On the labelling of polyhedral dice

© Applied Probability Trust 2014. We extend previous work which modelled the rolling of a typical playing die using a Markov matrix. The conditional probability of a roll's result (i.e. the final uppermost face) varies according to a standard n-faced die's initial position. However, for some standard ″-faced dice, we can label the faces in such a way that the conditional probability of a roll's outcome (i.e. the final score) does not vary according to the die's initial position. In such cases, we say that the die is fair under that labelling. Here, we derive general conditions that such labellings must satisfy. Using these conditions, we identify specific examples of fair polyhedral dice.