Theory:

Sherlock Holmes is a very famous private detective and the main character of the story. He was smart and intelligent. Till the end of the story, we can never find out his plans for solving the case.
 
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Dr. John Watson is Sherlock Holmes’s partner. He is a doctor by profession and Sherlock Holmes friend.

Jabez Wilson was a red-haired pawnbroker. He works for Red-Headed League for a few weeks.

John Clay/Vincent Spaulding is a criminal working at Jabez Wilson’s pawnshop under the anonym Vincent Spaulding. He was Wilson's assistant, also known as John Clay.

Archie/Duncan Ross/William Morris was also a man with red hair. He is John Clay’s associate
and is the head of the League.
  
Peter Jones is a Scotland Yard detective.
 
Dr. Watson goes to the home of his detective friend Sherlock Holmes where he saw him talking to the person who has a bright red hair named Jabez Wilson. Holmes invites Dr. Watson and wants him to hear the story of his client Mr. Wilson as he never heard of such a case before.
 
Mr. Wilson said he was a small businessman and run a pawnshop. He has one assistant named Vincent Spaulding. One day Vincent Spaulding drew the attention of Wilson by showing him an advertisement in the newspaper stating that there is an opening in the Red-Headed League in which all red-headed men are eligible to apply. The advertisement also stated that they are going to pay a salary of four pounds a week. Spaulding encouraged Wilson to apply for the job as he is red-headed. The fearful pawnbroker applied for the job to earn some money as he was not earning enough from the shop. Mr. Duncan Ross was the head of the league and selected Wilson into the league.
 
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Red- Headed man.

Wilson's only duty was to go to the office for \(4\) hours from 10.00 A.M till 2:00 P.M. every day and copy the Encyclopedia Britannica in longhand. After eight weeks of working for the league one day when he arrived at the office, he saw a notice on the door which say's that the Red-Headed League was closed. Wilson was not happy by seeing the notice and got suspicious about the Red-Headed League as they are playing some practical joke on him. This leads Wilson to come to Holmes to understand the reason behind why the league was closed. Holmes agreed to take up the case and promised Wilson that he would find the mystery behind it.
 
Holmes and his friend Watson went to Saxe-Coburg Square where Wilson had his pawnshop. Holmes was walking here and there outside Wilson's shop by tapping the ground with his walking stick. After watching outside the shop, Holmes knocked on the door of the pawnshop, and Spauldings opens the door. Holmes notices that the pants worn by Vincent Spaulding were dirty.
 
Spaulding asked if he could help Mr. Holmes? In response, Holmes instead if interrogating Spaulding he asked the directions to the Shore. Spaulding told Holmes has to go third on the right, and fourth on the left to go the shore. After giving directions to Holmes, immediately Spaulding goes inside the shop and locks the door. Holmes said to Watson that Spaulding was a smart young man.
 
Watson asked Holmes is Spaulding really smart and even asked him why he wants the directions to the Strand as he knew London thoroughly. Do you want to see how Spaulding look and is that the reason to knock the door and ask him the directions? Yes, the reason behind this is Holmes wanted to see the appearance of the Spaulding.
 
Holmes asked if Watson has noticed how the Spaulding's pants look. Watson replied that he didn't observe the trousers of Spaulding and asked why his trousers, but he said he observed Holmes tapping the ground with his stick. Holmes said he wants to go and see the road behind Saxe-Coburg Square and it is not the time for answering his questions.
 
Holmes asked if Watson had to go to the job. In response, Watson replies he has to go to the hospital. Holmes told Watson could proceed, but he wants his help that night and asked him to come at ten O'clock.
  
Holmes and his friend Watson were sitting in the study room. They were talking to each other about the episode that happened. Holmes tells Watson that there wasn't any Red-Headed League at all. It was just that the robbers wanted Wilson to be away from his shop for some time in a day. Clay saw the colour of Ross's hair and thought of the name Red-Headed League. So, the plan was when Wilson was away from his shop every day they can use that time to dig the tunnel. They were smart to think of such an idea.
 
Watson could finally understand that Spaulding was John Clay,the famous thief. He asked Holmes when did he come to know about that. Holmes said that he thought of everything that happened from the starting, first Spaulding was ready to work for Wilson for half-wages, which no one will ever do. Second, why was Spaulding often going to the cellar? It stuck to Holmes that maybe they are digging a tunnel; otherwise, no one will go to the cellar so many times. But if he is making a tunnel, then to what place, maybe to another building. That's the reason he tapped the ground to hear the hollow sound.
 
Watson understood now that Holmes wanted to check if the cellar was in front of the shop. Holmes said that by tapping, he understood that the cellar was behind and not in front of the shop. When he first saw Spaulding, his doubts about the tunnel was clear as his pants were dirty. It is because while digging the tunnel, pants get dirty because of the mud. Watson felt Holmes was intelligent.

Holmes said that he next went to see what lies behind the shop on the next street and to his surprise, it was the bank. Watson responded yes, there is a bank on the next street to the shop. So, Holmes was clear that they dug the tunnel to get into the bank's locker room to rob it.

Watson still had some questions regarding how Holmes knows that the robbery will be happening on the night of that day. He asked why did they choose that night. Holmes said that all the Red-Headed League offices are closed it means that the tunnel was ready. Holmes thought that as it was Saturday, it was good for robbery since everything will be closed on Sunday and robbers will get time to escape.
 
Watson understood the whole story and smiled as Holmes did not allow them to run away and said the same to Holmes. Holmes just said in a serious tone that he was indeed intelligent.