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 \(i=1,...,N\).


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

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