If you grew up playing board games like Monopoly or Ludo, then you must be familiar with the central object controlling the whole game — a die or dice. The recognizable square object with six sides decided your fate as you rolled forward. But what if someone told you that a dice need not be square all the times? No, not skew dice which are somewhat popular in certain circles but something that wasn’t a regular polyhedron at all. Can a sphere be a fair die?
Science-based YouTube channel The Action Lab recently took upon himself to establish if a spherical shape could ever work as a fair dice. To start, a fair die is defined when the chances of one of the six sides coming face up is equally probable on every roll. Examples of fair dice include the ubiquitous square/cube used everywhere from board games to casinos. Another fair die is Asymmetrical trigonal trapezohedron shaped skew dice which is slightly odd to look at but it still a legally fair die.
The host then takes a ping-pong ball. He labels it 1-6 at equidistant sites across the ball. However, when he rolls it, the ball falls down the table as predicted. He then attempts to roll the ball on a surface with more friction. This time the ball stops but at no particular number.
“It is because on a perfect sphere, the number of equilibrium points are infinite, making the chances of landing on one number nearly zero,” the host reveals.
He says it can be made into fair dice by some clever measures. If we take lines on the inside and make them have a corner that relates to these six numbers so it can act like an octahedron. Octahedron has eight faces and six vertices. By adding an internal cavity, the shape of octahedron inside the sphere, one can make a spherical die. This would create 6 different pockets or sides inside the sphere. If there is a form of weight, like a small metal ball, then it will land inside one of the pockets when the dice is rolled, and one of the numbers will face up. Now, the spherical die is fair.
Interestingly, all three shapes showcased in the video are “fair dice” but they are not equally fair. These can be manipulated to not be ‘fair’. Like many people can train themselves (and scientists have trained robots) to always land on heads when flipping a coin. So, depending on the symmetry of the dice, one can manipulate the roll of each of these dice.