So you want to spend your summer doing mathematics?

Here are a few places you should consider.  All but the U of C's YSP cost money, but all offer financial aid.  The YSP is also the only day program, but they do provide you with lunch.

The University of Chicago's Young Scholars Program, which runs daily for the month of July (no boarding).  In 2010,  students entering grades 7-10 will study number theory, and students entering grades 11-12 will study Galois theory.  After a one-hour lecture, students work in problem-solving groups and do computer exercises.  Mr. K was a founding counselor and returned three times.

Hampshire College's Summer Studies in Mathematics.  Six weeks in lovely Amherst, Mass: four hours of lecture-discussion, and three hours of low-pressure, high-math problem-solving sessions.  Mr. K was a student in '87 and liked it so much he returned as a staff member in '90.  Just about the best time you can have at mathcamp.

The USA-Canada Mathcamp.  Lots to choose from, including relatively laid-back and extremely hard-core advanced math.  Terrific visiting faculty.

University of Michigan's Math and Science Scholars Summer Program consists of two three-week sessions in Ann Arbor, Michigan.  A variety of topics is available at each session.

Ohio State University's Ross Program.  Very intense eight-week program--it's number theory, number theory, and more number theory for your first year.  Has produced many outstanding mathematicians.  One hour a day of lecture, six or more hours of homework, and three exams.  Inspired Tami Sagher's "Tales from Mathcamp" monologue--need I say more?

Boston University's PROMYS program is modeled on the Ross program but is shorter (six weeks), in Boston (not Columbus), and seems significantly more manageable.

Stanford University's SUMAC program is 3 1/2 weeks and includes a variety of summer topics.