Thank you 1001 times, @yebellz.
P.S.: Take a look at Jasiek’s First Life and Death.
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Week 1: The L-group (is dead)
Alright, let’s kick off this study group by looking at the L-group, which I think is a good place to start, since this pattern is very fundamental to understanding and evaluating other patterns.
Sensei’s Library: L-group at Sensei's Library
Cliff Notes:
- Black is dead (even if it is black’s turn to play)
- “L-group is dead” is a proverb to remember, since other positions can be reduced to the L-group
- However, still need to be able to read out how to kill under various resistances by black, and for some variations on the concept
- Still dead even in variations with hanes
Problems from Sensei’s library:
- Capturing the L-group at Sensei's Library (with links to a few problems)
- Beginner Exercise 229 at Sensei's Library
- Beginner Exercise 230 at Sensei's Library
Problem set from @antonTobi
Forum thread discussing further: Puzzle collection to practice killing the L group
Cheat sheet courtesy of @shinuito
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Ah-oh. I’m in trouble. I have learnt this one in this past… and then forgotten it… and then learnt it… and then forgotten it again. 
So when it comes up in games I am working it out fresh each time, though I’m confident I usually get it right.
Is it called the ‘L-group’ because of how it looks in this configuration?
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Yes, I believe it’s called the L group because it looks like an L in that orientation.
The basic thing to remember is that black is dead even if it is his turn to play. You don’t have to memorize every response to black’s futile attempts to live, but just practice related problems enough that you can read it out during a game.
It is fundamental since some related groups can be reduced into an L group in order to kill. As the defender, you need to avoid being reduced to this dead shape.
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Week 2: Rectangular Six in the Corner
I originally considered looking at one of the shapes related to the L-group for this week. The shapes listed under that link are slightly more complex descendants of the L-group, since analyzing them often makes use of what we’ve learned about the L-group (i.e., that it is dead).
However, before we move onto to those, I thought it might be worthwhile to look at another related shape (although it is more like a sibling to the L-group), the “Rectangular Six in the Corner”:
Sensei’s Library: https://senseis.xmp.net/?RectangularSixInTheCorner
Summary Notes:
- Life and death status depends on the number of liberties
- Zero liberties: unsettled, black lives by playing at a or b, white kills by playing at a
- One liberty: white can play at a to create an approach ko
- Two liberties: black is unconditionally alive
- Looks like an L-group, but with two descents closing off the corner space
Problems from Sensei’s Library:
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Week 3: L+1 Group (first type, “wider”)
Sensei’s Library: https://senseis.xmp.net/?L1Group#toc1
The L+1 group is a direct extension of the L group. There is an extra stone, giving the group a “leg” to stand on. The leg could be on either side, so there are two varieties. The above pictures and focus for this week is just one of the those varieties, which I am calling “wider”, since it makes the L group larger along the shorter dimension.
Summary: unsettled (white can kill, black can live).
Here are four interesting figures:
- The most straight forward way for white to kill is simply hane to kick out black’s leg. This reduces the shape to the basic L-group, where black is dead. Any resistances by black are answered the same way as for the L-group.
- White can also kill by playing inside at black’s vital point. This continuation reduces it to a bent four in the corner, where black is dead.
- If it is black’s turn, black can play on the vital point to secure life. The extra leg stone let’s black focus on making two eyes with move 3.
- Here is the corresponding continuation under the original L-group. Note what a difference the leg makes.
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Week 4: L+1 Group (second type, “longer”)
Sensei’s Library: https://senseis.xmp.net/?L1Group#toc6
Summary: unsettled (white can kill, black can live)
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Week 5: L+2 Group
Sorry, it’s been a bit longer than a week since the last post.
Sensei’s Library: https://senseis.xmp.net/?L2Group
Summary: alive
Hanes might threaten to reduce this shape to a dead shape, but each can be correctly answered with inside play in order to secure two eyes (marked with circles).
The L groups
This shape wraps up a main theme seen from the L group, through the L+1 groups, and onto the L+2 group:
- With no legs (L group), you are dead.
- With one leg to stand on (L+1 groups), things are unsettled and you need initiative in order to live, before your last leg is kicked out from under you.
- With two legs to stand on (L+2 group), you are alive.
Of course, there are a lot of complications along the way, with variations to each of these shapes.
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Shape 6: Bent Four in the Corner
I’ll stop calling these “weeks” now.
Here is the reference shape for “bent four in the corner”.
The reference shape is unsettled
Of course, black can make life:
However, white can start a ko fight:
If white wins the ko fight, black can be captured:
However, white might not want to start the ko fight if the ko situation in unfavorable.
If black has one external liberty, white must end the ko fight differently:
Although black is dead, black may later capture the three white stones (forcing white to play again at 5) as a ko threat should another ko fight arise elsewhere.
Lives with two external liberties
If black has two external liberties, then black does not need to fight a ko to live:
So, there is no urgency for black to play an additional stone inside until white has reduced external liberties to one.
Temporary sekis that can be reduced to bent four in the corner
The basic bent four in the corner shape is unsettled and hence may live or die depending on whether if black plays again to make life or if white can win the ko fight. This pattern also arises as a product of other positions that can be reduced to this shape.
For example, these two (temporary) sekis can be reduced to a bent four in the corner shape:
Above, white could play at A to force black to capture four white stones, in order to yield the bent four in the corner, with white’s turn to play. Similarly, white can play another stone inside in the below shape to do the same:
In these two sekis, black’s stones are usually dead, even if there are other big ko threats remaining on the board.
Under Chinese (and other area-scoring based) rules, white can wait until the end of the game to resolve this position. After other more valuable moves are finished, white can first remove ko threats and then initiate the sequence. The exception is if there are unremovable ko threats (like those that arise from a seki elsewhere). Under that exceptional case, it may turn out that black’s stones in the corner are strategically alive (see Life and Death under Chinese Rules for an example).
Under Japanese rules, white can simply leave the position untouched and then apply the special ko rules of the life and death resolution procedure, which essentially disallows any ko threats besides passing. Even though black’s corner may be uncapturable during normal play (due to large, unremovable ko threats), white can still argue that the stones are dead by not initiating the ko fight during the game.
It should also be noted that under the modern Japanese rules, it is not automatic that bent four in the corner is always dead either. Here is an example where two interacting bent fours in the corner produces a seki: http://warp.povusers.org/go/UndeadBentFours/
References
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I was hoping that this thread would start more discussion and participation. As a prompt:
- Would any one else like to make a post about a shape? Please feel free to.
- Also, please feel free to share problems or other comments about any of the shapes discussed.
- I made the most recent post a wiki, so any one can edit and add to it if they wish, or you can feel free to make your own post, if you prefer. Unfortunately, the older posts in this thread seem to be uneditable (I guess because of age) and I cannot convert them to wiki.
Shape 7: Six Die, Eight Live
Eight stones on the second line are alive
Six stones are dead
Seven are unsettled
Sensei’s Library: https://senseis.xmp.net/?SixDieButEightLive
Excercises
Status of the following shapes?
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Dead, first to play, alive.
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Why hadn’t I seen this thread before? I’ve made the L-thing mistake countless times. 
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(Haylee’s voice): “It’s daaaaad” 
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I don’t know why, but I’m able to edit all your posts in this thread that have an image in. I added a little dot to your last post to show this. First time I’ve been able to do this and I have no idea how.
Edit: Oh, you set them up as Wiki pages! Gotcha. I’m dumb.
Edit 2: I removed the dot again 
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What are wiki pages?
Pages any trusted user can edit, I guess.
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Users that have reached the “trust level” of “regular” can turn their own posts into wikis that can be edited by almost everyone (except for new users, I guess to prevent spam). Since you (@Gia) are a “regular”, you can do this as well. Click on the icon that looks like three dots on any of your recent posts, then click on the wrench icon, and you will get the option to turn your post into a wiki.
@bugcat, I’ve been making my more recent posts (unfortunately, it does not seem possible for me to edit or convert older posts into wikis) in this thread wikis in case others wished to add to them, or to correct any mistakes. I’m hoping that this thread would be more of a discussion where we all work through life and death shapes together, but I feel like my presentation style in the older posts might have made things more like a series of lectures. So, I’m trying something different to see if that might increase participation.
Update: thank you to @flovo for helping to convert the older posts in this thread into wikis.
Note: I’ve consolidated a couple of older posts into the original post.
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