TITLE: Interesting mathematical problems for 1st year university students
QUESTION [8 upvotes]: Can you explain some mathematical problems that you find the most interesting (NB: the problem must be accessible to a 1st year university student: that is, a problem for which there is an elegant solution that a student can find). Also, why do they interest you?

REPLY [4 votes]: the ~¾ century old Collatz conjecture is a great problem to experiment/study on esp with CS based approaches (writing code, visualizing results); its possible/ultimate/eventual(?) solution is regarded as very hard by experts. however there are many basic variants of exercises on it that have "correct solutions" that are accessible to undergraduates. (example: create a Finite state transducer to calculate iterates.) it has remarkable/amazing aspects of/connections to many areas of active mathematical & CS research:

fractals
chaotic behavior 
dynamical systems
number theory
(un)decidability
automata theory
random walks
pseudorandom number generators

see eg this Collatz conjecture experiments page for some basic starting leads & many links to standard/recent references