Tsumego elementary problem 157

I tried to solve this tsumego, black is on the move:

it seems to me that white can kill black only if he wins the ko two times. Is that correct? After first ko win he will connect at 1-3 and after second win he will take at 3-2.

Black lives:

\$\$B
\$\$ -------------------
\$\$ | . 1 . . X O . . .
\$\$ | O X . X X O . . .
\$\$ | . X X O O O . . .
\$\$ | O O O . . . . . .
\$\$ | . . . . . . . . .
\$\$ | . . . . . . . . .

White kills unconditionally:

\$\$W
\$\$ -------------------
\$\$ | . 2 1 . X O . . .
\$\$ | O X . X X O . . .
\$\$ | 3 X X O O O . . .
\$\$ | O O O . . . . . .
\$\$ | . . . . . . . . .
\$\$ | . . . . . . . . .

This doesn't work for Black, either.

\$\$W
\$\$ -------------------
\$\$ | . . 1 . X O . . .
\$\$ | O X 3 X X O . . .
\$\$ | 2 X X O O O . . .
\$\$ | O O O . . . . . .
\$\$ | . . . . . . . . .
\$\$ | . . . . . . . . .
• OK thanks, I didn't see the "black lives" sequence. I was only reading the sequence when black starts at 3-1 (right next to your "1"). I don't know why I missed this one. Commented Feb 10, 2015 at 15:45

I think black should play 2-1 then 3-1 and 1-3 are miai for b life, no need for ko