Math Digests September 2023

Here’s why mathematicians are so interested in cake cutting

Science News, September 8, 2023.

What’s the best way to cut a cake so that everyone is sure they got a fair share? In an article for Science News, Stephen Ornes takes a deep dive into the theory of cutting cake and the many ways that mathematicians think about the topic. The problem is simplest when just two friends share a cake. In that case, they can use a strategy called “I-cut-you-choose.” The first friend splits the cake into two pieces that she feels are equal, so that she would be happy with either piece; the second friend then chooses whichever piece he prefers. This strategy works great for two people, but for three or more friends, the problem is more complex. Ornes describes the development of the cake problem over time, from a 1948 paper to modern-day research.

Beyond dessert, the insights about cake cutting can be applied to real-world problems about dividing things fairly. Researchers have used the cake model to find more fair ways to design voting systems, admit students to schools, assign chores, and split rent.

Classroom Activities: ratios, fractions, optimization

  • (All levels) Draw a picture of a cake whose left half is chocolate frosted and whose right half is vanilla frosted. Try the “I-cut-you-choose” strategy with a partner. Take turns deciding how to cut the cake in each of the following scenarios. (There may be multiple right answers—just make sure you would be equally happy with each of the two pieces you cut.)
    • How would you cut the cake if you both have no preference between the chocolate frosting and the vanilla frosting? Give a few different options for how you could cut the cake.
    • How would you cut the cake if you both love chocolate, and you think that each bite with chocolate frosting is worth twice as much as a bite with vanilla frosting?
    • (Advanced) As in the last scenario, imagine that you both love chocolate, and you think that each bite with chocolate frosting is worth twice as much as a bite with vanilla frosting. How can you cut the cake so that one of the pieces is only chocolate, but you are still equally happy with either piece?
  • (All levels) The article discusses a strategy for sharing cake among three people called the “last-diminisher” method.
    • Read the description in the slideshow within the article titled “How the ‘last-diminisher’ method works.”
    • How could Carla believe that she has a piece that’s worth 1/3 of the value of the cake, but still be jealous of someone else’s piece? What would that mean about her perception of the other pieces?
    • Discuss with another student: what do you think are the benefits of this method? What did you find confusing? If you and your friends prefer different parts of the cake (for example, if one person loves fruit and another person does not), how might that affect the way you approach the cutting problem?

—Tamar Lichter Blanks

Were these stone balls made by ancient human relatives trying to perfect the sphere?

Science, September 5, 2023.

1.4 million years ago, a human ancestor known as Homo erectus accumulated 600 plum-sized stone balls in one corner of northern Israel. Researchers discovered these oddly symmetrical rocks beside hand axes at an archaeological site in the 1960s, and similar artifacts have appeared elsewhere around the world. Scientists have proposed that these could have occurred naturally, or that they could be debris left behind while making other tools. But now, one research team claims that these “spheroid” orbs were crafted intentionally—and the evidence is in their math. In this Science article, Phie Jacobs explains how archaeologists used 3D analysis software to measure angles of the stone balls, calculate curvatures, and find the center of mass. Jacobs writes that the practical purpose remains “enigmatic,” while suggesting another possible purpose: “the sheer joy of creating symmetry.”

Classroom Activities: archaeology, center of mass

  • (All levels) Do this Science Buddies activity about finding an object’s center of mass.
  • (High level) This research comes from a lab that studies “computational archaeology.” Based on the reading, describe how this field might differ from what people normally imagine archaeology entails.
    • Explain how the stone balls’ angles, center of mass, roughness, and curvature can suggest that they were intentionally crafted.
    • Watch this TED-ED video (What can you learn from ancient skeletons?), and list all the ways that mathematical analyses appear.

—Max Levy

Mathematician Solves 50-Year-Old Möbius Strip Puzzle
Scientific American, September 12, 2023.

The Möbius strip — the one-sided loop — has been around since the nineteenth century, and mathematicians love to trot out its counterintuitive features. But they still haven’t fully plumbed its mathematical depths. “Until recently, researchers were stumped by one seemingly easy question about Möbius bands: What is the shortest strip of paper needed to make one?” writes Rachel Crowell in this article for Scientific American. In August, mathematician Richard Schwartz of Brown University finally solved this question, announcing that the strip needs to be $\sqrt{3}$ times as long as it is wide.

Classroom Activities: geometry

  • (All levels) The writers of this article from The Conversation point out two counterintuitive properties of Möbius strips, both of which can be explored in class using construction paper, tape, pencil, and scissors. Create your own Möbius strips with these materials.
    • The authors write: “If you take a pencil and draw a line along the center of the strip, you’ll see that the line apparently runs along both sides of the loop.” Confirm this for yourself.
    • Now try to guess what will happen if you cut the Möbius strip along the center line. Then cut to find out if you were right. Are you surprised by the result? What experiments do you want to try next?
    • (High level) For more about Möbius strips, read the full article. For another example of a one-sided surface, read this article about the Klein bottle from Plus.
  • (All levels) In this video, mathematician Eugenia Cheng cuts a bagel along a Möbius strip. Before watching, try to figure out what will happen. Then, either watch the video or try it in class. What comes to mind after you see the result?
    • In the video, Cheng says that the bagel “can’t separate into two, because the cutting surface only had one side.” Does this make sense to you? Why or why not?
  • (All levels) Crowell is reporting on a new result about the aspect ratio of the Möbius strip — the ratio of its length to its width. Have students make Möbius strips in class using tape, scissors, and a flexible material like fabric. Compete to see who can make a strip with the shortest ratio of length to width. How close to $\sqrt{3}$ can students get?

—Leila Sloman

3 reasons we use graphic novels to teach math and physics

The Conversation, August 17, 2023.

Traditional math and physics textbooks can be boring and bombard you with information all at once. Mathematicians Sarah Klanderman and Josha Ho believe this information overload actually hinders learning. So, they propose an alternative: graphic novels that teach student’s concepts before calculations. “This approach helps build their intuition before diving into the algebra. They get a feeling for the fundamentals before they have to worry about equations,” they write. In this article for The Conversation, Klanderman and Ho describe the benefits of graphic novels as educational tools that make math accessible for students, parents, and people studying to become teachers. “It boosts their confidence and shows them how math can be fun—a lesson they can then impart to the next generation of students.”

Classroom Activities: storytelling with math

  • (All levels) Draw a short comic (stick figures are fine!) that tells a story while illustrating a concept you recently learned in math or science class.
  • (All levels) Suppose that you had more time to create a bigger comic or graphic novel. What scientific concept would you choose to explain?

—Max Levy

The story of zero: How ‘nothing’ changed the world
CBC Ideas, September 26, 2023.

In the beginning, there was no such thing as zero. Zero wouldn’t come about until the ancient Sumerians, says this episode of Ideas. Once humanity invented the concept of the number zero, massive progress in technology, medicine, engineering, and mathematics itself became possible.

Classroom Activities: calculus, asymptotes

  • (Mid level) The episode describes the mathematical issues that arise when you try to divide by zero. Without doing any calculations, what do you think the answer to $ 1 \div 0$ should be? What about $3 \div 0$? What about $ -1 \div 0$? Justify your answers.
    • Calculate $ 1 \div 0.5$, $1 \div 0.2$, and $1 \div 0.1$, and plot your results on a graph. What are the results hinting at? Do they match your prediction for the value of $1 \div 0$?
    • Repeat the exercise, this time calculating $1 \div (-0.5)$, $1 \div (-0.2)$, and $1 \div (-0.1)$. Do the results change your response?
    • Write down what you’ve learned from your calculations, and whether you’ve changed how you think about dividing by zero. What else do you want to know?
    • In the above exercise, you started to plot the vertical asymptotes of the graph $y = 1/x$. Learn about asymptotes with this online lesson by Richard Wright of Andrews University.
  • (High level, Calculus) In the episode, one mathematician describes calculus as “the art of properly dividing by zero.” What does that mean to you? Do you agree or disagree? What makes calculus different from the division-by-zero examples above?
    • In calculus, the derivative, or the instantaneous slope of a function $f(x)$ at the point $a$, is defined as
      $$\lim_{h \to 0} \frac{f(a + h) – f(a)}{f(h)}$$
      By plugging in $h = 0.1$, $0.01$, and $0.001$ to the following functions, estimate their derivatives at $a = 0$:

      • $f(x) = x$
      • $g(x) = x^3$
      • $h(x) = \sin(x)$
    • How well do your results match the derivative formulas $f’(x) = 1$, $g’(x) = 3x^2$, and $h’(x) = \cos (x)$? Can you prove that those formulas are accurate?
    • Can you think of a function whose derivative at zero is undefined? Why does this happen?

—Leila Sloman

Some more of this month’s math headlines