Math Digests June 2025

The Game Theory Hidden in the Mind of Sherlock Holmes

Scientific American, June 24, 2025

In mathematics, a “game” can be deadly serious. The field of game theory is dedicated to finding and analyzing winning strategies for all kinds of situations, including trading on the stock market—or escaping from your enemies. In this article, Manon Bischoff recounts one of the examples used in the early days of game theory, in which fictional detective Sherlock Holmes must evade an attack from Professor James Moriarty.

Classroom Activities: probability, optimization

  • (All levels) Read the first section of the article, up to the section header “A Battle of Wits.” Write down the four possible scenarios Holmes faces. Explain what you would do in his situation, and why.
  • (Mid-level) Continue the article, reading through the first paragraph of the section “Playing with Probabilities.” In this activity, we’ll recreate the solution discussed in the last part of the article.
    • Bischoff tells us: “Holmes chooses Dover with a probability of p and that Moriarty does so with a probability of q.” Write a formula for Holmes’ expected payoff, in terms of p and q.
    • Whichever value of p Holmes chooses, there are a variety of expected payoffs, depending on which q Moriarty has chosen.
      • For p = 0, 0.4, 0.8, and 1, find a formula for Holmes’ expected payoff in terms of q.
      • For each p, calculate the value of q that results in the worst outcome for Holmes. Calculate the corresponding expected payoff.
      • Now, consider more values of p, and plot Holmes’ minimum expected payoff as a function of p.
    • Based on the graph, which value of p should Holmes choose? Why? Are there other ways to choose p, other than the strategy outlined here?
    • If we change Holmes’ payoff table to the one below, what is the optimal value of p?

If both Holmes and Moriarty disembark at the same place, Holmes' payoff is -70. If Holmes gets off at Canterbury and Moriarty gets off at Dover, Holmes' payoff is 10. If Holmes gets off at Dover and Moriarty at Canterbury, Holmes' payoff is 30.

  • (High level) Read the rest of the article. Is it compatible with your answer? Explain the correspondence between Bischoff’s reasoning and yours.

—Leila Sloman 

Two Equations that Unlock El Niño

EOS, June 5, 2025

The Earth’s climate patterns are mathematically intertwined. This EOS article discusses a natural climate event called the El Niño Southern Oscillation, or ENSO. ENSO typically has two phases: El Niño, when the eastern Pacific Ocean gets warmer than normal, and La Niña, when it gets cooler. “Although ENSO originates in the tropical Pacific, its influence extends globally through atmospheric ‘teleconnections,’” wrote Jérôme Vialard. Vialard spoke with a scientist who recently presented a simplified mathematical representation of ENSO and its global connections.

An equation for dT/dt which depends on parameters T, R, F_1, h, sigma_T, xi_T, B, H(T), b, and c. An equation for dh/dt which depends on parameters epsilon, h, F_2, sigma_h, and xi_h.
Image used under NOAA Image & Licensing Guidelines.

Classroom Activities: mathematical modeling, calculus, nonlinear equations

  • (High level) Read the article. Explain in your own words what the recharge oscillator equations are meant to do.
  • (High level) The equation discussed in this new academic study is advanced. However, by unpacking what each variable represents, we can get a qualitative sense of each equation. For example, we know that $H$ represents “temperature” and $t$ represents time. The derivative ${dT \over dt}$ therefore represents how much temperature changes per unit time.
    • List all the variables and their meanings based on the text and Table 1 in the study.
    • The variable $h$ represents the amount of heat in the equatorial Pacific Ocean. What does the first term of the second equation ${dh \over dt}$ represent?
    • How does the negative sign in front of the term $\epsilon h$ affect the total heat?
    • $F_1$ corresponds to effects from the thermocline, the fact that ocean temperatures decrease rapidly with depth. When these effects are large, does the equatorial heat content affect ${dT \over dt}$ more or less?
    • Multiple terms contain the temperature variable, $T$. Some are raised to higher exponents than others. These terms can introduce more complex behaviors, such as regular oscillations up and down with time or “chaotic” effects. Using Google, research how else quadratic ($x^2$) and cubic ($x^3$) terms influence nonlinear systems. Discuss what you learn.
  • (High level) Examine the linear versions of the recharge oscillator equations. (Equations (1) and (2) in the study.) Suppose $R$, $F_1$, and $\epsilon$ are 1, while $F_2 = 0$.
    • Verify that the functions $T(t) = e^t T(0) + \frac{1}{2}(e^t – e^{-t})h(0)$, $h(t) = e^{-t}h(0)$ solve the equations.
    • Suppose $R$ and $\epsilon$ are 1, while $F_1$ and $F_2$ are both zero. Can you find a solution for $T(t)$ and $h(t)$?

—Max Levy

Lincoln City football kit features coded message

BBC News, June 9, 2025

The acclaimed logician George Boole is being honored by the soccer team in Lincoln, England, where he was born. Boole is known for Boolean logic, a system that converts logical statements into 0s and 1s. This makes it possible to apply the tools of algebra, like addition and subtraction, to logic. “The former teacher, who founded two schools, is commemorated throughout his home city,” writes Shirley Henry for BBC News. With new uniforms that display their slogan in binary code, the Lincoln City team is adding one more commemoration to that list.

Classroom Activities: logic, binary, algebra

  • (Mid-level) In these activities, you’ll learn about Boolean ideas from videos made by professor Hany Farid. Start by watching this video up to timestamp 06:35. Answer the following questions.
    • At timestamp 00:42, Farid shows a truth table for the logical expression “a AND b”. Describe in your own words what each row of the truth table means.
    • Farid shows truth tables that involve two variables, a and b. Sometimes, we want to consider logical expressions involving more than two variables. Construct truth tables for the following logical expressions: a OR b OR c, a AND b AND c
    • Discuss as a class the meaning of the expression (a AND b) OR c. Once you’ve settled on a meaning, construct a truth table for it.
    • Can you find an algebraic expression for the NOT and AND operations?
  • (High level) In Boolean logic, a statement can have two values: Either TRUE or FALSE. Because of this, there is a correspondence between Boolean logic and binary numbers—numbers whose digits have two possible values, 0 or 1. Watch another of Farid’s videos, this time about binary numbers. Try the following exercises.
    • Convert the number 353 in base 6 to base 10.
    • Convert the number 81 in base 10 to base 7.
    • Convert your birth year to base 1000.

—Leila Sloman

In Taxicab Geometry, Pi Equals 4 and Circles Aren’t Round

New York Times, June 9, 2025

Roads and hallways often restrict your direction of travel. For example, you might drive to school along straight roads that go either north-south or east-west, rather than driving diagonally through homes and fields. If you measure distance based on how far you drive along the roads, you get what mathematicians call “taxicab geometry.” It can make some math problems easier: In taxicab geometry, a walk of 5 blocks north and 2 blocks east is always the same length, no matter which streets you turn on. “You have to admit: Taxicab geometry has its advantages! But it also leads to surprises,” wrote Steven Strogatz in this New York Times article about the weird math of taxicab geometry.

Classroom Activities: geometry

  • (Mid-level) Calculate the following distances, assuming that city blocks measure ½ mile on each side:
    • How many miles would you walk if you follow the street grid 1 mile north and 2 miles east?
    • How many miles would you walk if you follow the street grid 20 blocks south and 4 blocks east?
    • For each question above, how many miles would a bird travel if it flew between the two points?
  • (Mid-level) Repeat the checkerboard exercise from the article using four red dots each 4 units away from the central black dot.
    • Explain how you find where to place the red dots between the four corners.
    • Explain in your own words why the “circle” looks like a diamond in taxicab geometry.
    • Calculate the taxicab value of $\pi$ in this arrangement.
  • (High-level) Aside from robots and computer chips, what other uses can you imagine for taxicab geometry in the real world?

—Max Levy

Can virtual reality solve Wisconsin’s school math problems?

Wisconsin Public Radio, June 11, 2025

It can be difficult to learn math just from textbooks and homework. Some educators hope that immersive games that include virtual reality can help. “Instead of reading and memorizing, students will solve problems using rich contextual experiences and kinesthetic sense-making tools,” according to a Wisconsin government proposal to provide at least 3,200 VR headsets to schools in urban, suburban, and rural districts. This WPR article describes the proposed legislation and a recent study conducted by a VR company.

Classroom Activities: scientific method, math education

  • (Mid-level) Look at the examples of VR in math found on Prisms VR’s website.
    • In what ways do you think virtual reality would (or would not) make learning these concepts easier?
    • Describe two different math concepts that may be easier to learn through VR.
  • (High-level) Read this summary of an education study conducted by the company Prisms VR.
    • What was the company’s hypothesis going into the study?
    • How did the company test their hypothesis?
    • What scientific or mathematical evidence does the company use to support their hypothesis?
    • What questions does this study not answer?
    • Read this article about the potential biases in industry-sponsored research. In what ways (if any) might this study and proposal be biased?

—Max Levy

More of this month’s math headlines