How does Sherlock Holmes solve the "A Scandal in Bohemia" mystery?

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Holmes solves the mystery of the whereabouts of the "compromising" photograph by faking a fire to provoke the owner to rush to retrieve it before leaving her house. He even hires a whole "cast" of actors and actresses to come out on the street in front of her house to make the "fire" seem real. The woman witholding the photograph (used in sort as blackmail) retrieves the letter at the alarm, realizes she has been outwitted, then returns the photograph of her own will since she has decided to not pursue the matter any further.

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