Usually, I start with the question "How would you be able to tell that this problem had been solved?" and repeat it two or twenty times in different words until someone actually tries to answer it.

What a true and hilarious depiction of life. I have the exact same problem doing web development. Because the people giving me projects are not IT people they tend to come up with totally dysfunctional solutions. Yet they almost always start by telling me how they want the problem solved. I have to dig to find out what the problem is first but I just ask them "What result do you want?" or "What purpose do you want this to serve?" and say "I can't make it serve the purpose without knowing what the purpose is." That works for me, without me having to ask them 20 times. Then again maybe you're doing projects in radically different contexts all the time, or with completely different people who vary in their ability to see the point in answering that question. I work with a limited number of people and contexts, all of which I understand pretty well, so my problem clarification process is pretty simple.