Why is it that even the best billiard players can't reliably plan combination shots (shots where the cue ball hits another ball which in turn hits yet another ball and so on) that have three or four ball collisions? Wouldn't it be possible to simply calculate the required angles of the collisions and then as a world-class professional execute the shot with precision?

In the world of mathematical billiards (billiards where balls don't have mass) this would be possible. However, when playing real billiards you have to take into account all the uncertainties around us. The initial direction of the cue ball might be off by just a tiny fraction. But since this error gets amplified with each collision, the direction of the last ball might be something completely unexpected. The small imperfections of our billiard table have to be also taken into account.

Picture of an early billiards game

Let's say we replace our human player with a robot. This robot has mechanical arms that are significantly more reliable at shooting the cue ball compared to our human player. We also find a better billiard table with less imperfections. Finally we bring in a computer that allows us to input different data about our environment and then calculate the initial trajectory of the cue ball.

There are quite a lot of equipment and people around our table. We have the robot and the computer. There are engineers and scientists in white lab coats fine-tuning the robot and entering data into the computer. The human player is still in the room, curious to see if the robot can beat her.

At which point do we have to take into account the gravitational forces of all these people and equipment? The answer is after the sixth or seventh ball collision. The limits of our abilities to predict the events on the table are much lower than we might have guessed.

In fact, if you want to predict the movement of an oxygen molecule reliably after roughly 50 collisions, you have to take into account the gravitational effect of a single electron located at the edge of our universe.

These calculations are based on mathematical physicist Michael Berry's work. You can read his interview about predictability and chaos taken from the book A Passion for Science here. I first came across Berry in Nassim Nicholas Taleb's book The Black Swan.

Why do you need to care about the predictability of billiard shots? The point of this story is to help us notice the existence of uncertainties around us; even the smallest unknowns will start to affect our predictions dramatically as the chain of interactions grows longer. The mathematical world is unfortunately a fantasy.