The Kites and Darts Game

Kites and darts are the names given to two tiles discovered by Sir Roger Penrose. These tiles have the very interesting property of tiling the plane aperiodically - that is, they cover the plane completely without repeating their pattern as do triangles, squares, hexagons, etc.
The tilings have many other properties of mathematical interest which can be read about in one of Martin Gardner's books, among other places.

Kadon sells a nice set of these tiles which can be used for making patterns and solving puzzles, but also includes a board and rules for a game that can be played with them.