128 tennis balls can be arranged into more combinations than there are particles in the universe

A seemingly bizarre problem has been solved by physicists at the University of Cambridge – how many different ways can you arrange 128 tennis balls? The answer – 10^250 (one with 250 zeros, or ten unquadragintilliard). To put that into perspective, it would exceed the total number of particles in the universe.

Tennis balls aside, the method used to calculate the conundrum and the implications of it are what is key. The method the team came up with allowed them to calculate configurational entropy, which describes how structurally disordered particles in a system are. For example, the water in a lake or the water molecules in an ice cube.

When a system changes (say because of the temperature) the arrangement of the particles does too. We can predict this on a molecular level, but when it comes to bigger things – like a sand dune – it becomes more difficult and we cannot predict about how systems will behave. To do so you would need to measure changes in the structural disorder of all the particles, and to do this you would need to know how many different ways a system can be structured.

Published in the journal Physical Review E, researchers carried out a calculation where the particles were 128 soft spheres, like tennis balls. They took the small sample of all the possible configurations and worked out the probability of them occurring. Based on these samples, they were able to work out how many ways the entire system could be arranged – and its configurational entropy.

They said their method should be applicable to a wide range of problems, including string theory, cosmology and how to make artificial intelligence more efficient. It could also allow them to predict the movement of avalanches, or work out how sand dunes in a desert will change over time.

Study author Stefano Martiniani said: "Obviously being able to predict how avalanches move or deserts may change is a long, long way off, but one day we would like to be able to solve such problems. This research performs the sort of calculation we would need in order to be able to do that.

"Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one. By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find."