Icosian game
This applet illustrates William Rowan Hamilton's icosian game (twenty = είκοσι in Greek). It is, in fact, a two-dimensional representation of a dodecahedron (with 20 vertices, 30 edges and 12 faces). The challenge is to find a path (along the edges) that visits each vertex exactly once. You may find the two shades helpful. Four of the 20 vertices are coloured green, along with another 'fake' vertex; these five can be moved to alter the representation. For more on the context of Hamilton's game, see www.mathsireland.ie/hamiltons-ireland.