Math Digests October 2023

The Game Theory of the Auto Strikes
Wired, October 1, 2023.

On September 15, the United Auto Workers — a labor union that represents workers from the automotive and aerospace industries — began a strike against Ford, General Motors, and Stellantis. The union was holding out for raises, pensions, and automatic cost-of-living adjustments to their wages. The companies, meanwhile, claimed those demands were too costly. In this article for Wired, Aarian Marshall walks through what the mathematical field of game theory — which studies scenarios in which rational players have competing interests — can contribute to the analysis of the strikes.

Classroom Activities: game theory, modeling

  • (All levels) Try this online lesson plan from Population Education simulating two simple game theory scenarios.
    • (Mid level) After playing each game, discuss in small groups what the two games in the lesson plan have in common with a strike scenario, and how they differ.
  • (Mid level) Read the Wired article.
    • Brainstorm in small groups everything that should go into “the pie.”
    • (Mid level) Come up with a game using poker chips that captures the key features of the strike scenario. First, brainstorm as a class the features that the game should include. If the discussion becomes too complicated, remind students that mathematical models balance realism with simplicity, and ask for their suggestions of how to do so.
      • How many teams will the game have? What roles will the teams play, and what roles will individuals within the teams play?
      • Who is in charge of distributing the poker chips? Is it one of the teams, or someone else?
      • What constitutes cooperation or defection? What are the effects of cooperation and defection?
      • What should happen in each round? When should the game end?

        Break into small groups to work out the game’s details. Have each group outline the best strategy for each side.

  • (Mid level) On October 30, the UAW entered a tentative agreement with the last of the three companies. (The agreement still needs to be approved by union members.) Read the linked articles about the agreements and compare to the prediction made by Marc Robinson in the Wired article. What did he get right? What did he get wrong? Do you think there was any information he missed?

—Leila Sloman

The mathematical theory that connects swimming sperm, zebra stripes, and sunflower seeds

Popular Science, September 27, 2023.

One peculiar pattern arises over and over in nature, appearing in plants, animal coloration, and even sand on the beach. Unique stripes and spots emerge mysteriously. The phenomenon occurs when two chemicals or forces compete, interact, and move around one another. This process is called “reaction-diffusion” for the way chemicals react with one another and diffuse, or spread, throughout the environment. What results are “Turing patterns,” named for mathematician Alan Turing, who discovered the principle at play. In recent work, scientists found a surprising new realm where reaction-diffusion appears—the movement of sperm. “Nature replicates similar solutions,” a researcher told Popular Science reporter Laura Baisas. The team noticed the same motion in bull sperm and green algae. “We show that this mathematical ‘recipe’ is followed by two very distant species.” In this article, Baisas describes how new mathematical models helped unpack the mechanics of sperm’s flagellar movement. They learn that the waviness emerges spontaneously, and the findings could have applications from robotics to fertility.

Classroom Activities: reaction-diffusion, simulations

  • (Mid level) Watch The Mathematical Code Hidden In Nature and describe, in your own words, how you think a “mathematical code” can explain zebra stripes.
  • (Mid level) Read this tutorial on reaction-diffusion models, then play with this reaction-diffusion simulator.
    • Recreate the “mitosis” and “coral growth” simulations described in the tutorial.
    • Describe in your words what the “kill rate” and “feed rate” would correspond to in leopard spots. Would the ratio of kill:feed be higher for leopard spots or zebra stripes?

—Max Levy

Coin flips don’t truly have a 50/50 chance of being heads or tails

New Scientist, October 17, 2023.

Flipping a coin seems like the perfect example of a fair random game, with each flip having a 50/50 chance of landing on heads or tails. But according to a recent experiment, there is a bias. In a study posted online this October, researchers found that a flipped coin lands on the same side it started on about 50.8% of the time. In practice, that means that if a coin starts with its heads side up, then you are slightly more likely to win if you bet on heads. To get their data, the researchers recorded the outcomes of 350,757 coin flips. This experimental result confirms an earlier theoretical model, as Matthew Sparkes describes in this article for New Scientist.

Classroom Activities: probability, coin flipping

  • (All levels) Have each student flip a coin 20 times. Each time, have them record the starting position of the coin and what side it lands on. They should end up with two sets of numbers: How many times the coin landed on heads vs. how many tails, and how many times it landed on the same side it started on vs. the opposite side.
    • Have each student calculate the percentage of their flips that landed on heads, as well as the percentage of their flips that landed on the same side.
    • Next, add up a tally for the whole class: How many heads (H) and how many tails (T)? How many flips landed on the same side (S) and how many on the opposite side (O)?
    • Have each student calculate the total percentage of flips that landed on heads, as well as the total percentage of flips that landed on the same side.
    • Discuss: How close to 50/50 were each of the results? What do you think would happen if everyone flipped a coin 1000 times instead of 10 times?
  • (All levels) Without actually flipping a coin, try to write down a random sequence of 20 imaginary coin flips (for example, HTHHT…). Then, watch this video about random sequences of coin flips from Numberphile.
    • Looking back at your sequence, can you detect any predictable patterns or trends?
    • Compare your imagined sequence to the 20 coin flips you recorded in the first activity. Are there any noticeable differences?
    • Discuss: What do you think are some challenges that make it difficult to generate randomness?

—Tamar Lichter Blanks

Don’t Worry About Global Population Collapse

Bloomberg Opinion, September 30, 2023.

Last year, the United Nations announced that the world’s population passed 8 billion people, and current projections predict a peak at 11 billion before next century. However, it’s hard to trust whether these estimates will prove correct. Innovations that help the planet cope with population growth are difficult to predict, and it’s essential to keep the margins of error in mind. It’s even harder to tell whether population reduction is good or bad. It could be good news for the future, since Earth’s resources like land, food, and water are finite. “And yet alarms are sounding,” writes F.D. Flam for Bloomberg. “While environmentalists have long warned of a planet with too many people, now some economists are warning of a future with too few.” In this article, Flam speaks with demographers and a mathematician about the debate around population growth.

Classroom Activities: error margins, growth rate

  • (All levels) Complete population growth activities from PopEd, such as “Stork and Grim Reaper”, or a free trial of this population growth game by Labster.
  • (Mid level) Watch this Khan Academy lesson on exponential growth. What other examples of exponential growth in nature or society can you think of?
  • (High level) Simulate population growth under the conditions below using spreadsheet software, then answer the questions that follow. Assume that we can describe global population as $P = P_0 e^{rt}$, where $P$ is total population (in millions); $P_0$ is initial population (in millions); $r$ = growth rate; $t$ = time in years.
    • If the initial population is 8 billion ($P_0 = 8000$), and the rate, $r$, is 0.01, in how many years will the population pass 11 billion?
    • Now suppose that the rate, $r$, decreases by a factor of 1% every year. In how many years will the population pass 11 billion? Discuss why.
    • Based on the reading, what would explain the rate decreasing with time?
    • The model we assumed here is greatly simplified. How would the math have to change if we wanted the population to peak and then fall?

—Max Levy

Joseph J. Kohn, Who Broke New Ground in Calculus, Dies at 91
The New York Times, October 24, 2023.

Scientists Say: Imaginary Number
Science News Explores, October 2, 2023.

In this obituary for the New York Times, Kenneth Chang covers the work of Princeton University mathematician Joseph J. Kohn, who passed away last month. Kohn studied functions that were defined over complex numbers — a set that includes familiar numbers like 2, $\pi$, or $\sqrt{5}$ as well as the imaginary square root of $-1$. “They’re called ‘imaginary’ because they don’t count or measure things, the way real numbers do,” Katie Grace Carpenter wrote earlier in the month in an explainer of imaginary numbers for Science News Explores. Despite their counterintuitive nature, imaginary numbers are crucial for studying electric circuits and quantum mechanics, she adds.

Classroom Activities: complex numbers, calculus

  • (Mid level) Read the Science News Explores article. Calculate the following numbers, leaving them in the form $a + bi$ where $a$ and $b$ are real numbers.
    • $(3i)^2$
    • $(1 + i)^2$
    • $i^4$
    • $(1 + i)^3$
  • (Mid level) The complex numbers include the real numbers, the imaginary numbers, and any sum of a real number plus an imaginary number. They can be plotted on a graph as pairs of coordinates. To see how, read the first two sections of this lesson from Maths is Fun.
    • On the complex plane, plot the numbers you calculated in the previous exercise.
    • Plot the numbers $1$, $i$, $-1$, and $-i$. Now multiply them all by $i$, and plot the results. What is the effect of multiplication by $i$?
    • Try the same exercise, but this time multiply the results by ${1 + i \over \sqrt{2}}$. What happens?
  • (High level) In this article for Scientific American, three researchers describe how essential complex numbers are in quantum mechanics, the physics of particles that are extremely small. Read the article and reflect: What did the researchers prove in the work they describe? What is the role of complex numbers in physics? What questions do you have?

—Leila Sloman

Some more of this month’s math headlines: