## Courtesy at stake ### Cycles of 3 dice

If, instead of cycles with four dice, we limit ourselves to cycles with three dice, say dice with 1 to 6 spots on each side, is there something interesting to mention? In this case we accept ties, but we agree that when a tie occurs it is ignored and the dice are thrown until one of them wins.

Analyzing the situation by using the Mathematica software again, it was possible to collect some information: in this case there are 462 different dice and 40666 non-transitive cycles can be built with them. But 68 of such dice do not take part in these cycles. For the remaining 394 dice, the frequency with which they take part in the non-transitive cycles is very variable: from two dice ($$\left\{ 1,1,2,2,2,4\right\}$$ and $$\left\{ 3,5,5,5,6,6\right\}$$) that appear in only one non-transitive cycle $$\left\{ \left\{ 1,1,1,2,4,4\right\} ,\left\{ 1,1,1,3,3,4\right\} ,\left\{ 1,1,2,2,2,4\right\} \right\}$$ and $$\left\{ \left\{ 3,3,5,6,6,6\right\} ,\left\{ 3,5,5,5,6,6\right\} ,\left\{ 3,4,4,6,6,6\right\} \right\}$$, respectively), to two other dice ($$\left\{ 1,1,1,5,6,6\right\}$$ and $$\left\{ 1,1,2,6,6,6\right\}$$) that appear in 1897 non-transitive cycles each.