# Number of ways to arrange 128 balls exceeds atoms in universe

By Jacob Aron Philip Lee Harvey/Getty How many ways are there to arrange 128 balls? Finding the answer is more difficult than you’d think, but could one day help us predict the shifts of avalanches or sand dunes. Mathematicians have long been interested in the most efficient way to pack spheres together, but there are many more ways to arrange a bunch of balls. If you chuck them into a box and jiggle them around until they jam in place, there are a multitude of possible arrangements – so many, in fact, that researchers thought counting them would require a computer larger than the universe. Now Stefano Martiniani of the University of Cambridge and his colleagues have found a clever way around the problem. They say there are 10250 ways to arrange 128 jammed spheres – far more than the 1080 atoms in the universe. So how did they do it? Think of each possible sphere configuration as sitting at a point on a vast energy landscape. As the balls jiggle loosely in the box, they have more energy, placing them at a higher elevation in the landscape. Settling down into a jammed state corresponds to the bottom of a valley, as the balls can’t move into a lower energy state. To investigate this landscape, the team simulated a jammed packing at random and then effectively gave it a few nudges, bumping the balls into a higher energy state. Repeating this allowed them to map out the valley, giving them an idea of how much of the landscape it takes up. Doing the same thing for a total of 1000 packings gave the team the average size of a valley, but it took a while – each valley required 300 hours. “We’re talking about a huge amount of computer time,” says Martiniani. Since they know the size of the entire landscape, dividing by the average gives a good estimate of how many valleys, and thus jammed packings, there are – leading to the figure 10250. “It’s a very simple statistical argument – the challenge was being able to obtain a distribution,” says Martiniani. It seems like a lot of trouble just to count some balls, but the technique could prove useful in developing a new kind of thermodynamics for real-life jammed systems, like sand and snow. Currently this is impossible due to the high numbers involved. “This idea was abandoned because it was thought to be numerically intractable, but we’ve shown it can be done,” says Martiniani. Journal reference: Physical Review E, DOI: