# Fair die

A fair die is a convex polytope which, if made of a homogeneous material and randomly thrown onto a flat surface, has an equal probability of landing on each facet. From symmetry considerations, all convex isotopic polytopes are fair dice, but there is no agreed-upon definition of fairness for non-isotopic polytopes.

Diaconis and Keller argued that a very flat regular-n -gonal prism has a low chance of landing on its rectangular faces and a very tall regular-n -gonal prism has a high chance, so there must be a prism between these extremes where the landing probability of a rectangular face is equal to that of the n -gon. Such dice are called "fair by continuity". However, the optimal edge ratio depends on the exact definition of fairness, which is not agreed upon. In sophisticated models, physical parameters such as the coefficient of friction between the surface and the die must be taken into account.

The dual of the elongated square gyrobicupola has been suggested as a possible non-isotopic fair die (e.g. a MathOverflow answer), but it is unclear whether it fits any reasonable definitions of fairness, and if so which ones.

There are also curved convex shapes which might be informally called dice. A bicone has two different ways it can roll on a flat surface, and it can land on either one with equal probability. Bowers suggested from symmetry considerations that the convex regular hard polytwisters are fair dice.

## In games

A standard set of seven dice used in tabletop role-playing games comprises the five Platonic solids and two pentagonal antitegums, one marked with the numbers 0-9 and the other with the multiples of 10 from 00-90. All of these are isotopic.

One common design for non-isotopic "barrel" dice with an even number of sides is to use isogonal prisms or antiprisms, then symmetrically augment the two base faces with pyramids so that the dice are unstable when landing on the pyramidal ends, and will always roll over onto the side faces. This results in a die that is fair for all practical reasons, although strictly speaking it does not have an equal probability of all facets.

Some dice that are intended to be fair by continuity are commercially manufactured, such as five-sided die (triangular prism) and seven-sided die (pentagonal prisms). Some experiments conducted on such dice have indicated that they deviate from fairness. Deliberately non-fair dice are also used in games, such as the Central Asian shagai and the commercial game Pass the Pigs.