DDK Fuseki Question

In the pictured situation, the AI is strongly recommending black to play E3. I don’t really understand the AI followup to that move.

To me, playing E3 leaves cutting points at E4 and D5. The E4 cut leads to the death of the E3 stone in return for some thickness along the left? And the D5 cut leads to the death of the C5 stone in return for some thickness along the bottom? How should I interpret this in relation to the rest of the board?

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I am not that much a fuseki person, and dont think I’d find E3 myself without knowing it in advance.
I think the idea is after E3-E4/D5-F3/C3 picks the two stones and E3-D5/E4-C4 (D4 may be better I guess if it is still a safe capture)/C3 and you exchange 1 stones, but the stone on B4 isn’t that useful?
And although I’m unsure black’s C3 could be delayed if there is bigger moves on the board, and if white doesnt cut immediately cut with E4 or D5, black can fix cut points with E5 and like white group wont have a way out easily.

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This isn’t a fuseki question (and not just because pedantically fuseki is not synonymous with opening but means arraying of stones so contact fights aren’t fuseki), it’s a local shape question. The rest of the board is irrelevant and the double hane is just the local move to make efficient shape. By offering one stone as a sacrifice the others will become well placed and lead to some inefficiency in White’s (b4 ends up in wrong place if white captures c5).

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White to play above, you wouldn’t play b4 now would you? Where is better?

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Hmm in this scenario as white, I would want to do something about the double-atari that black could play at C3. A tiger’s mouth looks to me like it’s a little too loose, I think I’d just connect at C3.

There’s also some potential for D2 to indirectly guard the C3 point and also peek under black’s stones a bit. Maybe it could be used to pincer those stones later and take away their base?

Is there a reason that white would naturally cut at the D5 point vs the E4 point?

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Right, c3 connect is a great move. It saves d3 whilst also ensuring c5 can’t escape because then you can block at b6 and b4 isn’t an atari anymore. As you say tiger mouth too loose because b4 is then atari after b5 b6. The problem with d2 is c3 is atari then c2 captures it.

b5 capture is good too, very clean to capture the stone (now e5 isn’t atari so black can’t build that centre wall for free), it lets black take d3 but then you get sente, and if black doesn’t take it then you can do d2 next to take the corner and undercut black as you said.

Not particularly, but this way offers a nice clean tewari lesson.

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