Can any 10 points on a plane always be covered with some number of nonoverlapping unit discs? It’s not immediately clear that the answer is yes. The most efficient way to pack circles together on the plane is the hexagonal packing shown above; it covers about 90.69 percent of the surface. But if our 10 points are inconveniently arranged, it’s not clear that we’ll always be able to shift the array of circles around in order to get them all covered. In this case there’s a neat proof that takes advantage of a technique called the probabilistic method — if,...