . X . . . . . . .
$$| b O X . X . . . . . . . .
$$| X O X . . . . . . . . . .
$$| 5 X . . . . . . . . . . .
$$| . . X . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .
$$| . . . . . . . . . . . . .[/go]

If Black goes after the vital point immediately, we get this. Black should start with the hane, reducing the eyespace first, before moving into the vital point.

As the proverb says: "there is death in the hane".

Well, the clamp actually takes away your second and third potential eye. Fortunately, the kosumi to the first line threatens to capture the clamp (making an eye) and to make the fourth potential eye. As for the damezumari, yeah, that's lurking in the back of my mind, but you gotta check. I do, anyway.

Switching moves is obviously a heuristic that I use. But to my mind making the one point eye is the simplest move. Later it's the first line kosumi that's a bit tricky.

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[go]$$B dfan's solution
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 5 2 X O . .
$$ | . . 9 . 7 6 4 8 . .
$$ +---------------------[/go]

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[go]$$B Bill's solution
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 . 2 X O . .
$$ | . . 7 . 5 6 4 . . .
$$ +---------------------[/go]

Speaking of that kosumi, why did you choose it over the atari? Compared to my line, Black now does have a small ko threat, but White has a big ko threat. I'm trying to figure out what Black gains from holding the last exchange in reserve. Do you want to keep the possibility of the throw-in (where my is)? I could have thrown in there first with my , but maybe we don't know yet whether it's a good idea. Or do you just not want to play forcing moves that you don't have to?

Can't white make a ko with the atari instead of the cosumi by playing your variation's 7?

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[go]$$B What ko?
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 5 2 X O . .
$$ | . . . 7 6 . 4 . . .
$$ +---------------------[/go]

I didn't see immediately. But looking at Black's shape it was clear white could easily reduce black into a shape too small to live - therefore there must be a vital point. seemed like a plausible candidate. White clamp seemed to be the most severe response. Then I read out from there to completion (not hard in this case because no interesting variations) and saw that because black has a sente move black can get the extra eye.

First, I tried, as Black, moves that increase the vital space. They didn't work.
Then I tried the central move on the first line. Didn't seem to work either.
Then I tried the right move, because it partitions the vital space, and prevents White stones on the left to connect.
I read this variation in my head :

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[go]$$B Variation
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X 3 X O . .
$$ | . O X 1 . 2 X O . .
$$ | . . 7 . 5 6 4 . . .
$$ +---------------------[/go]

came into my mind because it threatens to cut White. After 7, the connection of the black stones is protected by jachung (inducing the opponent into self-atari).
I've become familiar with Jachung thanks to the Level Up books. Now it pops up in my head.

I missed the fine print beneath and was reading how to kill

Things that popped up ( in order of appearance ):

( unclear; to be verified )

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[go]$$W var 1
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X . 1 . X O . .
$$ | . . . . . . . . . .
$$ +---------------------[/go]

( looks promising )

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[go]$$W var 2
$$ | . . . . . . . . . .
$$ | . . O . . . . . . .
$$ | . . . O O . . . O .
$$ | . O O X X O O . . ,
$$ | . O X . X . X O . .
$$ | . O X . . 1 X O . .
$$ | . . . . . . . . . .
$$ +---------------------[/go]

Quick reading said var 2 probably works;
then I went back to investigate var 1, to see if it also works.