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Why did the pins stay in place once they have been pushed into the correct position with the lockpick in the first example (for the old regular lock). I didn’t understand that and would assume that those pins fall back down once the lockpick moves to the next one?

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In a hypothetical perfect lock, they would fall back down. But in real life, every lock has slight imperfections, like not all of the pins being exactly the same diameter, and not all of the pin chambers being perfectly lined up. It's these imperfections that let previous pins bind and stay up when you move on to picking other ones.

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