It runs in the family

So at dinner on Wednesday, I noticed that my younger brother (age 14, non-Honors Algebra II, smart but unmotivated, pretty much what I would have been had I had an academically overachieving older sibling) was filling out 16-man brackets. But these weren't NCAA basketball brackets (what he would have been doing with those in December is beyond me anyway), or fantasy NCAA football playoffs, or any of the above. They were 16 competitors labeled from "A" to "P."

So I asked him: What are you doing?
Him: Filling out brackets.
Me: Why?
(Silence).
(A little later)
Me: Chris, why are you filling out brackets all over your paper?
(Silence).
(Later still)
Him: Harrison, I have a question.
Me: Shoot.
Him: If you have 16 people and you know their initial positions in a bracket, what's the minimum number of spots you have to fill in so you can fill in the rest of the bracket?

I'm pretty sure I've been more proud of a member of my family than I was of him at that moment. But I'm having trouble thinking of an example.

[PS: I told him he should guest-post the problem on this blog. He politely declined.]