<p>I've been meaning to write something about this for a while (I promised to answer questions about how Axlines are awarded and have been slow about it). I won't address the precise question being asked here about prodigies, etc. (Yes, every few years a prodigy comes along, such as Chris Hirata, who started Caltech at 15 and was one of the better math/physics guys, etc. But those are stories for another day.)</p>
<p>The Axline is a full merit scholarship that is awarded to less than the top one percent of the applicants each year, i.e. about 5% of the Caltech admitted pool. It is awarded without special application, on the basis of the application that applicants submit for the normal admissions process.</p>
<p>Axlines aren't particularly magical. They are, by virtue of the process that selects them, high school kids who stand out significantly even in the pool of smart, motivated admitted kids.</p>
<p>How they stand out is a complicated question. Those who get the top few scores on the International Math and Physics Olympiads are obvious candidates (there are a few such most years). Those who win Intel and Westinghouse are obvious candidates.</p>
<p>Other ways to stand out are less obvious. Taking courses at a well-regarded university like Princeton as a high school student and getting the top grade in a math class is one way to stand out. Sending a serious research paper along with evidence of passion for science is another way. Getting unanimous recommendations as "the best student in 30 years" from a few teachers -- even at a fairly mediocre school -- is a way. (It's surprising how rare that sort of thing is, which suggests it means something.) Distinctions of this sort usually come in twos and threes, which is a pretty clear signal.</p>
<p>So the short story is that to be an Axline you have to seem particularly accomplished or particularly smart compared to the rest of the admits. (I say "seem" because we evaluate applications, i.e. appearances; we lack epistemic access to "actual merit" or "actual intelligence".) If your application shows you can solve math problems which most people find very difficult (even most people admitted to Caltech), that's a way. If your application shows that you are particularly motivated to do scientific work (published in a real journal), that's a way. There are other ways.</p>
<p>Is it hard to become distinguished in these ways? Yes, for three reasons. Not everyone has the native talent to achieve successes of that nature. Of those who have it, not everyone is sufficiently motivated to achieve them -- most such people spend high school following the pretty standard route pursued by those who go off to elite colleges, not exceeding that high standard. </p>
<p>Of those who are motivated enough, not everyone has the same opportunities and information; some people's mathematician parents train them on Olympiad problems from the crib (and some of those children end up as very good mathematicians) and other people's parents stock groceries and don't know about Olympiad problems. </p>
<p>Since the ways to achieve distinction that I discuss above require fairly precocious and single-minded dedication, it's no mystery that most people don't achieve that distinction. Some of the barriers in the way are "fair" (need for talent) many are not (need for significant resources like time and money). </p>
<p>But for those who are curious about "what it takes," that's a pretty general outline of what it typically takes.</p>