BoingBoing reports on nontransitive dice: trick dice, weighted not physically but mathematically. Ivars Peterson explains the principle of nontransitive dice in his MathTrek column:

The game involves four specially numbered dice. You let your opponent pick any one of the four dice. You choose one of the remaining three dice. Each player tosses his or her die, and the higher number wins the throw. Amazingly, in a game involving 10 or more throws, you will nearly always have more wins.

Here’s what the dice look like:

Nontransitive Dice

Dice numbered in this fashion are known as nontransitive dice. They were designed by Bradley Efron, a statistician at Stanford University, to help illuminate probability paradoxes that involve the violation of a mathematical property called transitivity.

In general, transitivity is a binary relation such that if the relation holds between A and B and between B and C, it must also hold between A and C. For example, whole numbers follow such a rule: 5 is larger than 4 and 4 is larger than 3, so 5 is also larger than 3. Similarly, if A is heavier than B and B is heavier than C, then A is also heavier than C.

One place where transitivity doesn’t apply is in the children’s game of scissors-paper-rock. In the playground version, each of two players holds a hand behind his or her back. On the count of three, both players bring their hidden hands forward in one of three configurations. Two fingers in a “V” represent scissors, the whole hand spread out and slightly curved means paper, and a clenched fist signifies rock. The winner is determined by the following sequence of rules: Scissors cut paper, paper wraps rock, and rock breaks scissors (see Mating Games and Lizards).

In the case of the tricky dice, the mistaken assumption is that the relation “most likely to win” must be transitive between pairs of dice. It is not. No matter which die your opponent picks, you can always select a die that has a 2/3 probability of winning. That’s two-to-one odds in your favor!