Design Schemes for Fair Dice

TL;DR

This paper explores the design of fair dice with various shapes, modeling them through symmetry and sphere caving, and proposes a method to create fair n-faced dice, including coins.

Contribution

It introduces a novel approach to designing fair dice of arbitrary faces by modeling shapes via symmetry and sphere caving techniques.

Findings

Modeling symmetric dice shapes as sphere caving structures

Viewing coins as 2-face fair dice with thickness considerations

Proposing a general method for designing fair n-faced dice

Abstract

A cube is used as a fair die of 6 faces. However, there are many dice of different shapes on the market. To make them fair, most of them usually have some symmetric shapes. We here classify these variants of dice on the market into two groups. We first consider that a sphere as a model of a fair die with infinity faces. Based on this model, many symmetric shapes can be modeled as dice obtained by caving spheres. We also have a familiar fair device; a coin. That is, a fair coin can be seen as a fair die with 2 faces. However, a real coin has a thickness, and hence it is, in fact, an unfair die with 3 faces. From this viewpoint, we propose a way for designing a fair die with n faces for arbitrary n.