Tiling problems, like some of the great conjectures in arithmetic, have the delightful feature of being understandable to a five-year-old. In reality, they are staggeringly difficult!
The problem is easy to state: which convex polygons tile the plane? In other words, we take copies of a basic polygon, with no indentations or holes, and try to fit them together across the plane without overlaps or gaps. The square ceramic tiles decorating your floors or walls fit the bill. For regular polygons (polygons whose sides are all the same length and whose angles all have the same measure), it has been known since antiquity—since Aristotle—that only the equilateral triangle, the square and the regular hexagon tile the plane. But ceramic tiles are not all square: some, for example, are rectangular.
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A mathematical tile!
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In fact, every triangle tiles the plane, as does every quadrilateral. In his 1918 doctoral thesis at the University of Frankfurt, Karl Reinhardt proved that only three families of convex hexagons tile the plane. Furthermore, no convex polygon with more than six sides can tile the plane. That leaves only the pentagon to deal with!
