GokyoSeimyoVolume3Problem4/Attempts

Vital point  

is the key move (although it fails). Examining where it succeeds and then the one refutation will lead to the solution.

Ko  

If black captures with , white starts a ko in the corner to capture three stones. Black cannot connect because of oiotoshi.

Ko or double damezumari  

This , preventing the throw-in, results in ko again. Although defending with at is no longer oiotoshi, white will then play at a and it is double damezumari yet again.

Double damezumari  

If black plays to prevent the ko, results in double damezumari, and thus captures the stones.

Throw-in  

Preventing the ko with doesn't work either. White throws in with , and after it is double damezumari again.

Refutation  

However, after one black can play 2, and if white throws in at three black is safe after 4. Therefore..

Correct sequence  

White must make the throw in of 1 first, then play the vital point 3. Then no refutation is available.