Counting liberties didn't work

White A11 has more liberties than Black A10, but black wins the capture race when played out. How do you explain that?

Is it because white needs to respond to F16 and D12?

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That seems to be the case. Counting liberties works, except when a forcing move would take two liberties. This requires the loser to be sandwiched.

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When J13 and H13 are played black loses 2 liberties, but gains three by connecting with the group above. Your white group also loses a liberty from this. The net result of your move and black’s response is black’s group gets +1 liberty, and your white group gets -1 liberty.

You have to visualise any forcing moves or any required approach moves, and count the liberties from that position. Black D12–white E11, black F16–white G15, and now black and white each have three external liberties, plus one shared liberty => a win for the side to move. It all checks out.

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