Is there a way to navigate to random unanswered questions?

Sometimes, I want to just pop on MSE and try to answer an unanswered question. But I find it somewhat unhelpful sifting through the questions in the "pre-filtered" order(s).

I would love to have a button that presents a randomly-selected unanswered question (cf. the "random" link on OEIS, or the "I Feel Lucky" Google hit-link); even better if it observes other filters (e.g., "My Tags").

Thoughts?

It's possible to show just one question on that page by adding &pagesize=1 to the URL; but this has side effect of setting page size to 1 everywhere on the site, until you change it back. So I'd rather not do that. It's easy enough to look at the top question only, or maybe glance at the top three to find something interesting there.