What are the clues that helped Holmes to solve the problem in "The Red-Headed League?"

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In “The Red-Headed League” Holmes uses logic to solve a seemingly bizarre case of a man who is given a job copying the encyclopedia because he realizes that the suspects are trying to get the man out of his shop so they can make a tunnel to a nearby bank.

In this story a pawnbroker named Mr. Wilson is sponsored by the “Red-Headed League” an organization supposedly started by a red-head in America.  "Holmes possesses a nearly superhuman ability to read a person's background by observing small, seemingly-insignificant details" (enotes, characters).

The Strange Advertisement

Holmes’s first clue is the story itself.  The “Red-Headed League” is just too crazy to be real.  Holmes knows that there is some kind of ulterior motive.  Since the advertisement describes an American, there is no real way to check the story.

Holmes sees that the story is not meant to be checked, so it is likely not legitimate.

The Overly-Helpful Assistant

The second clue is the assistant.  Holmes is immediately curious about him, and believes that he is “as remarkable” as the advertisement for the Red-Headed League.  He is the one that brings Mr. Wilson the advertisement about the Red-Headed league and encourages him to apply, and he showed up just before the message.

You seem most fortunate in having an employee who comes under the full market price. (enotes etext pdf, p. 6)

Besides showing up out of nowhere, being too old to be an assistant, and working for too little money, Mr. Wilson also comments on other strange facts about his assistant.  He has a habit of taking pictures and then “diving down into the cellar like a rabbit into its hole to develop his pictures” (p. 6). 

This definitely catches Holmes’s attention, and leads him to realize that the assistant is involved and using the cellar in some way. 

The Strange Hours and Activities

For four hours a day, Mr. Wilson sits in an empty office and copies the encyclopedia.  Then one day he gets a note saying that the league is dissolved.  When he investigates he realizes that the tenant was a solicitor and he only rented the room on a temporary basis and there is no way to contact him. 

Holmes realizes that they need Mr. Wilson out of the shop for four hours a day.  They must be doing something there, probably in the cellar.  From this Holmes deduces that whatever the assistant has planned, it is happening soon.

The Assistant’s Pants

It’s time for a visit!  Holmes and Watson go to check out the shop.  Holmes asks the assistant for directions, and then declares that he is the “fourth smartest man in London” and he has “known something of him before” p. 13).  Holmes notices that the assistant has dirty knees, and concludes that he has been kneeling somewhere.  He beats the pavement and notices it is hollow.  Holmes tells Watson he knows something is up.

“A considerable crime is in contemplation. I have every reason to believe that we shall be in time to stop it.  But to-day being Saturday rather complicates matters. I shall want your help to-night.” (p. 14)

Holmes recognizes the assistant and realizes the importance of the dirt on his knees.  He checks for tunnels and finds one.  He knows they are tunneling into the building next door.  Since it is a bank they are after and it’s Saturday, they must act at once.

Holmes arranges for the police (and Watson) to help, and they catch the bank robbers in the act.  

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The Red-Headed League

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