|Ivars Peterson's MathTrek|
October 6, 1997
You can then invite your opponent to pick a different die, perhaps even the one that worked so well for you. You select one of the remaining dice. Again, in a game of at least 10 throws, you're very likely to come out the winner.
Indeed, it doesn't matter which die your opponent picks. You can always choose another die that will practically guarantee your triumph in a game of 10 or more throws.
Here's how the four dice are marked:
It turns out that the die with four 4s beats the die with six 3s, which in turn beats the die with four 2s, which beats the die with three 1s, which (completing the circle) beats the die with four 4s!
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!
The game can also work to your advantage when your opponent is allowed to select any pair of dice -- in effect leaving you with no choice but to take the remaining pair. Over the course of several matches involving games of 10 or more throws, you have a good chance of building a slight advantage into a comfortable win. Choosing blindly, your opponent may occasionally pick the 4-0 die together with the 3-3 die, or the 5-1 die together with the 4-0 die -- both losing combinations against the remaining pair.
Efron came up with two additional sets of four dice that show the same nontransitive property, but fewer numbers are repeated, making analysis of the dice more difficult. You can try dice numbered as follows: 2, 3, 3, 9, 10, 11; 0, 1, 7, 8, 8, 8; 5, 5, 6, 6, 6, 6; and 4, 4, 4, 4, 12, 12; or: 1, 2, 3, 9, 10, 11; 0, 1, 7, 8, 8, 9; 5, 5, 6, 6, 7, 7; and 3, 4, 4, 5, 11, 12.
There's plenty here to trap the unwary and baffle even the inveterate gambler!
For an update, see "Tricky Dice Revisited."
Copyright © 1997 by Ivars Peterson.
______. 1983. Nontransitive dice and other probability paradoxes. In Wheels, Life, and Other Mathematical Amusements. New York: W.H. Freeman.
One possible source of nontransitive dice is toy and novelty collector Tim Rowett. He offers a set of "Magic Dice" along with rules for several games at http://www.grand-illusions.com/magicdice.htm. You can find out more about Rowett's collection at http://www.grand-illusions.com/tim/tim.htm.