Math Digests March 2024

How do you celebrate Pi Day? DSU Mathematician explains why the number should be celebrated

WMDT, March 14, 2024.

The concept of $\pi$ is essential. How could you study geometry without marveling at how the circumference of a circle is always a factor of $\pi$ times larger than its diameter exactly? To approximate its value, students usually prefer 3.14, while others prefer 22/7, said mathematician Dawn Lott. “I tend to use ten digits, which is 3.14159265358.” This story from Leila Weah at WMDT shares the history of calculating $\pi$ and the differences in how people use and represent it.

Classroom Activities: pi, algorithms

  • (All levels) Read this Twitter/X thread from mathematician Alex Kontorovich about how to demonstrate $\pi$ using pizza crusts.
    • Identify and explain at least 3 limitations of this exercise that would lead to falsely high or low estimations of $\pi$.
  • (Mid level) Design your own pizza-based demonstration of $\pi$ that only uses pizzas but involves no cutting. (Think about other ways you might measure the circumference!)
    • Identify and explain at least three limitations of your exercise.
    • Write a concise step-by-step protocol so that someone not in this class could carry out the demonstration and understand what it says about $\pi$.
    • Compare and discuss your idea with other students.

NOTE: The article states that $\pi$ contains trillions of digits. The decimals of $\pi$ extend infinitely. The record for most decimal digits calculated by a human is 100 trillion.

—Max Levy

MSU Denver professor provides mathematical strategy to filling out March Madness bracket

CBS News, March 20, 2024.

How one fan came close to a perfect March Madness bracket

ESPN, March 19, 2024.

During the yearly college basketball tournament known as March Madness, fans often fill out a “bracket” to try to predict the outcomes of each of the 63 games. What are the odds of getting the entire bracket right? Around 1 in 120 billion, according to CBS News. But 5 years ago, Gregg Nigl of Columbus, Ohio made history by predicting the outcomes of 49 March Madness games in a row. An ESPN article by Ryan Hockensmith tells Nigl’s story.

Classroom Activities: probability, statistics

  • (Mid level) There are 63 games in the main March Madness bracket. Assuming every team has an equal chance at winning each game, calculate the following:
    • The number of possible brackets.
    • The probability of guessing a bracket perfectly.
    • Nigl’s odds of finishing the bracket perfectly, once he had correctly predicted the first 48 games.
    • The probability of correctly guessing 31 out of the 32 games in the first round.
    • According to this article, a top-of-the-line prediction model is accurate 75% of the time. Repeat your calculations from above with 75% as the chance of guessing a game’s winner correctly.
  • (All levels) Simulate a March Madness competition in class by breaking into 8 teams and running a tournament. The game you choose for the tournament should have outcomes that are somewhat, but not totally, random.
    • Ahead of the tournament, have students fill out brackets. Did anyone guess the entire tournament correctly?
    • By studying the class predictions, estimate the chances of correctly predicting the winner of each game.

—Leila Sloman

How A Guinness Brewer Helped Pioneer Modern Statistics

Forbes, March 13, 2024.

More than a century ago, a brewer from the Guinness company named William Sealy Gosset invented a far-reaching statistical method known as the $t$-distribution. Gosset was helping a fellow brewer who suspected hops with higher levels of a chemical called resin would improve the beer. Gosset’s colleague sought to prove his hunch with the scientific method: collect samples, measure resin concentration, and record his observations. But where was the line between “more resin in these samples” and “significantly more resin?” Gosset came up with the t-distribution to delineate between these two scenarios. “There is no doubt as to the significance of the method. The ‘$t$-distribution’ is featured in statistics textbooks and used in all fields utilizing stats from medicine to agriculture and much more,” writes Erik Ofgang for Forbes. In this article, Ofgang tells the story of Gosset’s work and influence.

Classroom Activities: T-distribution; scientific analysis

  • (Mid level) Read this guide on the Student’s $t$-distribution from Wolfram MathWorld. Then read Sections 3, 3.1, and 3.2 of Penn State’s “Statistics Online” textbook. Now suppose that the brewers collected the following data:
Batch Resin level Beer Quality
1 3 20
2 3 13
3 10 20
4 12 20
5 15 29
6 16 31
7 17 25
8 19 21
9 23 27
10 24 32
11 32 16
  • Calculate the $t$-score by hand.
  • Calculate the degrees of freedom and determine a $p$-value for the hypothesis that more resin improves the beer.
  • (High level) Read this WIRED article about a potential new treatment for Lyme disease and follow the link to the press release.
    • Describe what the scientists were testing specifically in 2 or 3 sentences, using your own words.
    • By looking through the technical report, describe how the scientists used the $t$-distribution test, also known as the “$t$ test” or “$p$-value test.”
    • What $p$-values does the scientific team report in their study? Does this seem relatively more conservative or less conservative as a threshold for significance?
  • (High level) Complete this calculator-based t-distribution activity and worksheet from TI Instruments.

—Max Levy

What is the three-body problem? The chaotic, cosmic mathematics behind the Netflix TV show

BBC, March 30, 2024.

Last month, Netflix premiered a new show featuring an old and famous scientific problem: The three-body problem. In the show, a group of aliens lives on a planet which orbits three suns at once. As physicists and mathematicians know, the movement of three objects acting on one another via gravity is nearly impossible to predict. Thus, the future of the planet is uncertain, and its inhabitants want to take over Earth instead. “The three-body problem is, then, the root cause of all the drama that plays out throughout the rest of the series,” writes Kit Yates for BBC.

Classroom Activities: physics, algebra

  • (All levels) Read the article. Devise your own premise for a book, movie, or TV show centered around a mathematical or scientific problem you have studied in class. Describe how the details of the problem and its solution lead to a high-stakes plot. For more activities on math-related stories, check out our September digest on graphic novels.
  • (High level) A two-body system follows paths described by conic sections. The equation $x^2 + y^2 = z^2$ describes a cone.
    • A conic section is the curve you get when you intersect a cone with a plane of the form $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Find and draw the following conic sections:
      • $x^2 + y^2 = z^2$ intersected with the plane $y = ½$
      • $x^2 + y^2 = z^2$ intersected with the plane $z = 1$
      • $x^2 + y^2 = z^2$ intersected with the plane $x – z = 1$
    • On your own, come up with a few more planes to intersect the cone with, and draw the results. Write down any observations you have.
    • A conic section is one of the following types of curves: An ellipse, a hyperbola, or a parabola. Derive the equations of an ellipse, hyperbola, and parabola in polar coordinates.
    • For more on conic sections, read this section from Lumen Learning and try the Section Review Exercises.

—Leila Sloman

A Mathematician On Creativity, Art, Logic and Language

Quanta Magazine, March 13, 2024.

To Claire Voisin, mathematics is something you feel, imagine, and meditate on while walking through Paris. Voisin is an award-winning French mathematician. Voisin appreciated math from a young age, thanks to its elegant proofs and definitions, but lost interest as a teen when math instruction felt more like a game. She turned to the creativity and structure of poetry, philosophy, and painting.  Eventually, however, Voisin found the same depth in math that she loved in her creative pursuits and began a four-decade-long career. In this article, Voisin spoke with Quanta Magazine’s Jordana Cepelewicz about her work and inspiration.

Classroom Activities: creativity in math

  • (Mid level) Based on the reading, describe what “deeper” things that Voisin eventually found in math. What do you think it has in common with the depth she found in philosophy, painting, and painting?
  • (All levels) Use this mathematical art resource to draw the “three bugs problem.”
    • How does the problem compare and contrast when you begin with a square versus a hexagon?
    • Now, draw a shape with a concave region and repeat the exercise. Discuss how this compares to the square and hexagon versions.
    • Next, follow the directions for “curves of pursuit” and sketch one that begins with a pentagon.
  • (High level) Voisin works in Hodge theory, a subfield of topology. Complete the Day 1 and Day 2 exercises from this topology introduction for high school students, which cover differences between topology and geometry. (Note for teachers: An introduction and more difficult lessons are available within the document. #6 of the Day 1 exit slip is incorrect.)

—Max Levy

Some more of this month’s math headlines