ATRACTOR

This text is about the following question: in a set of different dice, which is the best?

Fig 1

This question raises a mathematically interesting problem that has a somewhat unexpected answer. These two aspects motivated Atractor to build a module for the exhibition Matemática Viva¹, which highlighted this apparently odd behaviour.

Before describing this module, we clarify what we mean by a given dice being better than another one. Normal dice are cubes with each side marked with a different number of spots, from one to six. No dice is better than another, in the sense that two players, each throwing one of the dice, have the same odds of winning, and they may even end in a tie. If instead of considering two normal dice we assume that each side is marked with a number of spots between 0 and 6, with possibly two or more sides having the same number of spots, then the two dice can be different from each other, and in this case they may have different odds of winning, and the probability of a tie can be 0. The dice \(A\) is considered better than dice \(B\), if the probability of \(A\) winning is greater than the probability of \(B\) winning. For example, if the six sides of \(A\) have respectively 1, 1, 1, 3, 3, 3 spots and \(B\) has 4 spots on each side, a tie cannot occur and the probability of \(A\) winning is 0, which means that the probability of \(B\) winning is 1. If the two dice are equal, the probabilities of \(A\) and \(B\) winning are the same, but this value depends on the probability of a tie. For example, if \(A\) and \(B\) are equal and in addition all the sides have the same number of spots, the probability of a tie is 1 and therefore the probability of each dice winning is 0. The reader can check that this extreme situation only occurs under the described extreme conditions.

(*) Difficulty level: Upper Secondary, University