Which top tier schools admit younger students and allow them to live on campus?
currently age 13 (female)
Will graduate in either 2022 or 2023 at age 15 or 16.
Will want to live on campus.
Has had multiple successful grade accelerations in a public school. Currently 3 years accelerated in core subjects. 5 years accelerated in math.
Projected to take 6-8 AP classes and
9-11 college courses if graduation is 2023.
Will likely have 4.0
SAT 1500+ at age 13.
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Definitely check out Bucksbaum Academy: https://belinblank.education.uiowa.edu/students/academy/
I know someone doing that who loves it (he skipped last two years of HS). There is an honors residence hall and the Bucksbaum students stay on one of the floors together. Some are skipping last two years of HS, some are skipping three years of HS.
Also, there are other early entrace programs, too: https://cty.jhu.edu/resources/academic-opportunities/college-entrance/
By the time you reach 12th grade, you will have better test scores and accomplishments so you can basically apply anywhere.
But just because you can apply anywhere doesn't mean you will have a good time going to a top school at 15 or 16 because there are many other considerations that have to do with fitting in with undergrads who will be 18-23 no matter where you go.
Some of the pitfalls for a very young student is the social aspects of college. Part of the reason many students want to go away to college is to experience independence and various rights of passage whether that is discovering their own adult person with regards to sexuality, drinking, drugs, joining a sorority or fraternity, or having the freedom to do whatever they want. If you go to MIT, CalTech, Dartmouth, Brown, Princeton, Duke, Vanderbilt, and many others up to 50% of the undergrads are involved in Greek life and without going into details, many of the activities are not going to be age-appropriate for a 15 to 16-year-old. And regardless of what top school you attend, there will always be parties and social functions that will involve drinking and in some cases legal marijuana, etc, and even if you have access to attend, might not feel that's your scene. So try to avoid applying to schools that have a large Greek presence because you might be 2-3 years too young to find those purposeful to your college experience.
Also, keep in mind that at many top schools, the dorms are co-ed and in some cases include underclasspeople and upperclasspeople so there are men and women all living together, sharing common areas, bathrooms, and you may not feel that you have that much in common in such settings. So it might be easier to find a school that has segregated Freshman dorms.
When you are in high school everyone is a teenager so even though you think you are supported well by your teachers and admin and there isn't much of an age difference now. Once you finish high school you will be propelled into a completely different environment where no one is 15/16 and you'll really feel like an anomaly, which can be isolating.
My recommendation is that you connect yourself with some kind of network of young geniuses online and learn about their experiences and choices and adjustments they made to be successful college students. Remember, you have your entire life ahead of you and where you do your undergraduate work is not as important as what you become later in life as a thriving productive human, perhaps after a PhD. or two. I have watched some youtube videos of young geniuses and sometimes the academic part is the easy thing to figure out, deciding to leave mom and dad and the comfort of home where you get three squares a day and your laundry and room cleaned, is a harder decision.
The other option which might be interesting for you is to transition to college via a top boarding school like Andover, Exeter, Choate, or Deerfield, St.Pauls, Hotchkiss, Lawrenceville, Loomis Chaffee. These schools have some of the brightest students in America and from all over the world. Why I mention this is because going to boarding school is like going to a Pre-Ivy college. You can apply as a Junior for 2 years or you can apply as Post Grad for an extra year of HS after you graduate. The benefits of this environment are that you are essentially going away to school, living in a dorm, eating in a dining hall, doing all kinds of on-campus and off-campus ECs, and having access to a Harkness style of student-led discussion learning which is similar to how many Ivys' and top LACs teach. Plus unlike the HS curriculum, you currently have available, these types of schools have 250-450 classes to choose from taught by mostly Ph.D. professors. If you spend a year or two at a top boarding school, you will know if you are ready to attend an Ivy or Top LAC because you will have experienced what you need to know. I don't know what your financial situation is but most of them have generous financial aid similar to the top colleges so if you qualify, the costs are marginal.
Good luck with your high school career and I hope you can connect with other young geniuses.