### The optimal choice

A cycle of four dice such as the cycle (\(A\), \(B\), \(C\), \(D\)) considered before is said to be non-transitive. Now it is natural to raise the question of whether there would be other possible choices for the number of spots on the faces of the six dice that were also appropriate to highlight this unexpected property. If we want to choose four dice so that, as happens in the above mentioned cycle of dice, a tie never occurs in any of the throws between two of them, we have to choose them in such a way that any two faces of different dice have a different number of spots. If 0 to 6 spots are allowed in each dice, these seven possibilities are not enough to have two types of faces in each one of the four dice (\(4\times2=8>7\) possibilities), which means that at least one dice must have the same number of spots on all faces. By using a program written in the Mathematica software, we were able to select all the non-transitive cycles, that is, cycles where the probability of each dice winning with the next one is always bigger than \(\frac{1}{2}\). We found 37 non-transitive cycles. Twelve of them are shown in figure 5.

By evaluating the minimum of these probabilities for each cycle, all of them bigger than \(\frac{1}{2}\), we got only three possible minima: \(\frac{5}{9}\), \(\frac{7}{12}\) and \(\frac{2}{3}\), but only for one of the cycles, the first indicated above, is the minimum \(\frac{2}{3}\), which is also the value of the other probabilities associated to the cycle. Of the remaining 36 cycles, 3 of them have minimum \(\frac{7}{12}\) (2nd to 4th, in the figure) and 33 have minimum \(\frac{5}{9}\). The cycle with the biggest minimum is naturally considered the optimal choice, because it is the most appropriate cycle to highlight the above mentioned unexpected property. For a cycle with a minimum close to \(\frac{1}{2}\), which is the case of the cycles with minimum \(\frac{5}{9}\) or \(\frac{7}{12}\), we need a large number of repetitions to decide without any doubt which dice is the best. In conclusion, the cycle of "Matemática Viva" is the unique optimal cycle.