In "The Adventure of the Speckled Band," why does Holmes think the mystery is particularly intriguing?
Helen Stoner tells Sherlock Holmes that she is at present unable to pay him for his services. Yet he takes on her case for at least two reasons. One is that he feels sympathetic for a young woman who is so frightened and helpless. The other is that he finds one aspect of the mystery especially intriguing. Dr. Watson frequently mentions in his Sherlock Holmes tales that the great detective is mainly motivated by mental challenges, since he has been successful enough in his investigations to have no further cares about money. What intrigues Holmes is that the case presents what is commonly called a "Locked Room Murder Mystery." As Holmes explains to Dr. Watson:
“Yet if the lady is correct in saying that the flooring and walls are sound, and that the door, window, and chimney are impassable, then her sister must have been undoubtedly alone when she met her mysterious end.”
Holmes senses that if he could unravel the mystery of how Helen's sister Julia came to be killed while sleeping in such a room, he could also solve Helen's problem of believing that her life is in danger from the same source. This is what happens in the denouement of the story.
The prototype of the "Locked Room Murder Mystery" is probably Edgar Allan Poe's famous tale "The Murders in the Rue Morgue." Two women were horribly murdered in a room which appeared to be completely inaccessible from the outside. The protagonist in that story is an amateur detective named C. Auguste Dupin, who has many of the characteristics of Arthur Conan Doyle's Sherlock Holmes.