From the Exploring Data website - http://curriculum.qed.qld.gov.au/kla/eda/
© Education Queensland, 1997

A Special Set of Dice

If A is older than B, and B is older than C, it follows that A is older than C. The phrase ‘is older than’ is a binary relation, and this statement is an example of the transitive property of binary relations. Not every binary relation is transitive. For example if John is the uncle of Sam and Sam is the uncle of Jeremy, it doesn't follow that John is the uncle of Jeremy. It is obvious that this relation isn't transitive, but sometimes intuition says a relation is transitive, when in reality it isn't. This is the basis of a cute little trick, that can be explained using simple probability, using a set of non-transitive dice.

Bring out a set of these special dice, tell the students that you will be offering them the chance to have no homework for the next two nights, then explain the rules of a game -

I have this set of four dice. We will each chose a die; in fact I'll even let you choose your die first. We'll roll the two dice and the winner is the person whose die has the highest number. The first person to record five wins is the champion. Now if I am the champion, you have to do an extra hour of homework tonight. If you are the champion, then you are excused from homework for the next two nights. Any takers? After all, you've got to be in it to win it!

Of course, the student will be excused from doing the extra homework if the class can figure out why the teacher wins almost all of the time.

The Dice

Here are three sets of dice each with the property that A beats B, B beats C, C beats D and D beats A! (A ‘beats’ B means that over the long term die A will win more often than B.)

For the first two sets of dice the probability of A beating B (and similarly for the other pairs in those sets) is 2/3. In the last set the probability is 11/17, or .647.

Playing against a student, if the student chooses die A then the teacher choose D, if the student chooses B then the teacher choose A and so on. The probability that the teacher will win a particular game is 2/3. Finding the probability that the teacher will be the first to win five games takes a bit of thought, and is a suitable problem for more able Maths B students. [Solution]

A Beats B, and So On

Here is a table using dice from the first set that demonstrates that A beats B two-thirds of the time. For each possible outcome the winning dice is listed. A wins 24 times out of 36, or 2/3 of the time.

The Winning Player

A similar table can be constructed for B vs C, and C vs D, and D vs A.

Because of the repeated numbers on the first set of dice, it isn’t too hard to figure out the non-transitivity of the set of dice. The second set disguises this a bit better, and the third set best of all. With the third set there is the possibility of ties, so extra tosses may be required, and the odds aren’t quite as much in favour of the teacher.

References
Gardner, Martin, 1983. Wheels, Life and Other Mathematical Amusements, W.H. Freeman and Company, p 40 ff.