My work here at Aspen Research involves solving problems for clients. Sometimes they are simple analytical requests, sometimes they are failure analysis, sometimes they are product development, but one class of projects stands alone. These are what we call "Holy Grail Projects". These projects always seems to come from a new client, and they throw at us some problem that is industry wide and plague everyone in it, something along the lines of "make it twice as good at half the price without making any capital investments". They can even involve something that borders on being physically impossible or unreal.
(We certainly don't have a corner on the market for these projects. I've run into similar requests at previous employments with the requests usually coming from the marketing department.)
There are always lots of snarky comments that can be made ("Once we do that, you'll save even more money by eliminating your sales force and having my 7-year son sell it") but I've always had the good sense to force those ideas from my head and focus on the problem at hand.
Clearly the problems can never be solved as proposed. Everyone's been looking at it for years or decades and tried all the obvious solutions. So then it's a matter of redefining the problem. My usual approach is along the lines of something attributed to President Eisenhower: "Whenever I run into a problem I can't solve, I always make it bigger. I can never solve it by trying to make it smaller, but if I make it big enough, I can begin to see the outlines of a solution." That doesn't always work, but it certainly is a great way to start.