Math Digests October 2022

How to make a tastier chocolate? Use geometry.

Washington Post, October 5, 2022

What makes eating chocolate so satisfying? You may like the sweetness and creaminess of a milk chocolate. Maybe you like the sharp, rich bitterness of dark chocolate. To physicist Corentin Coulais, it’s all about the crunch. Coulais is a physicist who has conducted research on how the shape of a chocolate bar increases its brittleness, which in turn improves its quality. In a recent experiment, Coulais 3D-printed chocolate in various zig-zagging and swirling shapes. The most zig-zagging spiral shape was the most crunchy—and the one most preferred by Coulais’ ten volunteers. “A pleasurable eating experience doesn’t only take place in the mouth, but can be affected by the noises in your skull,” writes Galadriel Watson for the Washington Post. In this article, Watson describes the research project and why it may one day change the shape of the chocolate you can buy.

Classroom Activities: brittle geometry, chocolate math

  • (Low level) Watch this Infinite Chocolate math riddle for another example of tricky geometry with math. Discuss why this trick works.
  • (Mid level) Based on this image of the 3D printed chocolates, rank the shapes from most crunchy to least crunchy. Describe why you have ranked them this way.
  • Do you expect that an S-shaped chocolate would be more crunchy if it were thicker (more chocolate) or less thick? Why?
  • (Mid level) Geometry factors into the design of food in other ways as well. Imagine you are a chocolatier making a spicy chocolate entirely coated in chili powder. You make 1-ounce chocolates in three different shapes: a sphere, a flat bar, and a zig-zag (like Coulais’ printed treat). Each of these shapes has the same volume and weight of chocolate. Which will be the spiciest? Discuss what factor from its geometry most influences the overall spiciness of its chili coating.

—Max Levy

How a magician-mathematician revealed a casino loophole

BBC Future, October 19, 2022

In a friendly game of Go Fish, it might not matter how thoroughly you mix your deck of cards. But an incomplete shuffle can make all the difference in a high-stakes casino game. The question of how to randomize a deck so that gamblers can’t take advantage of patterns in the cards is practical, but also mathematical. In 1992, mathematicians Dave Bayer and Persi Diaconis proved that it takes seven riffle shuffles to fully mix a deck. With six or fewer rounds of shuffling, the deck is still biased, but seven or more shuffles basically randomize it. Diaconis, who was a professional magician before pursuing statistics, has continued to work on the math of mixing cards: he even traveled to Las Vegas to assess a new card-shuffling machine, as Shane Keating describes in an article for BBC Future.

Classroom Activities: probability, statistics, card games, computers

  • (All levels) Watch Persi Diaconis explain how to shuffle a deck in a video for Numberphile. 
    • (Probability/Statistics) In the video, Diaconis shows that if you take a standard deck of cards, mix it well, and ask someone to guess the identity of the top card, then do that for each card in the deck (where you reveal the top remaining card after each guess), you should expect the guesser to make about 4.5 correct guesses on average. With a partner, try playing this game with a set of 10 cards, such as the ace through 10 of a single suit. Record the number of cards you guessed correctly, then discuss your results with other students. Find the class average of the number of correct guesses from a deck of 10 cards.
    • (Probability/Statistics) Now, try a smaller version of Diaconis’s probability computation from the video: in a well-mixed deck of just 10 cards, what is the expected average number of correct guesses? Hint: Diaconis starts describing the 52-card version of this computation at time 2:17.
    • (All levels) If you watched the video above and are interested in learning more about the seven shuffles, watch this Numberphile2 follow-up video with more details.
  • (Probability/Statistics; Advanced) Some magic tricks use a special kind of shuffling, called the perfect shuffle or faro shuffle, in which the deck is cut exactly in half and the cards are interspersed one by one so that the two halves alternate perfectly. Read the explanation of perfect shuffles from Jim Wilson at the University of Georgia, then try to answer as many parts of Questions 1, 2, and 4 as possible. To model the shuffles, try experimenting with Excel spreadsheet linked to on the webpage or building your own spreadsheet.

—Tamar Lichter Blanks

Choose Your Own (Math) Adventure

Short Wave From NPR, October 18, 2022.

In early 2020, mathematician Pamela Harris and some of her students posted a new paper online about parking functions, which let you assign a collection of cars to parking spots along a one-way street. The paper, called “Parking Functions: Choose Your Own Adventure”, was structured as an interactive experience. At various choice points, readers can decide what kinds of parking functions they want to study — and at the end of their adventure, they’re rewarded with an unanswered question about parking functions. In this episode of Short Wave, Regina Barber interviews Harris about what inspired her and her students to write this unusual paper and the basics of the math involved.

Classroom Activities: combinatorics

  • (Mid level) Harris calls combinatorics “the art of counting”, and gives an example of counting the number of ways to make 37 cents out of change. For a slightly easier exercise, count the number of ways you can make 17 cents out of change. Write out your logic.
  • (All levels) For a more thorough introduction to combinatorics, check out the Art of Problem Solving’s online videos.
  • (Mid level) Read this description of the parking problem, adapted from Harris et. al.’s paper:Parking spots are lined up along one side of a one-way street, and cars arrive one at a time to park. A list of numbers represents the cars’ parking preferences: If there are 3 parking spots and 3 cars, the list (2, 2, 1) means the first car arriving would like to park in the second spot it encounters, the second car would also like to park in the second spot, and the last car would like to park in the first spot. Cars drive up their favorite spot; if it’s full, they keep driving until they find an empty spot. They can’t turn around, so that once they’ve passed a spot, they can no longer park there. If they pass all the spots without finding a place to park, they drive off and never return. The list of numbers is a parking function if every car is able to find a spot.
    • Harris and her students write that if there are 5 spots and 5 cars, then (1, 2, 4, 2, 2) is a parking function, while (1, 2, 2, 5, 5) isn’t. Can you convince yourself if this is true? If so, explain why.
    • For 3 spots and 3 cars, is (2, 2, 1) a parking function?
    • For 3 spots and 3 cars, is (2, 2, 2) a parking function?
    • For 3 spots and 3 cars, is (1, 1, 1) a parking function?

—Leila Sloman

DeepMind AI invents faster algorithms to solve tough maths puzzles

Nature News, October 5, 2022 

Think of the hardest math problem you’ve ever seen. Perhaps it’s a gnarly long division problem or a quadratic equation that seems impossible to factor. Maybe it’s a long, repetitive matrix multiplication, where you multiply large grids of numbers. Whatever the challenge, the sequence of steps you take to solve it are algorithms. Mathematicians and scientists have realized that they can rely on computers and artificial intelligence to carry out complicated algorithms for their hardest problems. In this article, Matthew Hutson describes a new AI for matrix multiplication that goes one step further. Rather than just fly through the standard algorithm for matrix multiplication, this AI discovers faster, better algorithms itself. And now, the math world is eager to find out which other math problems it can tackle.

Classroom Activities: matrix multiplication, algorithms

  • (Low level) Follow a real-life algorithm to make a paper plane (from Discuss what you think would happen to your paper plane if you skipped or messed up one of the steps?
  • (Mid level) Create your own algorithm for drawing a picture of your choice. You can come up with any picture you like, but make sure your algorithm takes no more than 10 steps, and that others can follow it easily.
  • (Mid level) Read more about how to perform matrix multiplication on this Math is Fun tutorial. Solve the example about selling pies, and do questions 1, 2, 3, and 8.
  • (Mid level) The week after this AI discovery, two people beat its record. Read more here.

—Max Levy

Huijia Lin proved that a master tool of cryptography is possible

Science News, September 29, 2022.

Researchers have shown that it’s possible to create a tool that allows different people access to different information, while keeping data they don’t need securely locked away. That concept, called indistinguishability obfuscation (or iO), contrasts with conventional digital security, in which those possessing a key or password have free reign over all the protected information. Elizabeth Quill covers the accomplishment — by Huijia Lin, Amit Sahai, and Aayush Jain — for Science News.

Classroom Activities: cryptography, zero-knowledge proofs

  • (Mid level) Quill writes that Lin was taken with cryptography as a graduate student, especially zero-knowledge proofs. Embedded in the article is a video of Amit Sahai explaining what a zero-knowledge proof is to 5 different people. Click here to watch the section where Sahai explains to a teen.
    • In the clip, Sahai illustrates the idea of a zero-knowledge proof by reading out Daila’s secret which she placed in a locked box. Come up with your own example of a zero-knowledge proof. Then partner up with a classmate and use your example to convince your classmate you know something, without revealing what it is.
    • After trading your zero-knowledge proofs, answer these questions about your partner’s example:
      • Did they convince you?
      • Did they reveal any information they didn’t mean to? If so, what?
    • Read your partner’s feedback and improve your proof if necessary. Now, change partners and repeat the exercise again.
  • (High level) For students interested in learning more about cryptography, Khan Academy has a cryptography unit online. After learning about the Caesar cipher and frequency analysis, encode a secret message with it using the Caesar cipher exploration tool. Then trade messages with a partner and see if they can crack it using this frequency analysis tool.

—Leila Sloman

Some more of this month’s math headlines: