How to stay academically active during a Mathematics Gap Year?

It turned out my answer was too long to be a comment

tbh, I'd try to cover the material in the first year during your year off. Unless you got an IMO medal and go to Trinity, I very much doubt you will find first year a breeze. This comes from a ex-Cambridge student. I think it is very hard to do any meaningful research without good foundations, unless it is something which does not need a lot of undergraduate knowledge like combinatorics or graph theory.

I am not sure how many weeks you spent in Cambridge, but like try to learn 1 course from a textbook and see how many questions you would be able to do on your example sheets or past exam papers. They are not easy....

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