
Packing Pennies in the Plane 
The hexagonal packing of discs in the plane is obtained by laying out
a row of discs in a line, then successively adding rows on either side
packed in as closely as possible. This coincides with
what you get by fitting discs tightly inside a honeycomb pattern of hexagons.
Thue's theorem. No packing of nonoverlapping discs of equal size in the plane has density higher than that of the hexagonal packing. It is really impossible to imagine how it could be otherwise. We can also build the hexagonal packing in this way: we start with a single disc in the plane, and then place around it six others. In contrast to the similar construction in 3D, where spheres are placed around a sphere, it is clear that no more than six can be so placed. Furthermore this continues on for each of the new discs etc. to give a global packing, which has to be optimal  doesn't it? But no straightforward proof of the Theorem has yet been found. 
