We consider a system of identically-sized spheres that coat a rod in a dense monolayer. We derive relationships that show how the number of spheres needed to cover a unit length of rod depends on the sphere and rod radii. The analysis could provide a stimulating exercise for students who have been introduced to the conventional examples of dense packing that are taught in many introductory physical science courses. The new class of liquid crystalline system which prompted this analysis may have applications in displays that can maintain stable liquid crystalline order over a broad range of temperatures.