Math Digests August & September 2025

August & September digests:

Students Find Hidden Fibonacci Sequence in Classic Probability Puzzle

Scientific American, August 7, 2025

Given three sticks, you can assemble a triangle so long as the longest stick is shorter than the two others laid end-to-end. But what are the odds that this occurs with randomly chosen sticks? Two young researchers recently discovered that the answer to this “pick-up sticks” problem involves one of the most famous mathematical patterns in nature: the Fibonacci sequence.  “We’d no reason to suspect that it would be,” one of the researchers told Scientific American, “but it was impossible that it wasn’t.”

Classroom Activities: probability, geometry

  • (All levels) Which of the following combinations of stick lengths can create a triangle? If there are more than three sticks, you may choose any three of the options for your triangle.
    • 1, 1, 1
    • 1, 6, 10
    • 10, 10, 2
    • 1, 2, 3, 15
    • 10, 20, 40, 100, 120, 500
  • (Mid-level) Read the article and answer the following questions.
    • What are the 5th, 10th, and 15th numbers in the Fibonacci sequence?
    • What is the sum of the first 5, 10, and 15 Fibonacci numbers?
    • What is the product of the first 5, 10, and 15 Fibonacci numbers?
    • What is the probability that you can build a triangle from 5 sticks of random length between 0 and 1?
    • What is the probability that you cannot build a triangle from 10 sticks of random length between 0 and 1?

—Max Levy

Exploring Barcelona’s Architecture Through a Mathematical Lens

The New York Times, September 24, 2025

For The New York Times, Katrina Miller covers a mathematical walking tour of Barcelona, Spain. Developed through a collaboration between college student Nicolás Atanes Santos and the Catalan government, the tour invites visitors to examine the city’s geometry and symmetry. Around every corner is a math problem that allows people to see the real-world importance of the math they learn in school. “Mathematics, it seems, is anywhere you choose to find it,” Miller writes. These activities build on some of the tour’s math problems.

Classroom Activities: geometry, trigonometry

  • (Mid-level) Imagine that two slanted columns 80 feet apart meet at the center to form a triangle with an apex 100 feet from the ground. How long is each column? (Hint: see the example in the article.)
    • Use trigonometric functions to calculate the angle measurements of each right triangle.
    • Now imagine that only one column is slanted and the other stands upright, with height 100 feet. How long must the slanted column be if it reaches the top of the upright column? What angle measurements form as a result?
  • (Mid-level) An ancient bullring has a radius of 100 meters. Find the area and perimeter of the bullring.
    • If the audience sits in bleachers that continue from the edge of the bullring out 100 more meters in every direction, what is the area of the seating area?
    • If each person takes up 0.25 square meters and the venue is 70% full, how many people are in attendance?
  • (Mid-level) Read the example about window shutters of Casa Batlló in the story. How many different arrangements of shutters would exist if Casa Batlló had 15 windows?
    • What if the shutters can be opened and closed individually?
  • (All levels) Write a math problem about a building in your hometown.

—Max Levy

Amazon Supposedly Asked Job Applicants to Solve This ‘Hanging Cable’ Problem

Popular Mechanics, August 21, 2025

In this article, writer Mike Darling discusses the following math problem: Two ends of an 80-meter cable are held up by 50-meter poles. Given the cable’s minimum height from the ground, can you figure out how far apart the poles are? According to a YouTube video by the channel MindYourDecisions, this question was posed during a job interview at Amazon. The seven-year-old video, which presents the solution, has been making the rounds on social media.

Classroom Activities: calculus, hyperbolic cosine, physics

  • (All levels) Try to solve the problem experimentally. Cut a length of string 80 cm long, and four pipe cleaners: two that are 50 cm long, one that is 20 cm, and one that is 10 cm. By mimicking the setup described in the interview problem, estimate the answer.
  • (High level, Calculus) Suppose as you move along the $x$-direction between the two poles, the height of the cable varies according to the function $h(x)$. Find an equation involving $h(x)$ that states that the length of the cable is 80 m. (Assume that $x = 0$ at the midpoint between the two cables.)
    • With the help of an online calculator, estimate the answer to the interview question if $h(x) = x^2 + h_{min}$ and when $h(x) = |x| + h_{min}$.
  • (High level) In actuality, the function that describes the cable height has the form $h(x) = a\cosh(x/a) + b$. Solve for $d$, $a$, and $b$ if $h_{min}$ is 20 m. Watch the viral video mentioned in the article. Does your solution match?
  • (Mid-level) In class, present the demos from Section V of this write-up by Tatsu Takeuchi.

—Leila Sloman

The math on LA rent control doesn’t add up for actual tenants

LA Public Press, July 22, 2025

Despite policies that set an annual limit on rent increases, landlords in Los Angeles can raise rents by as much as 10 percent every year. For many renters—some of whom spend more than half their earnings on rent—a 10 percent bump is equivalent to paying hundreds more every month. “Landlords are pocketing most of the extra money as profit while renters still can’t afford to stay,” writes Elizabeth Chou for LA Public Press. This article describes the economic math of rent stabilization, and efforts to better protect tenants in Los Angeles.

Classroom Activities: rent math, data analysis

  • (Mid-level) Assuming your rent begins at $2000 per month, answer the following questions with spreadsheet software.
    • Calculate your new monthly rent payment if your landlord increases rent by 5% and 10%.
    • How much extra will you pay over one full year for each of those rent increases?
    • If your rent increases at the same rate (5% or 10%) every year, how much extra will you pay in each scenario over the course of five years?
    • If a new city policy limits rent increases to 2% annually, how much less will you spend over five years? (Compare the change to both 5% and 10% increases.)
  • (Mid-level) Your landlord budgets \$1000 per month to cover their ownership expenses—\$500 for property tax, \$200 for insurance, \$200 for regular maintenance, and \$100 for miscellaneous expenses.
    • How will the landlord’s total monthly expenses increase with the following changes: 1% increase in property tax; 4% for insurance; 2% increase in miscellaneous expenses.
    • How will the landlord’s net profit (income minus expenses) change next year if they increase your rent by 0%, 2%, 5%, and 10%?

—Max Levy

Faculty, Students, and Researchers from Around the Country Celebrate 100 Years of Black Mathematics Ph.Ds.

The Dig, September 30, 2025

This September, mathematicians gathered at Howard University to celebrate the centennial of Elbert Frank Cox’s Ph.D. Cox was the first Black person ever to earn a Ph.D. in mathematics. (As recently as 2019, only 1 percent of math Ph.D.s in the US were awarded to Black students.) The event, titled “Cox Centennial Celebration of 100 Years of Black Ph.D. Mathematicians,” included talks about Cox’s life and work. Danny Flannery reported on the event for the Howard University news hub.

Classroom Activities: linear algebra, differential equations

  • (High level) Cox’s dissertation focused on difference equations, the discrete version of a differential equation. For each of the following difference equations, identify and solve the continuous analog. Plug in the values $x=1,10,100$ to test how good the approximations are.
    • $f(x + 1) – f(x) = nx^{n-1}$ for $n=2,4$,
    • $f(x + 1) – f(x) = e^x$,
    • $f(x + 1) – f(x) = 2.$
  • (Mid-level) If a function is evaluated at the discrete points $x=1,2,3$, then you can write it as a vector$$\mathbf{f} = \begin{bmatrix} f(1) \\ f(2) \\ f(3) \end{bmatrix}.$$
    • Find a matrix $\mathbf{D}$ that satisfies
      $$\mathbf{D}\begin{bmatrix} f(1) \\ f(2) \\ f(3) \end{bmatrix} = \begin{bmatrix} f(2) – f(1) \\ f(3) – f(2) \\ – f(3) \end{bmatrix}.$$
      Is this matrix invertible? What are its eigenvalues?
    • Cox studied a more general class of difference equations that have the form
      $$a f(x+1) – bf(x)= (a – b)x^v,$$where $a$ and $b$ are any nonzero numbers and $a \neq b$. Find a matrix equation that captures this difference equation. Is your matrix invertible?

—Leila Sloman

At 17, Hannah Cairo Solved a Major Math Mystery

Quanta Magazine, August 1, 2025

Teenager Hannah Cairo recently shocked the world of math research by disproving a 40-year-old mystery. Cairo was homeschooled for most of her life and joined a graduate class at the University of California, Berkeley—one of the world’s leading math departments—before finishing high school. In that class, Cairo encountered a homework problem about the problem she solved. It mesmerized her. Soon after, she defied the odds and found an answer.

Classroom Activities: trigonometry, Fourier series

  • (High level) Cairo disproved a conjecture in harmonic analysis, a branch of mathematics that represents any function as a sum of sines and cosines. After reading the article, watch this video introducing Fourier series and explain in your own words what a Fourier series is.
  • (High level) According to the article, Joseph Fourier introduced Fourier series to help solve differential equations. One of the simplest differential equations is$$\frac{d^2f}{dx^2} + f = 0.$$
    • Show that $f(x) = \cos(x)$ and $f(x) = -\sin(x)$ satisfy the above equation.
    • What differential equation does the function $f(x) = \cos(2x)$ solve?
  • (High level) We can represent a square wave (as shown in the video) in terms of sine functions:$$\text{Square Wave }\approx \sin(2\pi t) + \frac{1}{3}\sin(6\pi t) + \frac{1}{5}\sin(10\pi t) + \frac{1}{7} \sin(14\pi t) + \dots$$
    • Write out the next three terms in the series.
    • Why are the coefficients preceding the sine functions reciprocals of odd numbers? (Hint from Wolfram Mathworld.)
    • Explain the qualitative difference between representing a square wave with just the first term, $\sin(2 \pi t)$, versus the first five terms shown above. (Hint from Wolfram Mathworld.)
    • Read this introduction about Fourier transformation. Explain why analyzing waves might be useful in the real world.

—Max Levy

The Math of Climate Change Tipping Points

Quanta Magazine, September 15, 2025

In some mathematical models of the climate, small changes in temperature or ice cover can produce huge effects. One of these so-called “tipping points” predicts catastrophe if global warming continues unimpeded: “It could freeze fertile landscapes in the British Isles and Scandinavia, disrupt monsoons throughout the tropics and dry out rainforests,” writes Gregory Barber for Quanta Magazine.

Classroom Activities: stability, calculus, algebra

  • (Mid-level) Watch this example of bifurcation theory and read the caption. Based on the article, think about why this is an example of a tipping point.
    • What causes the tipping point, and what changes at the tipping point?
  • (High level) The tipping point described in the article shows up in models of ocean currents. Assume that we can represent the strength of an ocean current with the equation$$C(t) = \frac{100}{1 + e^{0.3(t-20)}},$$where $t$ represents time in years, and $C$ represents strength in Sverdrup (Sv), a unit equivalent to one million cubic meters per second.
    • Graph $C(t)$ for $t$ between –10 and 30. Include the units on your axes. (Bonus: What units do the constants 100 and 0.3 have? Why?)
    • Interpret the graph in your own words. How does it change, when, and what does this mean about the ocean current?
    • Now assume that ocean current strength changes with air temperature $T$ according to the formula$$C(T) = \frac{100}{1 + e^{0.3(T-4)}}.$$Graph $C(T)$ for $T$ between 0$^{\circ}$ and 7$^{\circ}$ Celsius. (Hint: make sure to examine small increments of $T$.)
    • What does the graph show?
    • Now assume that $T$ changes over time according to the formula $T(t) = 4.1t$. Graph $C(t)$ and explain your work.

—Max Levy

Math Masters

Off 90, August 4, 2025

In this segment, the public television station in Austin, Minnesota profiled Math Masters, an extracurricular program in which fourth-, fifth-, and sixth-graders compete in teams and individually. Math Masters was begun 36 years ago in Austin, and now hosts dozens of events all around Minnesota—even a few in Wisconsin and Colorado. “It’s exciting to see kids that don’t think they are good at math do well,” teacher and Math Masters coach Laurie Herman told Off 90.

Classroom Activities: arithmetic, teamwork

  • (All levels) The first round of a Math Masters competition is the “facts round,” in which students must answer 75 questions in five minutes.
    • For a similar style of practice, try to solve fifty of these “Minute Math” problems in one minute.
    • Try to answer the following five questions in thirty seconds or less:
      • $(10 \times 8) \times 8 \times 8 – 17 =$ ?
      • $(14 \div 2) \times (7 – 4) \times (10 – 0) =$ ?
      • $50 – (31 – (19 – (50 – 49))) =$ ?
      • $78 \div (3 \times 13) =$ ?
      • $(25 – (4 + (19 – 3))) \div 5 =$ ?
    • In solving the preceding questions, did you use any shortcuts or tricks? If so, which ones? Can you think of any ways to optimize your strategy?
    • Do you think “Minute Math”-style practice is helpful? Why or why not? Write down your thoughts, and then discuss as a class.
  • (All levels) In the next round of Math Masters, students must solve difficult word problems. In pairs, come up five multi-part arithmetic problems of your own—like the ones above—and then write them as word problems. (Example: If your arithmetic problem is $8 \div 4$, your word problem might ask about how to divide eight cupcakes amongst four friends.) Trade problem sets with another pair, and answer their word problems as fast as you can.
    • Reflect: Which was easier—the facts round or the word problems? Which did you enjoy more? What else did you notice about your experience?
  • (All levels) For an even better sense of what Math Masters competitions are like, try this Math Masters prep packet.

—Leila Sloman

People Around the World Want Political Change, but Many Doubt It Can Happen

Pew Research Center, September 15, 2025

In 20 of 25 countries surveyed by the Pew Research Center, a majority of people want major changes to their political systems. In the United States and France, the youngest people surveyed said that few elected leaders possess any of the following traits: honesty, ethics, being qualified, focus on important problems, and understanding the needs of ordinary people. This report from Pew explains the survey data and apparent trends from around the world.

Classroom Activities: data analysis

  • (Mid-level) Use the data provided in the first figure of the article to answer the following questions.
    • Calculate the mean, median, and mode proportion of respondents who say they are “not confident” their country’s political system can change.
    • Which countries had an above average proportion of respondents feel this way?
    • Which countries (if any) had a “not confident” proportion greater than one standard deviation from the mean? Show your work.
    • In one sentence, summarize your conclusions from this analysis.
  • (Mid-level) Use the data provided in the third figure on page 3 (“Younger adults more likely than older to say that few or none of their elected officials are honest”) to answer the following questions.
    • List three factual observations about the data.
    • Write down three conclusions you can draw based on the measure “youngest-oldest diff.”
    • Share your observations and conclusions with a classmate.

—Max Levy

On 9/16/25, celebrate a date of mathematical beauty

NPR, September 16, 2025

This September, we experienced a date with wondrous mathematical properties. September 16, 2025 is commonly written as 9/16/25. 9, 16, and 25 are (respectively) 3, 4, and 5 squared. What’s more, 9 + 16 = 25, so that 3, 4, and 5 satisfy the Pythagorean theorem, one of the most important theorems in geometry. Ari Daniel reports for NPR’s All Things Considered.

Classroom Activities: algebra, geometry, mathematical dates

  • (All levels) Can you spot any patterns in today’s date?
  • (Mid-level) In your own words, explain why 9/16/25 is a once-in-a-century date.
  • (Mid-level) Daniel points out that the full year, 2025, is also a square: 452. Without consulting any outside resources, find a Pythagorean triple where the hypotenuse length is 45. (Hint: $45 = 3^2 \times 5$.)
    • Can you find any other Pythagorean triples involving 45? How many?
    • Complete the following Pythagorean triples: (6, 8, __), (8, __, 17), (__, 40, 41)
    • (High level) The Pythagorean quadruples generalize the idea of a Pythagorean triple. Can you figure out what 3D shape’s side lengths appear in a Pythagorean quadruple?

—Leila Sloman

More math headlines from August & September