Life In 19x19

Why does analysis suggest R3 when White can just play S3 and kill Black?



also if black plays C4, cant white just ladder the stone and kill it?

this is the game in which the post-game analysis suggested R3 as the best move for Black - https://online-go.com/game/42351415

R3 forces white to choose between playing R2 and S3. If white plays S3, white can just capture like you say. But P1 becomes sente for black. If white plays R2 and white extends at S3, white can still play S2 to kill the stones, but then S8 is sente.

Basically R3 makes white choose, and black can have the resulting sente later.

C4 is a preparation for an endgame tesuji around C2.

Probe:

There are big endgame moves at P1 and S8. However each is gote. The cut sacrifices a few stones at no cost (unless it makes W alive when W wasn't), for a larger threat, not just a boundary play but also threatening the entire W group. It turns out that whichever way W responds, either P1 or S8 becomes sente. So it maximises the profit.