Edit: I made a mistake with some of the language here. A comment from a true graph theorist:
“Hamiltonian” usually means there’s a hamilton/hamiltonian cycle. Graphs with a hamilton path are “traceable”. Hamiltonian implies traceable, but not conversely.
Thus, I have edited below accordingly.
There was a nice video up on Numberphile about a problem which could easily be explained to a school student, and yet we don’t yet know the answer to it. See the videos here:
The game is the following: Given a sequence of consecutive integers, draw a graph where the nodes are the integers and there is an edge between each integer if their sum is a square number.
If we take the numbers from 1 to 12, then the following would be the associated graph (note that there are three disconnected pieces of it).