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This is a shape that comes up quite often in real games. Can black kill the white corner?

See J-group

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It seems so. After , is normal, but the placement at creates the possibility of a snapback at . White is dead here.

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If white pulls back, instead, he still cannot manage two eyes.

With the hane

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What if white also has the marked stone? Now can black kill?

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The tactic from before no longer work. Black can try to play the same move (to create the same dead shape), but white has a recourse with .

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If black tries the placement directly, white also lives. It seems there is no way to kill the white group when it the hane is in place.

With a turn

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How about when white has turned, but black has the hane?

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It seems black can kill white in this situation. If white jumps back instead and plays at , black simply pushes at .

With a turn, but a solid connection on top

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Notice that in this position, with the solid connection instead of the marked point, white can live.

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If black tries to reduce the space white can block. Black's cut doesn't work.

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Neither does this placement.

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Black has to throw in after white captures, so white has time to make the second eye.

The point of it all

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Consider a joseki like this 3-3 invasion after a pincer on the keima approach move to the 4-4 point.

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How do you choose between the squared points?

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If white expects to move with and , he better play and not the marked point. White's corner is alive so he can continue on the outside.