Joe Buhler - Wildly Nontransitive Dice - G4G14 Apr 2022

Summary

This talk explores paradoxical dice, an intriguing extension of non-transitive dice theory. The presenter uses a magic square as an example, demonstrating how rolling dice multiple times can reverse the expected outcomes. The concept is expanded to include higher powers, revealing eight possible tournament outcomes. The lecture concludes with a theorem on the existence of 'wild dice' and a brief proof, engaging the audience with the fascinating mathematics behind these paradoxical results.

Takeaways

• 🎲 The talk introduces paradoxical dice, which are an extension of non-transitive dice.
• 👨‍🏫 The presenter uses a real-life example involving hiring women in university departments to illustrate paradoxical effects.
• 📊 A classic example of non-transitive dice is presented using a 3x3 magic square, where the outcomes of rolling two dice (A, B, and C) cycle in dominance.
• 🔄 When each die is rolled twice (A squared), the cycle of dominance reverses compared to a single roll.
• 🔢 The phenomenon of rolling dice multiple times (cubed, to the power of five, etc.) can lead to complex and varied outcomes in dominance cycles.
• 🏆 The concept of 'tournaments' is used to visualize the outcomes of dice rolls, with arrows indicating the winner over the loser.
• 🔄 It's possible to have different outcomes (tournaments) depending on the power to which the dice are rolled.
• 🤔 The number of possible tournaments increases with the number of dice and can be calculated based on the dice's properties.
• 🧮 The existence of 'wild dice' is proven, which can lead to all possible tournaments occurring without a final winner.
• 📝 The proof for the existence of wild dice involves the concept of 'nice' dice, where the sum of values and cubes are zero modulo M.
• 📘 The detailed explanation and proof were published in the Monthly magazine in 2018.

Q & A

What is the main topic of the talk?

The main topic of the talk is paradoxical dice, which is an extension or intensification of the idea of non-transitive dice.

Who are the co-authors mentioned in the talk?

The co-authors mentioned are Ron Graham and L Hales.

What is the paradoxical situation presented in the talk involving the dean of Sciences?

The paradoxical situation is that each science department has a higher fraction of women than every other department in the Athletic League, but when looked at collectively, the division has fewer women than all other departments in the Big Ten.

What is a non-transitive set of dice?

A non-transitive set of dice is a set where one die (A) beats another (B), B beats another (C), but C beats A, creating a cycle of dominance.

What is a magic square and how is it related to the dice in the talk?

A magic square is a square grid of numbers where the sums of the numbers in each row, column, and diagonal are the same. In the talk, a 3x3 magic square is used to represent the outcomes of rolling three non-transitive dice.

What does 'a squared' refer to in the context of the talk?

'A squared' refers to the set of outcomes obtained by rolling die A twice and adding the results, which is used to explore the intensification of the non-transitive effect.

How many tournaments are possible with n players?

With n players, there are 3^n possible tournaments, as each player can either win, lose, or tie against another player.

What is the significance of the different tournaments (Alpha, Beta) mentioned in the talk?

Alpha and Beta represent different outcomes of tournaments when rolling the dice different numbers of times. Alpha is the regular tournament outcome, Beta is the reverse, and others indicate different cyclical outcomes.

What does the theorem mentioned in the talk state?

The theorem states that for any number of dice, there exists a set of dice such that all possible tournaments occur without a final winner, meaning the cycle of dominance is fully cyclic.

What does a 'nice' die mean in the context of the talk?

A 'nice' die is one where the sum of its values is zero and the sum of its cubes is also zero. It also refers to a die that is mod M, meaning all of its values are congruent to a mod M.

Where can more details about the proof mentioned in the talk be found?

More details about the proof can be found in the Monthly magazine from 2018.