Good advice in engaging consultants, from a recent LinkedIn discussion, is to be specific about what you want. However, sometimes good results occur with some misdirection.
A client asked me to help them with an operational organization issue (a touch outside my specialty, but they had been happy with earlier work I had done for them). They had entered a new market area and wanted me to outline how they would serve it. In their home market they used many branch locations, which had been developed over many years. They wanted a more targetted approach to selecting locations in the new market, they said. "Help us find the cities that make sense, and then another consultant will be brought in to help with siting details."
As I got into the project, it appeared that the model they used in their home market made less sense in the new market. (In fact, it made less sense in the home market than it used to, but that was outside my project.) I reported to the CEO that I wasn't so sure that the original consulting charge was quite right. The CEO told me to follow my instincts and report back when I had a substantive recommendation.
I came back to him with a radically different delivery model from what they had used in the home market, but which had been used with success by other companies in their industry . The CEO asked me to explain that recommendation to the board.
The board meeting was a success, but over the course of the discussion it became clear to me the CEO was smarter than I had thought. He had suspected that their traditional delivery model was not right for the new market area. He had engaged me knowing that I would push back if I thought their direction was wrong. He knew that his board was expecting a plan to duplicate the old delivery model in the new territory, but they respected me and would listen to well-documented statements.
The end result: the client saved millions of dollars by shifting to a more cost-effective delivery model. Along the way, the CEO gained the confidence that his consultant was not just telling him what he wanted to hear.