4 ideas
The old man holds that there is no such thing as a mystery in any crime, provided intelligence is brought to bear on it. What looks mysterious is a failure of reasoning by police and public, who fix on the obvious suspect or the dramatic clue and stop thinking. The facts needed for a solution are usually already public, printed in the inquest reports and newspapers.
Many of the old man's solutions turn on impersonation or substitution: a person everyone assumed to be one individual was actually another. Identifications by witnesses, relatives, and clothing are treated as settled facts, so a disguise or swapped identity can survive the whole inquest. The method is to ask whether the named person in each testimony has actually been proven to be that person.
The old man treats crime as an intellectual contest. He admires the clever murderer more than he pities the victim, and he almost never takes his solutions to the police. The cases show detection split off from justice: understanding a crime and bringing its author to account turn out to be separate acts. His nervous knotting of string while he talks marks the solving as a private, compulsive game rather than a public duty.
In the collection's final story, 'The Mysterious Death in Percy Street', the old man explains the killing of Mrs. Owen, found dead in her lodging-house room. He leaves the teashop and never returns, which recasts every earlier session as a criminal lecturing on how crimes should be done.