All you need to know about Carpenter's Square - 20

Diagram 20.1: White to play
Does Black square help in any way?

Diagram 20.2: False hope
Black 2 is incorrect. Black is expecting this sequence similar to Diagram 19.2. Black square would help in this case. However......

Diagram 20.3: Black fails
White 5 is the correct answer. Note that white can use this sequence in Shape 19 as well.

Diagram 20.4: Continued (12=*bs*, 13=*bx*)
Black is one liberty too short and cannot play 'a'.

Diagram 20.5: Black fails
So black has to block at *b2*. However, Black 4 is incorrect. This result is the same as Diagram 19.3 and black fails to make use of *bs*.

Diagram 20.6: Solution 1
Unlike Diagram 19.4, Black 4 tesuji now works. Thanks to Black square black can throw in Black 10 to start a ko fight.

Diagram 20.7: Variation
White 5 may atari from the other side, forming a different ko. But note that black takes the ko first so this result cannot be good for white.

Diagram 20.8: Solution 2
Since Black 4 in the previous diagram appears to be the key point, White 3 wants to occupy that location first. To avoid the failure in Diagram 20.5, Black 4 is a must. White 5 is a nice idea. Black has to go back to Black 10 to avoid the Bulky Five killing shape. So the result is a ko. If black wins the ko fight by connecting at *w5*, white can still play 'a' to form a seki, destroying 9 points of territory, although black also has a yose at 'b' that can get a few points back. Overall this solution seems to be slightly worse than Solution 1 for black.

Diagram 20.9: White fails
White 1 is a thoughtless move, hoping black to block at 'a' so that knowledge in Shape 7 can apply. But Black 2 is good - please verify that black is unconditionally alive now.
To summarise: black corner is quite resilient because of the hane move. Diagram 20.6 and Diagram 20.8 are the right solutions, resulting in ko.