Math Digests February 2022

These are the best starting words to use to play Wordle, according to a math expert

USA Today, February 8, 2022

What’s the best strategy for Wordle, the viral word-guessing game? If you’re a frequent player, maybe you start by figuring out many vowels as possible, or by ruling out some of English’s most common consonants, or maybe you have a favorite go-to word as an opening guess. Many people have used math and statistics to analyze the game. This USA Today article highlights math YouTuber Grant Sanderson, who developed a strategy based on information theory. For his computer algorithm, SALET was the starting word that led to the right answer in the smallest average number of guesses—but Sanderson emphasizes that the best strategy for a computer and the best strategy for a human are not the same thing.

Classroom activities: information theory, logarithms

  • (Pre-calculus, Calculus, Computer Science) Watch the 3Blue1Brown video on solving Wordle with information theory (and the follow-up video with a correction—both videos are long but give an excellent introductory lesson on information theory). Let’s explore some other basic examples.
    • How many bits of information are contained in one roll of a standard six-sided die? What about in two consecutive rolls of a die? Why does the answer not depend on the numbers that you roll?
    • How many bits of information are contained in the choice of a random integer between 2 and 12 (inclusive)?
    • If you roll two dice and take the sum, what is the expected information (i.e. entropy)? (Hint: count the number of ways you can roll each number between 2 and 12, then use the equation onscreen at 11:38 in the first video.) Why is this number smaller than the previous answer?
    • Discuss the Wordle strategies that you like to use and how they connect to information theory.
  • (All levels) Read more about the history and applications of information theory in this Quanta Magazine article.

—Scott Hershberger

A simple mathematical model can account for lizard’s green-and-black pattern

Ars Technica, February 3, 2022

As an ocellated lizard grows up, some of its scales change from green to black or black to green. By adulthood, it has a two-toned pattern of scales that looks like a miniature maze. This complex design resembles something unexpected: a tool from physics. Researchers recently showed that this lizard’s pattern aligns with a simple mathematical model, called the Ising model, that is also used to represent magnets, melting sea ice, and more. In an article for Ars Technica, Jennifer Ouellette describes how the Ising model works and how it describes the lizards’ scales, including the way the colors flip over time.

Classroom activities: physics, combinatorics, lattices

  • (High school) Learn about the Ising model by reading the first part of the section “Definition of Ising Model” in these course notes by Stanford graduate student Jeffrey Chang.
    • How many possible arrangements of up and down arrows are possible in the $5\times 5$ grid pictured?
    • How many possible arrangements are there for an $n \times n$ grid, where $n$ is a positive integer?
  • (High school, Geometry, Algebra II) Given any two points $p=(x_1, y_1)$ and $q=(x_2, y_2)$ in the Cartesian plane, the set of points of the form $ap+bq=(ax_1+bx_2, ay_1+by_2)$, where $a$ and $b$ are integers, is an example of a lattice, similar to the lattices used in the Ising model.
    • By drawing dots on graph paper, sketch the lattice for $p = (1, 0)$ and $q = (0, 1)$. Make another sketch for the lattice corresponding to $p = (2, -2)$ and $q = (1, 3)$. Which types of lattice are they? Which type most closely resembles the scales on the lizards?
    • What is the area of the parallelogram whose vertices are $0, p, q$, and $p+q$ when $p = (2, -2)$ and $q = (1, 3)$?
    • Give an example of points $p$ and $q$ for which all of the lattice points lie on a single line in the plane.

—Tamar Lichter Blanks

The (Drag) Queen of Mathematics

NPR Short Wave, February 10, 2022

Mathematics and drag: It’s a unlikely combination, but TikTok star Kyne has captivated viewers with her stunning makeup and outfits, flair for performance, and easy-to-follow explanations of math concepts. Kyne Santos studied mathematical finance at the University of Waterloo and now creates math drag videos full-time. For an episode of the podcast Short Wave, Emily Kwong interviewed Kyne about how she both brings math to a wide audience and represents STEM in the drag scene. “My whole message is just that math can be really interesting,” Kyne said. “Math can be beautiful. Math can be fun. And math can be extremely relevant to our world.” (Also see an article about Kyne in the January 2022 issue of Notices of the AMS).

Classroom activities: exponential growth, exponential decay, history of mathematics, diversity and inclusion

  • (Algebra II, Pre-calculus) In one of her most popular videos, Kyne explained that if you folded a piece of paper 42 times, it would be as tall as the distance from the Earth to the Moon. Let’s explore this.
    • Find a stack of paper. How many times can you fold a single sheet in half? Can you figure out a way to use a ruler to determine the thickness of a single sheet?
    • How thick would the sheet of paper get if you folded it 20 times? What about 42 times?
    • Each time you fold a sheet of paper in half, its horizontal area decreases. If you started with a sheet of paper the size of a football field (360 ft by 160 ft), what would its area be after 20 folds? What about after 42 folds?
    • No matter how you fold a sheet of paper, its volume (length times width times height) should stay the same. Do your calculations agree with this statement?
  • (All levels) Kyne mentions that Euclid is one of her math heroes. Discuss the stories of inspiring mathematicians past and present.

—Scott Hershberger

An Ancient Geometry Problem Falls to New Mathematical Techniques

Quanta Magazine, February 8, 2022

Can you turn a circle into a square? Don’t fall for the simplicity of the question—it has baffled mathematicians for almost 2,500 years. The problem, known as “squaring the circle,” challenges you to cut a circle into any (finite) number of pieces and reassemble them into a square of equal area. It quickly turns philosophical. You’ll wonder how to make a corner out of something round, and why those two things are so fundamentally different. Researchers have made some headway in the last century by imagining cutting unorthodox shapes impossible to create with real scissors. Around 1990, one researcher showed how to “take a circular space and make it straight,” a result described as “jaw-dropping.” And this year, mathematicians found an even better way to solve the problem. In this article, Nadis describes the long-running journey and fascinating result.

Classroom activities squaring the circle, transcendental numbers

  • (All levels) To learn more about circle squaring, watch this Numberphile video.
  • (All levels) Try to square the circle yourself. Use a compass to draw a circle. Use scissors to cut it out. Now, using only that circle—and every part of that circle—try to cut and reassemble pieces into a square. How close can you get?
  • (Pre-calculus) The ancient problem originally asked if you could construct a square with the same area as a given circle using just a compass and straightedge (rather than cutting the circle into pieces). This is impossible because $\pi$ is what’s known as a transcendental number. Watch this Numberphile video for an introduction to transcendental numbers.

—Max Levy

NFL Kickers Are On Fire This Postseason

FiveThirtyEight, February 8, 2022

People sometimes joke that American football is poorly named. The sport is mostly a game of hands—throwing, catching, carrying, and tackling. But it is often a foot, in fact, that decides football games. Kickers will boom the ball from over 50 yards away in the most critical moments. And this year, according to data analyzed by Alex Kirshner, kickers have done so better than almost ever before. “There’s been no recent postseason in which NFL teams have asked so much of kickers in such fraught situations,” writes Kirshner. “They’ve responded almost perfectly.” In this article, Kirshner uses metrics like field goal attempts, makes, and distance to compare the past 22 years of NFL playoff data, revealing what he deems “wild success.”

Classroom activities: sports statistics, data analysis

  • (All levels) Let’s analyze the data from the table in the article titled “Kickers have been nearly automatic this postseason.” First, sort the table by season, and then copy the values from the top-left corner (2000) to the bottom-right corner (77.8). Paste them into a spreadsheet (e.g. Microsoft Excel or Google Sheets).
    • What is the average number of field goal attempts in a postseason in the years between 2000 and 2021? (See here for help with Excel formulas)
    • (Mid level) What is the average distance of field goal attempts over that entire span? (Hint: you’ll need the numbers in columns 2 and 4.) What is the range in the postseason averages over that same span? Does this value suggest that distance varies a lot or only a little from year to year?
    • (Mid level) Plot the average distance data on your spreadsheet as a scatter plot (x-axis=season; y-axis=avg. distance). Discuss whether it looks like postseason kicking is improving. Add a trendline and the $R^2$ value to the plot—how strong is the evidence?
  • (High level) Have students create and analyze their own data in groups with an online typing test. (Alternative options: Pacman or a simple mouse-based field goal kicking game). If they want to participate, each student can test their typing speed and share their words-per-minute score. Collect all the data. Calculate the average and standard deviation, first by hand, and then via Excel or a graphing calculator. (Find basic statistics formulas here.)

—Max Levy

Some more of this month’s math headlines: