Idioms & Phrases at cross purposes With aims or goals that
conflict or interfere with one another, as in I'm afraid the two
departments are working at cross purposes. This idiom, first
recorded in 1688, may have begun as a 17th-century parlor game
called "cross-purposes," in which a series of subjects (or
questions) were divided from their explanations (or answers) and
distributed around the room. Players then created absurdities by
combining a subject taken from one person with an explanation
taken from another.