Taking Liberties

This Go problem was presented to me by my Tsumego Pro mobile app this morning.  It's not particularly difficult, but I think it nicely demonstrates some ideas about liberties.

It's Black's turn to go, and it's pretty obvious that a race is on, and the winner is going to take some stones.  If Black wins it'll take the 3 white stones in a row, and if White wins it'll take the six connected black stones just below.

A connected group of stones is taken prisoner when the other player removes it's last "liberty", i.e. the last place into which that connected group can grow.  At the moment White has 4 liberties and Black has 3.  So it would seem Black is on to a loser, since the players can just take turns removing one liberty at a time from each other's stones.  However, it's not quite as simple as that.

Black's move can leave White's liberties unchanged or reduce them by one. It can also leave it's own liberties unchanged or reduce them by one.  This presents 4 combinations for the effect of Black's next move:

Black's next move and White's response for each of the four categories

What should Black do?   We can write off any move that reduces it's own liberties leaving White's unchanged (e.g. bottom left).  That would be mad.  The obvious thing to do is top right: remove a White liberty and leave Black's liberties unchanged.  The problem is White will respond in the same manner, bringing the game closer to the end, with the overall liberty difference unchanged (and still in White's favour).

The two possibilities that remain are moves which do not change the liberty difference: either a move which removes both a White liberty and a Black liberty (bottom right), or a move which leaves both player's liberties unchanged (top left).  On the face of it, both of these seem like bad moves since, although the liberty difference is unchanged, White has been handed the next mover advantage.  However, the move shown top left, which leaves all liberties unchanged, is in fact a winning move.  That's because it leaves White with a choice of moves which does not include the ideal "one for nothing" option, whilst Black keeps "one for nothing" moves in it's own arsenal.   This more than compensates for handing White the next mover advantage.

I'm not great at Go, but this app has really helped me develop my intuition for endgames.  I don't usually analyze problems in this much detail, but for some reason today I felt the need to understand how it was possible Black could win given the apparent losing position it started with.  There's probably nothing special about this particular problem (I'm just procrastinating) but the analysis is instructive, I think.