How do I chose invasion or reduction points?

Question

I have the following board:

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . a . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . b . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . c . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Now, white territory is rather large. So I like to decrease it!

However, I have the following problems/questions/considerations:

1. I will lose every KO battles elsewhere on the board (I have around 70 points hanging that I will win, but my opponent can play 5-10 stones that I must react at.)
2. I can invade both a, b and c. Does invading at a make b impossible? Is b even possible? Is c even possible?
3. How many points should I estimate for a invasion at a, b and/or c?
4. Should I start this invasion as soon as possible or delay it?

Answer 1

First off, count the current score. If you're winning, do not invade. Just make sure you keep white small enough that you still win.

Let use further presume that black would be losing. You counted, so you know by how many points. A move around c could be parried with a sacrifice of two white stones while sealing off the rest of the upper right corner completely. So, if you're only behind a bit, you should probably play around c.

If you need to invade deeply, you probably made a mistake earlier - why do you need to invade that late, after leaving such a large area for the opponent? Try to find a better invasion or reduction time when you review the game, and look out for similar situations in the future. Good invasion and reduction timing is very important, and you'll soon learn to avoid trouble like this beforehand.

But maybe you could not prevent it (or made a honest mistake), and arrive at the current board, far behind. You need to invade deeply. You also already identified a few usual spots at a and b, but there are many more to consider.

When invading against that many opponent stones, you cannot expect a good result. Considering it's roughly 10 vs 1, even living small in ko should be considered a success. That black always ends in gote is understood. I'm aware that, especially for weaker players, it may sometimes seems more like a gamble, but don't worry about that. We'll try to find a few useful ideas how to deal with this. This is actually a very, very broad topic and I'll try to stick to the most important things.

In advance, I apologize to the reader if the following is too short an explanation. I'm writing this mainly for players in the 4 kyu to 12 kyu range, as that's where these problems typically become most prevalent, though weaker players may hopefully be able to profit from it, too.

Direct invasion at 3-3

If black invades at a, I would definitely want to kill it as white:

$$cm1
$$ -----------------------+
$$  . . . . . . . . . . . |
$$  O . . . . . 8 6 . 5 . |
$$  O . X . . O . 7 1 . . |
$$  . X . . . . . O 2 3 . |
$$  O O O . . . . . . 4 . |
$$  X X O O . O . . . . . |
$$  . . X . . . . . . . . |
$$  . . . . . . . O . . . |
$$  . . X X . O . . . . . |
$$  . . . . X . . . . . . |

A useful technique can also be black 4 in the following diagram, which kind-of-reverts to the above. You may want to remember this move, for if applied at the right time, it can throw off an unaware opponent completely.

$$cm1
$$ -----------------------+
$$  . . . . . . . . . . . |
$$  O . . . . . . . . . . |
$$  O . X . . O . 2 1 . . |
$$  . X . . . . . O 3 . . |
$$  O O O . . . . . 5 4 . |
$$  X X O O . O . . . 6 . |
$$  . . X . . . . . . . . |
$$  . . . . . . . O . . . |
$$  . . X X . O . . . . . |
$$  . . . . X . . . . . . |

You should consider other points for invasion, too. I'm sorry that I won't show many variations, for that would inflate this post too much. Please research them on your own - it will be worthwhile, as these will surely come up in your games sooner or later.

Attaching the corner

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O 1 . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

This is a variant that is usually played as a regular joseki, however, it is unlikely that a joseki would develop on this board - white has too many stones that allow her to deviate for a better result.

But it is not all moot yet. Its basic idea is: When you're weak, you want to touch the opponent and force him to either draw back strongly (white grants you additional stones in the area for almost free), or touch back (typically with a hane), which often leads to cuts in their shape that you can try to use to force an exchange. This is actually an extremely important idea to keep in mind in such situations.

Please research the (too) many possible variations on your own. Joseki books should give a good points which moves are important, but remember that white is free to deviate.

Submarine, straight below hoshi

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . a . . |
$$ | . . . , . . . . . X . . . . . O b 1 . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . c . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

This is a rather common joseki move, too, though again, everything is different on this board. White usually replies at a, b or c.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . 2 . . |
$$ | . . . , . . . . . X . . . . . O . 1 . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 3 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white protects the corner with 2, black would surely extend to somewhere around 3, and hope for living or connecting somehow.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . b . |
$$ | . . . , . . . . . X . . . . . O 2 1 . |
$$ | . . . . . . . . O O O . . . . . . a . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O d . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white bumps down at 2, black can decide between a and b, which are probably miai. If black a and white b, black would again play at or around d next.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . b . . |
$$ | . . . , . . . . . X . . . . . O . 1 . |
$$ | . . . . . . . . O O O . . . . . a . . |
$$ | . . . . . . . . X X O O . O . . . 2 . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

One of the strongest moves is this white 2. Note that if black a, white b, and black has extremely poor eye shape, which is not acceptable on this board. So black must play at b next, and the corner becomes a tsumego, of which I don't know the resu..--- eh, I mean, which I assign to you as homework.

Submarine, low keima from the hoshi

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O a b . |
$$ | . . . . . . . . O O O . . . . . c 1 . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

This is another usual point, and white can reply at a, b, c, or possible elsewhere.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O 2 . . |
$$ | . . . . . . . . O O O . . . . . . 1 . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 3 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Nobi defense at 2 would revert to a similar situation as before, and black would look forward to play at or around 3 next. Life is not guaranteed here, though. It seems the 1-2 exchange benefits white more than black, because it removes a lot of options in and nearby the corner.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O 3 2 . |
$$ | . . . . . . . . O O O . . . . . a 1 . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Tsuke tobi at 2 is unlikely on this board as it seems to give black too much wiggle room: Refer to the joseki after the moves 1-2-3-a.

$$cm1
$$ -----------------------+
$$  . . . . . . . . . . . |
$$  O . . . . . . . . . . |
$$  O . X . . O . . 4 6 . |
$$  . X . . . . . O 3 2 . |
$$  O O O . . . . . 5 1 . |
$$  X X O O . O . . . . . |
$$  . . X . . . . . . . . |
$$  . . . . . . . O 7 . . |
$$  . . X X . O . . . . . |
$$  . . . . X . . . . . . |

White would never play atari from the inside at 4, as then black could play 5 in sente and extend to 6, which is too good for him.

Low keima approch

You suggested a regular low approach. White would probably defend the corner solidly, or kick.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O a b . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O c . . |
$$ | . . . . . . . . . . X X . O . f d e . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white defends the corner at a or b, black would probably move back to play c and hope for a double hane (c-d-e) (or possibly crosscut with c-d-f) and good exchange. As a and c are miai, if white blocks at c, black would play at a (or possibly 3-3) next, and hope to live locally.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . 2 . . |
$$ | . . . . . . . . X X O O . O . 3 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 4 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white kicks, black cannot afford the usual move at 3, because white would immediately come back and play 4, after which black is heavy. Note that the 3-3 offers a lot of aji, but here white is so strong that even that won't suffice.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . 4 . |
$$ | . . . . . . . . O O O . . . . . 2 3 . |
$$ | . . . . . . . . X X O O . O . a 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 5 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Black would instead hane at 3, after which white would surely play 4 (alternatively, white a may be interesting and not expected by any opponent). Black would then attach back at 5, and it would not be easy for white to capture the whole group.

Various exchange offers

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . a b . . |
$$ | . . . . . . . . . . . . X . . c d . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

The moves around a to d give white the option to seal off the corner, while black captures 2 white stones. If this is acceptable for either player, and which of the moves is best, requires just a lot of normal, boring reading and counting.

Hybrid approach

If you remember the attachment idea from above, you may think about this move:

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . a c . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . b d . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white draws back solidly (at a or b), black got an almost free stone, which will help him live locally. This is too soft in this case, so white would surely hane at c or d.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . 2 4 . |
$$ | . . . . . . . . . . . . . . 6 O 1 . . |
$$ | . . . . . . . . . . X X . O . 5 3 . . |
$$ | . . . . . . . . . . . . X . 7 . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

A very, very simple sequence could be black drawing back, then white just protecting the corner. Note that this would be better than black 3, white 1, which we considered earlier.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . 2 5 . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . 4 3 . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white pushes above at 4, black will definitely hane at 5 and try to live locally. This is dangerous for white, as she risks a lot more points if black lives (and he likely will, as white must spend another stone below 4 to connect her two stones and prevent black from moving in that outwards direction.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . 2 . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . 3 . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Black may also entertain the idea of counter hane, which may reasonable, as he's not much weaker than white on the right side - the two white stones are weak! The counter hane is a bit complicated, and will lead to an exchange, the quality of which depends on the players' fighting abilities.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . b . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . a 2 c . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white outside-hanes at 2, black can think about a, b or c, which again depend on the players' fighting abilities. Black has several thinkable continuations here, because white's two stones are not strong.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . c . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . 3 2 . . |
$$ | . . . . . . . . . . . . X . . b a . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If black crosscuts, white's primary options seem to be drawing back at a, which increases liberties and also enables black 1 to be caught with c, so black is under some pressure.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . d O 1 . . |
$$ | . . . . . . . . . . X X . O b 3 2 a . |
$$ | . . . . . . . . . . . . X . c 4 e . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white ataris at 4, black would not counter hane at a, because if he wanted to do that, he should play a directly instead of 3. So black will move out at b, and white can continue with c, d or e, depending on her reading of the variations. White is in control of the direction more than black.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . a 3 . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . c 2 b . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If black pulls back at 3, the obvious replies are a, b or c. White a would be followed by black b, which seems to let b live or connect out eventually.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . 8 . |
$$ | . . . . . . . O O . X . . O . . 6 7 . |
$$ | . . . , . . . . . X . . . . . O d 5 . |
$$ | . . . . . . . . O O O . . . . 9 . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . e 3 . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . a 2 4 . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

In response to white 4 or a, black would probably slide towards the corner at 5, and push for eye space with the sequence to 9. Considering d (and likely e, too) is sente, black may live there - if he believes he would not, he should chose different moves earlier.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . b . . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . a 2 3 . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

The counter hane at 3 would likely be followed by white a or b.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . c . |
$$ | . . . . . . . . . . . . . . . O 1 . . |
$$ | . . . . . . . . . . X X . O . 4 2 3 . |
$$ | . . . . . . . . . . . . X . . . d 5 . |
$$ | . . . . . . . . . . . . . X . a . . . |
$$ | . . . . . . . . . . . . . X . O O b . |
$$ | . . . . . . . . . . . . . X X X X . . |

Magari at 4 invites black 5, which connects out or likely captures the two white stones with a, if white resists with the descend at b. Black can also consider playing c instead of 5 first, after which white would atari, black connect, white extend, and finally black d to connect out.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . 4 . . |
$$ | . . . . . . . . . . . . . . . O 1 5 . |
$$ | . . . . . . . . . . X X . O . a 2 3 . |
$$ | . . . . . . . . . . . . X . . . 6 . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

White 4 is a strong reply to 3, and up to 6, black is in danger. Black would look for an exchange, using aji such as the cut at a.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . 4 5 . |
$$ | . . . , . . . . . X . . . . . O 3 . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . 2 O a . |
$$ | . . . . . . . . . . . . . . . O X X . |
$$ | . . . . . . . . . . X X . O . . O X . |
$$ | . . . . . . . . . . . . X . . . O . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

For instance, if white replies to black's clamp at 1 with 2, black would play tsuke at 3 (instead of just a) to increase his options, and white becomes quite busy. Black should be able to live somewhere.

Note that black should not just play 1 at a, for white would then play 1 and black would be short of eye space.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . 5 O 2 . |
$$ | . . . . . . . . . . . . . . . O X X . |
$$ | . . . . . . . . . . X X . O . . O X . |
$$ | . . . . . . . . . . . . X . . . O 3 . |
$$ | . . . . . . . . . . . . . X . . . 4 . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white just cuts down at 2, black would white cut at 5, possibly exchanging 3 to 4 first. Black is still looking for an exchange, and the several sente moves he gains while white is trying to live the 2 stones should be sufficient.

Answer 2

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . a . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . X X . O . . c . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

b looks good as a probe, it makes miai of following up with either a or c. Note the symetry in white's shape. Below a possible follow up. After black 7, a and b are miai. White needs to be careful as there's still a weakness at c.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O 2 . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . c . . . . |
$$ | . . . . . . . . . . . . . . . O 4 b . |
$$ | . . . . . . . . . . X X . O . 6 3 . . |
$$ | . . . . . . . . . . . . X . a 7 . 5 . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

If white blocks on the other side, a san-san invasion becomes possible with the help of the R14 stone. Black gets a smaller territory than in the previous diagram, but still has options around R10 for later.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . 8 7 . |
$$ | . . . . . . . O O . X . . O . 6 3 9 . |
$$ | . . . , . . . . . X . . . . . O 4 5 . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 2 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

The attachment at R16 is usually a good stabilization move, but doesn't work so well here because of the surrounding white stones. The black shape is left without eyes even if Q16 gets captured.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . 8 . |
$$ | . . . . . . . O O . X . . O . . 4 5 . |
$$ | . . . , . . . . . X . . . . . O 3 6 . |
$$ | . . . . . . . . O O O . . . . . 7 . . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 2 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Starting at the san-san is another interesting idea. White might kill the corner, as explained in mafu's answer, but the aji of the cut at a will force white to pull back after black 9. To avoid this, white can choose to let the corner live small. If white starts a ko, then black will have ko threats around Q11.

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . 8 6 . 5 . |
$$ | . . . . . . . O O . X . . O . 7 1 . . |
$$ | . . . , . . . . . X . . . . . O 2 3 . |
$$ | . . . . . . . . O O O . . . . . a 4 . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O 9 . . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Whether you start with R14 or R17, the key idea is to use that first stone as a probe and be ready to abandon it.

Answer 3

I decided to revisit this post with the help of Leela Zero.

LZ's assessment is that there is no good move for black above the 11th line. The sequences are fascinating, here is the start of one:

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . 6 . |
$$ | . . . . . . . O O . X . . O . . 3 . . |
$$ | . . . , . . . . . X . . . . . O . 5 . |
$$ | . . . . . . . . O O O . . . . . . 4 . |
$$ | . . . . . . . . X X O O . O . . 1 . . |
$$ | . . . . . . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . O . 2 . |
$$ | . . . . . . . . . . X X . O . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O . . |
$$ | . . . . . . . . . . . . . X X X X . . |

Black's best move is at R11. The sequence would then depend on the value of the initiative, here is one that is similar to what I posted 3 years ago:

$$cm1
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ | . . . . . . . O O . X . . O . . . . . |
$$ | . . . , . . . . . X . . . . . O . . . |
$$ | . . . . . . . . O O O . . . . . . . . |
$$ | . . . . . . . . X X O O . O . . . . . |
$$ | . . . . . . . . . . X . . . . . . 8 . |
$$ | . . . . . . . . . . . . . . . O 2 7 . |
$$ | . . . . . . . . . . X X . O . 4 1 . . |
$$ | . . . . . . . . . . . . X . 6 5 . 3 . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . . . . . . . . . . . X . O O 9 . |
$$ | . . . . . . . . . . . . . X X X X . . |

Answer 4

Many question in one post ...

An invasion on a should work, because the corner is not yet 100% secure. White will probably build up strength to the outside leaving no potential on b. White will probably get the initiative afterwards.

An invasion on c looks very big, probably bigger than the corner invasion because of the two white stones to be captured. Maybe c is not the correct point to start the attack because a capping move by white (on the point left to c in the diagram) is a strong response, but the region is certainly interesting

An invasion on b is probably not feasible, but it can act as a question move which side white wants to defend.

Answer 5

Given the size of White's territory, you are probably behind. Then the question is, can you catch up, and if so, how? I'll save the first question for last, and answer the others in order:

2) You have the choice of ONE invasion point, a, b, or c (or some alternatives mentioned by other posters). Choosing one strengthens White sufficiently to preclude the others. Of three choices, b is "neither here nor there," and is least likely to work. If you want to invade around there, use the second-line "submarine" tactics suggested by others.

3) You can hope for a 10-15 point reduction by invading at c. That's not small, and is enough to decide an otherwise close game in your favor. An invasion at a) is potentially larger, 25-30 points, but it will be in ko, which is to say that you might net only the 10-15 that you would get "for sure" with c.

4) The decision should have been made earlier, but "better late than never." This is your "last clear chance" to win the game.

1) The ko that would result from an invasion at a is quite large. If your position is so shaky that you would lose every fight for a 25-30 point ko, you're probably lost anyway. If c doesn't work for you, your best hope is that you win the ko at a, by losing only 5-10 points to a ko threat, and winning 25-30 points as compensation.