I am teaching a course on proof. We have learned the methods of proof: direct proof, proof by contrapositive, by contradiction, by induction, etc. We have also done cardinality, modular arithmetic, functions, and a tiny bit of logic. A student approached me and asked if she could do a project on figurate numbers. These are the sequence of numbers you get by increasingly large triangles (triangular numbers), squares (square numbers), pentagons (pentagonal numbers), and so forth. All the exercises I could think of beyond find and prove general formulas for the $n$th number are too elementary. Are there any neat results I could ask her to prove? Thanks.