Rectangular six in the corner

No outside liberties  

If there are no outside liberties:

Black can live with 5 points of territory by playing either a or b,

White can kill Black unconditionally by playing at a.

White to kill  

After , Black can't play at a or b because of shortage of liberties. Black is dead.

One outside liberty  

When there is exactly one outside liberty, the best White can do is to make it an approach ko. The vital point is at a.

Main line  

Black's throw in at is necessary: if White gets to play there she makes it a bent four in the corner. starts the ko. If Black wins this ko, by playing again at and then playing at a capturing the two White stones, the position is settled with Black living in the corner. Should White win this ko by connecting at . Black captures four stones at a and White sets up a direct ko for life as shown in the next diagram.

Continuation  

After (Black a in the previous diagram), White sets up another ko with and . She can finish the ko by connecting at , creating a bent three nakade shape. So, White has to win the ko twice to kill the corner and Black only once to live. ^[1]

White's failure  

With one outside liberty, this fails. Black can play because in this case he is not in atari.

Two outside liberties  

When there are at least two outside liberties, Black is unconditionally alive.

if White is stubborn  

through is copied from the one-outside-liberty main line. Now Black can play the surprising move of (oshitsubushi) because he still has at least two liberties. White cannot connect at so Black will play there to make two eyes.

kevinwm: Aside: Even in New Zealand rules where w can play a9, it is not even a ko threat to make a ko for the corner. (That is, a9 does not make ko for the corner, and does not even make ko for the corner if b does not respond. if w a9, b tenuki, w a8, b a9, w b9, and b a7 makes oshitsubushi again.)

 

Suppose there are exactly 2 outside liberties.

External kos after the other liberty is filled  

If external kos are more likely to occur when the corner has no outside liberties than when the corner still has an outside liberty, then White filling one of those liberties directly is better than White starting inside.

After , White has no ko threats as long as is empty, but will get two ko threats when is played.

External kos before the other liberty is filled  

If external kos are more likely to occur when the corner still has an outside liberty than when the corner has no outside liberties, then White should play like this instead of starting with .

and each threaten more than starting with would threaten, and after , threatens at least as much as starting with it would've threatened.

After , White has one ko threat regardless of whether is empty or filled.

Weakness  

If is in place, the weakness of the connection enables White to create a ko. White takes first with .

Black first: It is alive with at .

White first: It is ko after as shown in the diagram.

Black is dead  

In this diagram, Black is dead.

No, this is not a seki. Why?

Explanation 1  

After and fill up all the outside liberties, the stones are caught in a connect-and-die.

Explanation 2  

If prevents the connect-and-die at any point of the game, he turns his group into a straight three big eye instead and dies in gote.