1. Getting the answer right is only half the problem here. They are prodding you for your thought process—take that as a sign that that's what they're actually looking for, even if you're wrong. They also want to hear it just in case they need to steer you in the right direction.
Regardless of pairing policies, companies might ask you to collaborate on a problem with someone else. It is typically more productive if both developers are engaged in conversation about it and then working as opposed to doing their own work separately.
2. It's incredibly awkward and tense as an interviewer to sit there for 10 minutes. Also, how do I know it'll only be ten minutes?
If I wanted to know if you could solve a problem in your ideal setting, then I would email you a problem to solve on your own time. During an interview, however, I want to have a conversation. I only have one hour with you and your ten minutes of silence is a wasted opportunity.