Tennis Ball Arrangement

Source: The Conversation, Feb 2016

say you have 128 tennis balls. How many different ways can you arrange them so that each ball touches at least one other? You can stack them, lay them out in various grids, stack the layers and so on. There are probably a lot of configurations, right?

This question was answered recently by a team of researchers at Cambridge University. The number of possible arrangements is on the order of 10²⁵⁰; that’s a 1 with 250 zeroes after it. To give a sense of how large this number is, note that there are only about 10⁸⁰ atoms in the universe. In fact, if we packed the known universe with protons, there would be only about 10¹²⁶ of them. So if we could somehow encode each configuration of the tennis balls on an atom (or even a subatomic particle), we would be able to get through only about the cube root of the total number of possibilities.

Since it’s impossible to actually count all the arrangements of the balls, the team used an indirect approach. They took a sample of all the possible configurations and computed the probability of each of them occurring. Extrapolating from there, the team was able to deduce the number of ways the entire system could be arranged, and how one ordering was related to the next. The latter is the so-called configurational entropy of the system, a measure of how disordered the particles in a system are.