# Terribly Non-Transitive Dice

October 15, 2018

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James Grime is a British mathematician known to the (mainly English-speaking) public for his work with the journalist Brady Haran on the YouTube channel Numberphile. Despite his youth (37), he has already given his name to a mathematical curiosity: a set of atypical dice. There consist of five cubic dice of different colours. The six-sided dice are balanced, meaning that each side has the same probability of appearing when rolled. What is special are the numbers engraved on the sides; the red die has five sides showing a 4 and one side with a 9, the blue one has three sides showing a 2 and three with a 7, the green has five sides showing a 5 and one blank side, the yellow die has three sides showing a 3 and two with an 8, and the purple one has one side showing a 1 and four sides with a 6.

Let’s roll the red and blue dice together. The probability that the red die will beat the blue die must be greater than 1/2. In other words, the red die “is stronger than” the blue one. Similarly, the blue beats the green, which beats the yellow which in turn beats the purple… which beats red. If you are following, you can see a surprising situation of non-transitivity, which can be shown as: red > blue > green > yellow > purple > red.

This is a well-known phenomenon and a mainstay of popular science. James Grimes’ curiosity is even more surprising; if the dice are rolled in pairs and the results added together, the result obtained with the two red dice is, with a probability greater than 1/2, smaller than the result obtained with the sum of the two blues. In other words, the two blue dice beat the two red dice.

Continuing the calculations, we can see that the order is completely reversed, summarised as 2 x red < 2 x blue < 2 x green < 2 x yellow < 2 x purple < 2 x red.
Of course, the question you are itching to ask is what happens when we play with three sets of the same colour? Four? More? Do these “orders” alternate? Does one get the upper hand after a while? Is another order possible? In this case, as all the values on the dice are known, we just need to calculate all the values for the sum of dice of the same colour and the corresponding probabilities. The existence of such paradoxical situations motivated the American mathematicians Joe Buhler, Ron Graham and Al Hales to study whether even more extreme sets of dice existed. Their idea was to show that, for any integer k, there is a set of k dice of different colours (always balanced but with potentially exotic values on the sides) so that when all possible tournaments are tested (for one die, for the sum of two dice, for the sum of three dice, for the sum of four dice, etc.), all the possible “orders” can be found an infinite number of times. All power balances are possible between these combinations of dice!

The combinatorial complexity of the problem is not easy to perceive: remember that, to establish an “order”, for each choice of two colours, you have to determine which is stronger. Thus, with only three dice, there are already 2 power 3 = 8 possible orders; with four, there are 26 = 64; with five, there are already 2 power 10 = 1,024... and the game of dice shown by Joe Buhler, Ron Graham and Al Hales must use each of these possibilities an infinite number of times. The proof, published in the spring of 2018, is technical but not extremely difficult. It is based on a fine understanding of the difference between the arithmetic mean (used to calculate average grades in class) and the median (values separating the possible results into two sets of the same probability) for the sum of throws of n dice when n becomes large. Given the difficulties relating to these indicators in the generalist media, the demonstration is all the more appreciable.

> AUTHOR

Roger Mansuy

Mathematician

Roger Mansuy teaches at Louis-Grand Lycée in Paris, and is a member of the French Committee for Maths Teaching (CFEM).