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I didn't even look at the 'door' aspect of this group as it makes one eye elsewhere if White tries to capture it by saving the stone preventing it connecting out.
The word 'elsewhere' throws me - as far as I'm concerned Black lives in situ if White blocks the outside connection.
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So it's at least alive in gote on my analysis and therefore the White move can't be worth more than a few points and therefore not worth more detailed analysis at this stage of the game... Not sure what this says about the difference between our approaches to analysing games though.
For a short while after realising what was afoot, I was also surprised that White didn't take free points (quite a few) but the next few moves showed me that White was rather more concerned with keeping flexibility for his still weak group than I would have been. The older I get the more I realise that strong players worry about their groups ALL the time. I now believe the most important go proverb is "a stitch in time".
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Perhaps knowing the door group was a hindrance here. Given that you mentioned the door group that was the first think I looked at but, of course, if that were all there was then black is already dead. Had you not mentioned that I think I would have first looked at the connection possibilities and their threat to produce an second eye.
Not too sure that I follow this, as I knew the door group but didn't think of it and so didn't apply it here, so it was hardy a hindrance. The hindrance, as I saw it, was lack of a trigger to actually see the door. But possibly the door group means different things to you and me. For me it's a dynamic entity and so includes the forcing hane. Perhaps you just define it by the empty space.
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You mention subconscious thinking, but what is its relevance here, IYO? Hopefully not "trust it without conscious thinking":)
Being more than a tad pedantic, I didn't mention subconscious thinking. I referred just to the subconscious. If I look at the L-group, my subconscious tells me instantly, so there's no real thought involved, that it's dead. In a real game I may need to think consciously if my opponent plays an unusual move or a threat nearby, but otherwise I do trust my knowledge of the L-group without conscious thought. However, when it comes to the carpenter's square, I know the supposed result instantly but I do NOT trust myself to play it out in a game without careful thought. The door group is somewhere in-between for me. I know the outcome, but I need to double check the plays (get the right vital point) but feel that's trivial.