I then ask them a fairly broad open-ended set of questions to see how they describe the process of solving said problem.
This usually involves talking about tools, their process, their libraries of choice(something more generic here for other fields) and generally helps to assess their personable skills.
If I like them, I will ask them if they have any personal projects they want to share or something they are enthusiastic about in their domain of knowledge.
I think the biggest thing is that a lot of companies doing technical interviewing only reserve a small period of time to determine their new candidate. For people long in the industry, their resume should speak for itself. For younger people, it's kind of like, would you sleep with a woman you just met? How does that experience usually turn out? It's a 50/50 or worse chance you end up with a nightmare. Spend some time to interview your candidates, or end up with the one-night-stand.