Origin of cross-purpose
British Dictionary definitions for at cross purposes
Idioms and Phrases with at cross purposes
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.