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# Which topics/problems could you show to a bright first year mathematics student?

I am teaching a one semester course (January to June) to first year students pursuing various different degrees. Because there are students studying actuarial science, physics, other sciences, other management or business sciences etc., the course has to be a generic one. By this I mean that we teach Calculus almost exclusively. Sure, there are topics like the Binomial Theorem and general remarks of proving theorems, but students that are interested in mathematics don't find the material particularly interesting. What is more, I can relate to them since I found the first two years of university mathematics somewhat boring. This included calculus, linear algebra, convergent/divergent sequences, multiple integrals etc. Real analysis (except for sequences), complex analysis, abstract algebra, topology and even elementary number theory do not appear until the third year.

The sad thing is that many students take mathematics only up to second year and do not get to see any of the "cool"/"interesting" mathematics even though they might be interested in mathematics. So I wondered: Is there any way of introducing "interesting" mathematics to them? ("Them" - in particular first years, but this question is also relevant for second years, who might have a higher degree of maturity.)

Some things that I (and the other lecturers of this course) have thought of are:

Adding challenging questions to tutorials (e.g. IMC or Putnam, though these are harder than we would like)

Writing short introductory "articles" about fields or groups or perhaps the Euler characteristic (as an introduction to topology) etc. Of course, this is very idealistic, since one often doesn't have time or energy to do this.

Referring them to library books where some of these things are explained. Also, quite idealistic, but how many will actually go to the library.

The best solution is probably to combine the three. Have a question which has a strange answer or solution, which can be explained by some interesting mathematics. Shortly explain how this is done, and have a reference where the student can go if he is interested enough to pursue it further.

Are there any other ways of achieving this goal? Do you know of any questions to which this (combined) procedure can be applied?