Solution:  5/9, 5/9, and 5/9.  

Because each die contains three different numbers, there are 3 × 3 = 9 possible outcomes when two of these dice are rolled.

As shown in the tables below, A beats B on 5 out of 9 rolls, 2-1, 4-1, 9-1, 9-6, and 9-8, so the probability that A beats B is 5/9. Similarly, B beats C on 5 out of 9 rolls, 6-3, 8-3, 6-5, 8-5, and 8‑7, so the probability that B beats C is also 5/9. Finally, C beats A on 5 out of 9 rolls, 3-2, 5-2, 5-4, 7-2, and 7‑4, so the probability that C beats A is, once again, 5/9.

In math, we’re used to the inequalities A > B and B > C implying that A > C. This relationship is known as the transitive property. But with these dice, that relationship does not hold, so these dice are often referred to as “nontransitive dice.” Like the game Rock, Paper, Scissors, they don’t obey the laws of inequalities that apply to individual numbers.