Recently I've encountered the following sequence is a joseki app:

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . 6 4 3 . . . . . . . . . .
$$ - . . 1 5 X O . . . O . . . . .
$$ - . . 2 X . O . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

My question is... what would happen if B2 is at W5 instead? It seems that white will be killed, e.g.,

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[go]$$W
$$ -------------------------------
$$ - . . . 0 . . . . . . . . . . .
$$ - . . 7 5 6 . . . . . . . . .
$$ - . . 1 2 X O . . . O . . . . .
$$ - . 9 3 X . O . . . . . . . . .
$$ - . 8 4 . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

and it becomes a L group, and according to http://senseis.xmp.net/?LGroup, white is dead in the corner.



Another question is.... in addition to the 3-3 invasion, is there any method for white to invade black's fairly large corner?

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . X O . . . O . . . . .
$$ - . . . X . O . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

Good questions!

In response to your first question, I think the usual response is to tenuki and leave aji in corner. Later on moves like "a" could live in corner (alternatively, white may opt to get forcing moves from the left side instead of living in corner). Usually, black 2 here is considered a bit wishy-washy and either b or c is more common.

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . . b . . . . . . . . . .
$$ - . . 1 2 X O . . . O . . . . .
$$ - . a c X . O . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

To your second question, the local shape is the 3-3, but it depends on the actual board. For example, if white has a move on the left side at a below then b, c, or d might work for white. Also, e and f are potential moves depending on the situation.

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . e X O . . . O . . . . .
$$ - . b . X . O . . . . . . . . .
$$ - . d f . . . . . . . . . . . .
$$ - . c . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . a . . . . . . . . . . . .[/go]

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . a . . . . . . . . . . . . .
$$ - . 5 4 8 . . . . . . . . . . .
$$ - . 7 1 2 X O . . . O . . . . .
$$ - . 3 6 X . O . . . . . . . . .
$$ - . 9 . . . . . . . . . . . . .
$$ - . 0 b X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

Now W can live with a. If B tries to prevent him from living after a then W at b is problematical and usually not worth the risk for B

_________________
Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

Actually, no. I don't consider that empty triangle block wishy-washy and it's not uncommon: according to GoGoD just in the corner shape (ignoring the position of the extension from white's 2 stones) it is more common than c with 189 hits to 100. b is most common with 249. I thought I had already written a detailed post about the various answers to this 3-3 invasion and what situations to use them but I can't find it . But in summary b cuts off white from the outside group, but makes it quite easy for white to live in the corner so is usually best when you can profit from attacking the outside group after white lives in the corner in gote. The block at 2 is the move which makes it hardest for white to get an unconditional success of something in the corner as, should black get additional stones on the left side, he can kill it if white tries to live later (sequences likes DrStraw's). The timing of when to activate the 3-3 stones and the related life-and-death problem is the key issue here. If white does live then the block of 2 can end up inefficient (compared to b). White will often tenuki like illluck said, the thinking being it was a good exchange to give black an empty triangle, but white does lose some other aji (e.g. d isn't sente now, but that's not often important). The block of c is focused on keeping the corner territory, but means white can get b in sente which helps the top group a lot. Something to remember is that black has 2 answers to that: as well as the tesuji of the first post there's this:

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[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . a 3 . . . . . . . . . .
$$ - . . 1 4 X O . . . O . . . . .
$$ - . . 2 X . O . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

It is a common beginner mistake to always play 4 at a because it's a cool tesuji, but it gives white sente reduction of the corner and white gets stronger on the outside. The connect of 4 is actually usually the best move (48 to 27 in GoGoD) so should be your first instinct. The cool tesuji is good if the black group is surrounded and in danger of dying or you are in endgame and territory considerations trump group strength and sente. And after that white should not unthinkingly push at a (though it can be a nice yose move later) as depending on the top group if white gets a stone at a or b later he can play c to save the corner stone and make a gote eye in the corner (this can be very useful if the top group is under attack).

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . c . O b a . . . . . . . .
$$ - . . O X X O . . . O . . . . .
$$ - . . X X . O . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . X . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

Perhaps more important than the above is understanding why this push at 3 is bad, and a common mistake that comes up a lot. Stop doing this and you will go up one rank, guaranteed or your money back . You correctly saw black will block at 4. The reason 3 is bad is that now if white plays a black can block at b and the marked stone is in a perfect place to defend the cuts. Go is all about efficiency: you should try to play moves which make your opponent's existing stones end up in bad places rather than good places. In fact now white can live in ko rather than die, can you see how?

But anyway, consider this situation from invading under a lone 4-4 stone:

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[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . 1 2 . . . . . . . . . . .
$$ - . . 3 X . . . . . . . . . . .
$$ - . 5 4 . . . . . . . . . . . .
$$ - . 7 6 . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

Here push at 3 is good because when black blocks at 4 white can hane at 5 and black usually extends so white can crawl again and make some eyespace and territory. Black can hane with 6, but then because of the cuts white can capture the outside stone which is ok. You can see that your position is like black already has a stone at a so this doesn't work.

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . 1 2 . . . . . . . . . . .
$$ - . . 3 X . . . . . . . . . . .
$$ - . 5 4 8 . . . . . . . . . . .
$$ - . 6 7 a . . . . . . . . . . .
$$ - . 9 . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

Another important shape that comes up a lot is if white has an approach stone at the top. This helps the 3-3 invasion considerably. I hope you can see white shouldn't push at a now because that makes the marked stone perfectly placed. Instead it is best for white to hane connect in sente with 3-6 and then slide to 7 (black has the choice to play 4 at a and let white connect). This way the marked stone is not so efficient for black and white can live more comfortably.

Click Here To Show Diagram Code

[go]$$W
$$ -------------------------------
$$ - . . . . . . . . . . . . . . .
$$ - . . 5 3 4 . . . . . . . . . .
$$ - . . 1 2 6 O . . . . . . . . .
$$ - . . a X . . . . . . . . . . .
$$ - . 7 . . . . . . . . . . . . .
$$ - . . . B . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .
$$ - . . . . . . . . . . . . . . .[/go]

One of my online lectures for the British Go Association's Shodan Challenge about the 3-3 invasion with the approach stone in place was recorded; you can see it here:

https://www.youtube.com/watch?v=csIMlfMzA_k

It covers some of this stuff in more detail (though not so much the 3-3 invasion in the kick shape iirc) with game examples of what situation to use different variations etc.