2.1 Predicting the dice roll

If you’re going to predict how a rolled die will land, you need data. There are numerous factors here related to the die itself, and to the environment in which it is rolled.

Here’s Marcus to discuss a few of the key considerations.

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Transcript: Video 3 Predicting a dice roll

MARCUS DU SAUTOY

If you’ve ever played a dice-based game such as Snakes and Ladders, Ludo, Monopoly or Backgammon, you may have wondered whether dice rolls can be predicted. This question was explored in 2012 by a team of mathematicians in Poland, who used a high-speed camera to capture the trajectory of a die roll at a rate of 1500 frames a second. They found that in addition to knowing the initial position and the velocity of the die, that the most important factors are the friction and bounciness of the landing surface. Air resistance, on the other hand, can be disregarded.

On a high-friction rigid surface, dice tend to bounce around a lot more than on a low-friction or soft surface. The more a die bounces, the harder it is to predict the outcome. Energy is dissipated each time the die hits the surface, and at some point there’s little energy left, and the die comes to rest. If the amount of energy dissipated on impact with a surface is quite high – so, not many bounces – and the initial conditions can be established with sufficient accuracy, then the outcome of the throw can be pretty predictable. Furthermore, this predictability implies that the die lands on the surface that was lowest when the die was rolled. But if less energy is dissipated on impact with a surface – which means we’re going to get more bounces – then the outcome is more chaotic.

So, what do you think: do we use baize-covered tables for games because they look nice, or to increase the number of bounces, which therefore increases their chaotic behaviour and makes it much harder to predict the outcome?

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Video 3 Predicting a dice roll

Even with the data and the mathematical machinery in hand, there is still something missing – the scientific or physical theory that ties it together.

This comes in the form of Newton’s three famous laws of motion which relate an object’s motion to the forces acting upon it. Newton explained these laws in his Principia Mathematica, first published in 1687. (This is one of the most important scientific books ever to be published, and it nearly didn’t make it to the press. Newton was always extremely reluctant to publish his work, and it was published only through the intervention of Edmund Halley, who personally financed its publication after the Royal Society had spent its book budget on a History of Fishes!)

Applying Newton’s laws to physical systems (like the rolling of a die or the motion of a planet) can be written as mathematical equations. The differential calculus mentioned earlier is the key to solving these equations, and giving us the position of the object in motion at any desired time. However, as you shall see, solving these equations can turn out to be an extremely difficult problem. Sometimes it can even be impossible, or at least impossible to the necessary degree of accuracy.

Newton’s laws can nevertheless be used to make some remarkable deductions. Newton himself applied his mathematics to the solar system, calculating the relative masses of the large planets, the motion of the moon, and much more. He was even able to deduce the shape of the earth as a ‘squashed’ sphere (somewhat like a grapefruit) rather than perfectly spherical, as had been previously believed. Experiments carried out around the globe after Newton’s death proved his deduction to be correct.