Book 6: The Tattooed Potato and Other Clues by Ellen Raskin
Odd. That's the one word to describe this mystery by the author of The Westing Game.
Dickory Dock seeks out a job as an assistant to "a well-known portrait painter" whose only requirements are that the assistant be "a native New Yorker, neat, well-organized. Quiet! Observant!" The artist, Garson, is an eccentric phony, a not so good painter, in Dickory's opinion, but the pay is good.
Turns out that Garson is often visited by the chief of police to help solve crimes. Through a little role-playing (Garson as Sherlock Holmes and Dickory as Watson), the two solve some of the chief's supposedly hard to solve cases. Figure in a deaf-mute, a couple of blackmailers, a derelict and a blind man, and you have one unusual mystery book that would only be enjoyed by a more sophisticated young adult reader.
Or someone that likes just plain odd books.