Fig 8

(click on Figures 8 and 9 to see an interactive version)

The lowest values for the probability of non-transitive cycles are $$\{$$ $$\frac{18}{35}$$, $$\frac{17}{33}$$, $$\frac{16}{31}$$, $$\frac{15}{29}$$, $$\frac{14}{27}$$, $$\frac{13}{25}$$, $$\frac{12}{23}$$, $$\frac{11}{21}$$, $$\frac{10}{19}$$, $$\frac{19}{36}$$, $$\frac{9}{17}$$, $$\frac{17}{32}$$, $$\frac{8}{15}$$, $$\frac{15}{28}$$, $$\frac{7}{13}$$, $$\frac{13}{24}$$, $$\frac{19}{35}$$, $$\frac{6}{11}$$, $$\frac{17}{31}$$, $$\frac{16}{29}$$, $$\frac{5}{9}$$, $$\frac{19}{34}$$, $$\frac{9}{16}$$, $$\frac{17}{30}$$, $$\frac{4}{7}$$, $$\frac{15}{26}$$, $$\frac{7}{12}$$ $$\}$$ and they vary from $$\frac{18}{35}$$ to $$\frac{7}{12}$$, appearing with different frequencies as shown in Figure 8.

To get an idea not only of the lowest values for the probabilities of each non-transitive cycle but also of the three probabilities associated to each cycle, we can make a graphical representation of those triplets of probabilities. Note that the representation of a triplet by a point involves the (arbitrary) choice of the firts element of the triplet. The other choices correspond to a rotation of $$120^{\circ}$$ and $$240^{\circ}$$, about the right line through the origin with the direction defined by the vector $$(1,1,1)$$. Those three sets of points are represented in different colours in figure 9. In Figure 10 the stereoscopic pair represents a similar set of probabilities but corresponding to cycles with three dice with 1 to 4 spots on each side.