I sent the sub-optimal solution, then I checked their puzzle page again that changed and asked for a better than O(n) solution. Then I thought about it for a while and got to the point I described. I didn't send it because I knew I could do better; I just hadn't figured it out yet.
Then I got busy with some of my other side projects since I got an email from Facebook saying they were finished finding Summer interns (that's what I was going for). Solving the solution, while fun, would not be efficient use of my time.
I guess the point in asking about posting the solutions is that it sort of undermines the effectiveness of the puzzle as a recruitment device. Plus it takes the fun out of figuring it out for ones' self. And by fun, I mean stress. ;)
Jesse, I got as far as linking the problem to the binomial coefficients and drawing some correlations to Pascal's triangle. I got stuck at the part for coming up with an algorithm for all numbers, rather than for powers of 2.
Why would you ever post the solution? I guess you're not interested in a job at Facebook.