I ran across this one, which has had me stumped all day.

Suppose you have a bowl of k noodles. You randomly reach in (with both hands) and extract two noodle ends, then glue them together. You do this k times. What is the expected number of noodle loops ?

There has to be a cute, simple way to do this one, but I haven't been able to come up with anything. I can write down explicit expressions for the case with N = k, N = k-1, N = 2 and N = 1, but they involve !! (double factorials), on account of the fact that one noodle has two ends. It's also true NOT forming a loop is the same as taking one of the noodles out of the bowl. It's hard to see how anything I write down will be analytically soluble.

There's some little trick I'm missing, for sure...