Fans of The Hitchhikers’ Guide to the Galaxy know that 42 is “The Answer to the Ultimate Question of Life, the Universe, and Everything” although, as it turns out, we don’t actually know what that question is. Mathematicians may have found part of the question. Think about packing tennis balls (don’t panic – there’s a connection). Tennis balls commonly come in a cylindrical tube of three balls. The question is whether less packaging could be used if the balls were arranged in, say, a triangle. More generally, if you have n balls, what arrangement of balls requires the least packaging? Two possibilities are a “sausage” that stacks the balls up in a cylinder and a “cluster” that nests the balls in a pyramid-like shape. In three dimensions, it turns out that the sausage is better for 56 balls or fewer, but beyond 56 the cluster is better. (The abrupt change at 56 is known as the “sausage catastrophe.” I’m not making this up.) The Sausage Conjecture states that for dimensions 5 and higher the sausage is always optimal, for any number of balls. Curiously, this has only been proven for dimensions 42 and higher. So, if you are in charge of packing 42-dimensional tennis balls (perhaps a problem of universal significance?), stack them up in a sausage and know that you have chosen the ideal shape.
