A classic tsumego that occurs when a hoshi with double-ogeima is invaded a 3-3. I found a random pro game on Waltheri (from 2012 between rather strong Chinese pros) in which this shape occured:
Though there are other ways to invade a hoshi+ogeima, the 3-3 is the most obvious invasion point when there is the double ogeima, as the formation’s symmetry suggests.
The sequence you should be familiar with. What would you do next?
In this game, Black chose to make a ko.
Ko seems to be the correct solution, but did you know that this invasion can live unconditionally, despite the second ogeima stone?
Please try to find an alternative sequence before reading on!
So again, ko is the correct solution in praxis. You will probably never see the unconditionally living sequence in a game.
I am showing it below and you can learn it as a random trivia knowledge. But I am not responsible for anything that happens, should you try it at home. 😡
Behold, the tsumego solution is…
…the tigermouth at 5. You may stop beholding now.
Again, this is a totally impractical move and only mentioned here for your amusement.
Evidently, If Black takes one stone, White is alive in the corner.
If Black wants to try to kill, Black has to attach inside. What is the status of the corner?
When Black kills off the eyes inside, White can use the cutting point outside to capture three stones on the tail.
But the point is, Black can let White live with two points. Black does not need to play 10 immediately, any stone outside that prevents the cutting sequence is sente. And then Black still has a6/a7 sente. So this is too miserable a life.
Compared to White playing ko, Black is actually relieved about this position. If you do see this sequence in a game, either White is not very strong or White is already winning by a lot.
Let’s see what happened in the pro game:
(There are some missing numbers in the diagramme, and I guess you can figure out that once every three moves, the ko is retaken.)
After a series of ko threats, White was first to run out of threats and contended himself with the hane at 24, a sort of reverse-sente with the implied continuation 26 and 28.
Meanwhile, Black won the ko with 25 and the difference with the previous diagramme is very dramatic. White’s shape on the outside is so broken that White still felt like adding a move at 30 (that does not fix everything yet).
Basically Black won the ko in sente so he gets the next move at 31. However, in the end White won the game by a nice attack on Black’s upper group.