How do I get a random YouTube video with the YouTube API?

I need a way of getting a completely random YouTube video. No restriction.

How can I do it with the YouTube API?

Attempts:

1. I went through the api and examples at the YouTube dev site. http://www.youtube.com/dev/, but had no luck finding the correct api or a way of doing it there.

2. I searched with Google, of course, and got http://randomyoutubevideo.net/, but they only offer an api from THEM to use in between me and YouTube (this gives me hope that it IS actually possible to do this).

3. I even checked the YouTube app gallery http://youtube-gallery.appspot.com/ to see if anyone is doing it. and HOW.

What I will also do is ask on the YouTube discussion pages. Perhaps someone there can help.