## Courtesy at stake

### Increasing the number of spots

For a virtual version of this module, it is interesting to have several different non-transitive cycles, all optimal (in the sense that the lowest probability associated to the cycle is the greatest possible4). This is not feasible for dice with 0 to 6 spots on each side because we have seen that there is only one optimal cycle. But using dice with a larger number of spots on each side, and keeping the other restrictions of the problem (in particular, 2 to be the maximum number of different sides on each dice) it is possible to obtain more optimal cycles. Using the Mathematica software it was possible to find 13 optimal cycles for dice with 0 to 7 spots on each side, and 81 optimal cycles for dice with 0 to 8 spots on each side (among a total of 527 and 3495 non-transitive cycles, respectively). In figure 6, the 13 optimal cycles for dice with 0 to 7 spots on each side are shown.

Fig 6

4 Trybula and Steinhaus proved that the minimum probability can not exceed the value found in that cycle: $$\frac{2}{3}$$. See [1] and [2].