Getting into Harvard, Princeton, or Yale as an undergraduate is just super tough. A sufficient condition is to give the university, say, $10 million and be sometimes awake. Having a parent who went can help but not necessarily enough. Being a straight A student with 800s on the SATs is not sufficient.

I never got anywhere trying to get into Princeton as an undergraduate, with average SAT scores over 700, but did get accepted in the math (or statistics) department as a graduate student. I'd read lots of John Tukey's work -- uniformity in topology (in the back of Kelley's 'General Topology'), Tukey's lemma in axiomatic set theory, stepwise regression, power spectral estimation (Blackman and Tukey), the FFT (Cooley and Tukey), and 'Exploratory Data Analysis', had been working in applied math and computing, wrote G. Watson, and got accepted. I also got accepted in the Division of Applied Math at Brown and in Operations Research at Cornell. I went to Hopkins.

Anecdotal evidence to be sure, but I went to an Ivy League School (Penn) on just straight-As and 700s (!) on the SATs. I was not a "legacy" student of any kind. I knew many such people there, and only about 10% of my colleagues admitted to being a legacy -- maybe it was just my social circle? Or that I didn't major at Wharton, which is stereotypically legacy?

Your point about graduate school is spot on, though. The game completely changes (though I think it's still mostly a game). I especially love the "don't buy this muffler..." quote!

Maybe you were just a more distinguished candidate during graduate admissions? Also, if you're international, PhD admissions are a lot less biased against internationals compared to undergrad admissions (most top universities have 8-10% internationals for undergrads vs. ~50% for PhDs)

Getting into an Ivy as an undergraduate is very tough if you aren't a potential legacy admit, that is, a child of an alum. Once all those slots are filled, there aren't all that many left over.

Out of couriosity, is this true for other majors, too? Because if it is, it sounds like a great "backdoor" to an Ivy League degree.

Kudos for being accepted by all the big names!

>Since in many cases PhD is basically free

In many cases an Ivy League Bachelors is basically free as well.

Between tuition waivers for lower income students, and parents paying for higher income students, an Ivy League Bachelors is effectively free to most of the time.

The "free" Bachelors is still mostly the privilege of American students (with a few truly exception international students sprinkled in).

For PhD, As sidww2 pointed out, and what I have experienced as well, the admission pool becomes much more international; and now you are competing with the world. There is a phrase about China saying that underlines the scale well: "in China when you are one in a million -- there are 1,300 other people just like you".

Moreover, overall the number of spots in graduate schools is by far smaller than for undergrad; making competition even tougher.