Bizarro takes on English idioms involving a carrot and a stick:
There’s been a lot of passionate argument about whether “carrot and stick” or “carrot on a stick” is the original and/or “correct” variant of the expression. The arguments are mostly about the inherent plausibility of one or the other of the images; the textual evidence doesn’t come down clearly on one side or the other. For some recent surveys, look at Paul Brians here. Michael Quinion here, and Jan Freeman here.
Jan Freeman suggests an amicable resolution: the possibility that the two expressions arose independently, both building on the idea of a carrot as an inducement.