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?