Cubes and Eulerian graphs
A South African teacher, known on Twitter as @Pyfagoras, sent me this photo of his board. It shows how to turn a cube into an Eulerian graph by adding in the diagonals across each face of the cube. In an Eulerian graph, every vertex has even degree (that is, an even number of edges coming out of it) and it is so called because Euler proved that for such graphs it is possible to start anywhere and traverse every edge in the graph exactly once, returning to the start. Furthermore, such a circuit is impossible if the graph contains a vertex of odd degree. For the cube on the board, there are 6 edges coming out of each vertex, meaning it is not only Eulerian, but also a regular graph. You can read more about it in its incarnation as the 16-cell.