What methods does Sherlock Holmes use to solve the murder in the story The Hound of the Baskervilles?

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Holmes uses his usual analytical and empirical methods to solve the crime. He dismisses from the start any superstitious notion of a legend of a mysterious demonic hound and applies logic to his search for a murderer and motive.

One method Sherlock Holmes uses to solve the crime is simply...

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Holmes uses his usual analytical and empirical methods to solve the crime. He dismisses from the start any superstitious notion of a legend of a mysterious demonic hound and applies logic to his search for a murderer and motive.

One method Sherlock Holmes uses to solve the crime is simply background research. He discovers that Stapleton has only a few people between him and inheriting the Baskerville estate, which makes him a very likely suspect.

Holmes also uses subterfuge. He doesn't want a mastermind like Stapleton to know that he, Holmes, is on his tail, for fear it will make him extra cautious. Therefore, he sends Watson up alone to investigate, but also sneaks up himself in disguise to do sleuthing, unbeknownst even to Watson.

Finally, near the end, Holmes makes bold moves to confront the hound and manages to do so, saving Sir Henry in the process and revealing that Stapleton has been using a very large dog painted with phosphorus to kill the other people in the way of his inheriting.

After the fact, Holmes processes what has happened and fills in remained loose ends by speaking with women closely involved in the case who suspected Stapleton of evil-doing.

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First of all, there technically was no "murder" to be solved in Sir Arthur Conan Doyle's The Hound of the Baskervilles. Sir Charles died under mysterious circumstances that indicated his ancestral curse may have been to blame, but the cause of death was a heart attack. Although Stapleton, who sent out the glowing hound to scare Sir Charles, was the cause of that heart attack, whether Stapleton could actually be convicted of murder for Sir Charles' death is uncertain since "no signs of violence were to be discovered on Sir Charles's person." Nevertheless, Holmes is able to discover the underlying foul play in Sir Charles' death and prevent a more overt murder of Sir Henry by his excellent detective work. Some of the methods he uses to solve the case are interviews and research, occult observation, first-hand observation of the suspect and the setting, and a "sting" operation. 

Interestingly in this story, Holmes is absent for multiple chapters because he sends Watson to be the on-site attendant of Sir Henry, the client who is in danger. Holmes uses this time when Watson takes the attention off of Holmes to do a lot of background research and interviews which are not revealed to Watson or the reader until later. At the same time, he engages in occult, or hidden, observation of the setting by living in one of the Neolithic huts on the moor. Watson notices a mysterious "man on the Tor" who turns out to be Holmes. Holmes then makes himself available outwardly at the home of Sir Henry, where he is able to observe that the portrait of one of the Baskerville ancestors bears a striking resemblance to Mr. Stapleton, confirming his suspicion about Stapleton. Holmes and Watson then arrange with Sir Henry to take a dangerous walk home in the dark from the Stapletons, which is designed to lure Stapleton into making his move against Sir Henry. Although this risky sting operation imperils Sir Henry, Holmes is able to shoot the attacking hound before it harms his client, and the perpetrator of the crime runs off into the moor, presumably to be sucked into the Grimpen Mire. At the end, Holmes reveals to Watson any remaining loose ends. 

Holmes' behind-the-scenes research, use of Watson as a distraction while he observes the suspect from a distance, on-site observation to confirm his theories, and final sting operation are some of the main techniques Holmes uses to clarify the circumstances of Sir Charles' death and to prevent the untimely demise of Sir Henry.

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