## Courtesy at stake ### Conclusion

The non-transitive cycles with three dice including a given dice $$D=\left\{n_{1},n_{2},n_{3},n_{4},n_{5},n_{6}\right\}$$ with 1 to 6 spots on each side, can be obtained from the non-transitive cycles including the dual dice of $$D$$ given by $$E =\left\{m_{1},m_{2},m_{3},m_{4},m_{5},m_{6}\right\}$$, where $$n_{i} + m_{i}= 7$$, for $$i=1,...,6$$ (of course if $$E$$ is the dual of $$D$$, $$D$$ is also the dual of $$E$$). For example, the dual of dice $$D = \left\{1,1,2,2,2,4\right\}$$ is $$E=\left\{3,5,5,5,6,6\right\}$$, and, as we have seen before, each one of these dice appears in only one non-transitive cycle. The non-transitive cycle involving $$E$$ is $$\left\{\left\{3,3,5,6,6,6\right\},\left\{3,4,4,6,6,6\right\},\left\{3,5,5,5,6,6\right\}\right\}$$ and therefore the only non-transitive cycle involving $$D$$ is $$\left\{\left\{1,1,1,2,4,4\right\},\left\{1,1,1,3,3,4\right\},\left\{1,1,2,2,2,4\right\}\right\}$$. Another example: the dual of dice $$D = \left\{1,1,1,5,6,6\right\}$$ is $$E=\left\{1,1,2,6,6,6\right\}$$. We have seen that these dice appear in 1897 non-transitive cycles each. Moreover, each non-transitive cycle involving $$D$$ can be obtained by considering the dual of the dice involved in each non-transitive cycle involving $$E$$.

Remark: This procedure may be generalized to dice with $$1$$ to $$N$$ spots on each side by defining the dual of a dice $$D =\left\{n_{1},n_{2},...,n_{N-1},n_{N}\right\}$$ as the dice $$E =\left\{m_{1},m_{2},...,m_{N-1},m_{N}\right\}$$ where $$n_{i} + m_{i} = N+1$$, for $$i=1,...,N$$.

REFERENCES

 H. Steinhaus, S. Trybula, "On a paradox in applied probabilities", Bulletin of the Polish Academy of Sciences 7 (1959), 67-69.

 Usiskin, "Max-min probabilities in the voting paradox", Annals of Mathematical Statistics Vol. 35, 2 (1964), 857-862.