bleep wrote:I've just started reading a book on the opening, and it tells me that invading the corner is the standard answer to the pincer (Opening Theory made Easy). That makes sense, but the problem I have is reconciling that to invading the corner early, which I've always believed to be a poor choice. Don't the two ideas directly contradict each other, and if that's the case, what makes it suddenly ok to leap into the corner early in the game?
Despite his self-deprecating comments that accompany the following...
Boidhre wrote:...The answer on whether you go straight to 3,3 or jump out depends on the rest of the board...
...Boidhre has the essential idea correctly. Unfortunately, defining and interpreting 'the rest of the board' is often a complicated proposition.
I'm going to make a superficial pass at that that huge task.
As a preliminary matter, let us remember that 'invading the corner' is often a misnomer. A 4-4 stone does not own the corner. It is not his, therefore you can't invade it.
I'd prefer to phrase it as 'taking the corner'. When the opponent has a 4-4 stone, taking the corner is something you can do, and it is also something that he can do, because - despite that 4-4 stone - nobody has done it yet.
As a second preliminary matter, let us recall that pros tell us that a wall is generally worth about 2.5 to 3 points per stone, so that in a situation like this:
Bc
$$ | . . . . . . . . . . . . . . . . . .
$$ | . . . X X X X . , . . . . . , . . .
$$ | . . . O O O O . . . . . . . . . . .
$$ | . . . . . . . . . . . . . . . . . .
$$ | . . . . . . . . . . . . . . . . . .
$$ ---------------------------------------
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[go]Bc
$$ | . . . . . . . . . . . . . . . . . .
$$ | . . . X X X X . , . . . . . , . . .
$$ | . . . O O O O . . . . . . . . . . .
$$ | . . . . . . . . . . . . . . . . . .
$$ | . . . . . . . . . . . . . . . . . .
$$ ---------------------------------------[/go]
...white is getting 2 points of territory per stone, thus black is showing a net profit of 1/2 to 1 points per stone by playing on the fourth line.
1) With the preliminaries out of the way, let us look at the simplest example, the 'invasion' of the lone 4-4 stone:
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . W . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . W . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
Black can make a wall with
, which threatens to kill the white stone, so white replies with
. Black is showing a net profit here already. Next he can hane, which forces white down one row, making black's net profit about 1.5 to 2 points per stone. White makes a slight change of direction to ensure eye space with 4 thru 7, then continues with 8, 9, and 10, and black finishes with 'a'. ( This is the simplest variation. There are more complex variations, but let's stick to the natural play for the sake of example. )
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . 7 . . . . . . . . .
$$ | . 5 . . . . . . . . . .
$$ | . 4 1 X . . . . . , . .
$$ | . 6 W 2 3 9 a . . . . .
$$ | . . . . 8 0 . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
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[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . 7 . . . . . . . . .
$$ | . 5 . . . . . . . . . .
$$ | . 4 1 X . . . . . , . .
$$ | . 6 W 2 3 9 a . . . . .
$$ | . . . . 8 0 . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
If the rest of the quadrant is empty as shown in this example, this is a net profit of something like 7 points for black. (
and
are partially masked by their neighbors, so they count a bit less, whereas 'a' is worth a bit more because of it's influence in two directions. Counting exactly is hard, but I seem to recall a pro saying about 7 points. )
In short, this is a bad play for white because black gets more in influence than white gets in territory.
To underline the point, let's look at a slightly contrived counter example:
$$Bc
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . O . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . W . . . . . O . . O
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . O . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . W . . . . . O . . O
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
Here, white can take the corner territory at a net profit, for black's influence is nearly neutralized by the nearby white stones.
$$Bc
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . X . . . O . . . . .
$$ | . X . . . . . . . . . .
$$ | . O X X . . . . . , . .
$$ | . O O O X X X . O . . O
$$ | . . . . O O . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . , . . . . . , . .
$$ | . . O . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . X . . . O . . . . .
$$ | . X . . . . . . . . . .
$$ | . O X X . . . . . , . .
$$ | . O O O X X X . O . . O
$$ | . . . . O O . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
In the resulting situation, white does very well. Indeed, black may have to struggle just to live.
2) Next is the 4-4 stone with 2 extensions:
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . c . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . d . X . . . . . X . .
$$ | . . a . . . b . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . c . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . d . X . . . . . X . .
$$ | . . a . . . b . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
There are three general areas to invade in black's framework: the corner 'a', and two sides 'b' and 'c'. ( There are also hybrids like 'd', part corner, part side, but in practice they tend to resolve to one or the other. )
The problem with a side invasion...
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . 8 . . . . .
$$ | . . . 3 . . . . . 9 . .
$$ | . . . . . . . . . . . .
$$ | . . . X . 2 . 4 . X . .
$$ | . . . . 1 W . 6 5 . . .
$$ | . . . . 7 . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
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[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . 8 . . . . .
$$ | . . . 3 . . . . . 9 . .
$$ | . . . . . . . . . . . .
$$ | . . . X . 2 . 4 . X . .
$$ | . . . . 1 W . 6 5 . . .
$$ | . . . . 7 . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
...is that with rather simple moves, black can chase white and secure both the corner and some of the other side. And white is still not clearly alive. ( These may not be best moves by both sides. They just illustrate the general nature of the position )
By contrast, the corner invasion is absolutely alive.
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . a . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . X . . . . . . . . .
$$ | . X . . . . . b . . . .
$$ | . O X X . . . . . X . .
$$ | . O O O X X X . . . . .
$$ | . . . . O O . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . a . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . X . . . . . . . . .
$$ | . X . . . . . b . . . .
$$ | . O X X . . . . . X . .
$$ | . O O O X X X . . . . .
$$ | . . . . O O . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
...and there are still invasions or reductions of the sides with moves like 'a' or 'b'.
Again, the benefits of an invasion are apparent when contrasted with a diagram in which black gets the 3-3 first:
$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . c . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . X . .
$$ | . . X . . . b . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------
- Click Here To Show Diagram Code
[go]$$Bc
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . , . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . c . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . X . . . . . X . .
$$ | . . X . . . b . . . . .
$$ | . . . . . . . . . . . .
$$ | . . . . . . . . . . . .
$$ ----------------------------[/go]
Now the corner is out of reach, and even the side invasions can be attacked vigorously.
This is all just my attempt to knit together bits and pieces of theory that I've picked up here and there. I'll leave it out here to be critiqued by stronger players. If they don't trash it entirely, I'll come back after lunch and continue.
stone is questionable. With the three space pincer it is correct.
, and a Black pincer, 
invades the corner. 