A few weeks ago, the 2012 crop of the 10 winners of the Intel Science Talent Search have been announced. As usual, these are great projects in a wide variety of fields (cancer research, robotics, geophysics, ecology, …).
Several math projects were in there too: 18 year old David Ding came 4th with his work on Infinitesimal Cherednik algebras of (done during a PRIMES program at MIT— which seems to be a fantastic setting indeed for insterested high-schoolers — under the supervision of Sasha Tsymbaliuk).
And 17 year old Anirudh Prabhu came 7th for his first non-trivial analytic lower bound of odd perfect numbers (his arxiv papers are here). Apparently he has been taking accelerated courses for a number of years, and did his research on his own without a mentor.
That’s impressive in both cases, and shows that there is not one single recipe for success.