High school and undergrad research paper topics

Here are some interesting topics for a high-school or undergrad research paper:

  • Which graphs are currently known to be graceful?
  • Which theories are currently known to be decidable?
  • Which functions are currently known to terminate? For example, primitive recursive functions.
  • Which types of primes have been named? For example, twin primes, Mersenne primes, etc. What is interesting about each type?
  • Which properties of relations are named? For example, reflexive, symmetric, etc. Which operations do you know of that have those properties?

  • Which properties of functions have been named? For example, commutative, associative, etc. Which operations do you know of that have those properties?
  • Which algebraic structures have been named? For example, monoid, group, etc. Can you think of an example of each structure?
  • Which shapes tile the plane? What methods are available for proving a shape tiles the plane?