## Non-standard fair die

Goal: A dice is a polyhedron with different numbers on its faces. When you throw a dice, the result of the throw will be the number facing the ground. A dice is fair if the probability law on the set of numbers written on the dice resulting from the throw is uniform. A common way to create fair die is to use the most symetric polyhedrons a.k.a. Platon's polyhedrons:

But there are many other ways to create fair die! For instance in an article by K. Robin McLean (Dungeons, Dragons and Dice The Mathematical Gazette, Vol. 74, No. 469 (Oct., 1990), pp. 243-256) the author draws the following polyhedron which leads to a fair dice:

The above example is still very symetric in a specific sense: it is isohedral, i.e., every face can be obtained from the other by applying a rotation.

The goal is to conceive many different kinds of fair die, especially non-isohedral ones. Ideally these die would be 3D printed and have their fairness studied.

Questions: What are the different 'kinds' of fair die? How 'stable' are these fair die?

Schedule: Winter semester 2018/2019, starting Friday the 21st September.

Supervision: Clément Guérin. Hugo Parlier.

Difficulty level: EML 2.

Prerequisites: Algèbre linéaire, Structures mathématiques, Géometrie.

Tools : Software to use 3D printers. Sage for modelization. R for statistics.

Expected results: Whereas a complete classification of these fair die is out of reach, the students will be expected to develop a geometric intuition in understanding different ways of conceiving a fair dice. Concretely, the students will have to test their construction of fair die either by modelizing the die using Sage or by 3D printing the die and throwing the dice to test the actual fairness. It should be noted that the outcomes should ideally be processed using statistic tools (R for instance).