Math Digests July 2025

This month’s digests:

New Sphere-Packing Record Stems From an Unexpected Source

Quanta Magazine, July 7, 2025

To solve a stubborn old problem, it often helps to consult an outsider. That’s true even of problems that only a few experts have dared to try, such as the mathematical question about how densely spheres can be packed. Sphere packing is like trying to load as many ping-pong balls as possible into a shoebox, only in higher dimensions we can’t easily visualize. The classic techniques that solve this problem involve patterns of points in space, called lattices. But the newest breakthrough in sphere packing comes from a mathematician with a different specialty. This Quanta Magazine article tells the story of that mathematician’s discovery.

Classroom Activities: geometry, lattices

  • (Mid-level) Consider “sphere” packing in two dimensions. How many circles of diameter $1$ can fill a $10 \times 10$ square? Sketch your answer. (Hint: Look at the first figure in the text of the article.)
    • What percent of the space is filled by circles?
    • Is the “filled” percentage equivalent to packing one circle into a $1 \times 1$ square? Explain.
  • (High level) Look at the example in the text, “How to pack spheres using ellipsoids.” Answer the following questions.
    • As precisely as you can, describe a lattice that matches the one in the first panel.
    • Draw your lattice, as well as the biggest ellipse that surrounds $(0,0)$ without extending past its neighbors, as in the example. Which points on the lattice does the ellipse touch?
    • The major and minor axes of the ellipse lie along two perpendicular lines, which can be written as $k_1x + k_2y = 0$ and $\ell_1x+\ell_2y=0$. Calculate $k_1,k_2,\ell_1,$ and $\ell_2$.
    • Calculate an equation for the ellipse, of the form$$\frac{(k_1x+k_2y)^2}{a^2} + \frac{(\ell_1x+\ell_2y)^2}{b^2} = 1.$$
    • What is the equation of the ellipse after it is halved, as in the second panel?
    • What is the equation for one of the neighboring copies of the ellipse, as in the third panel?
    • Repeat for the lattice of points made by intersecting lines of the form $3x \pm 2y = 6n$, where $n$ is an integer.
  • (Mid-level) Calculate how many 0.5-inch diameter marbles you can fit in a box measuring $1 \times 0.5 \times 0.4$ feet$^3$. (Hint: refer to this video for more explanation.)
    • Check your calculations by building the box yourself out of cardstock and fitting real marbles inside.

—Max Levy

Cool your car quickly with this fluid dynamics trick

TikTok, June 25, 2025

After a car sits in a hot parking lot all day, the black seats can seem like hot coals and the air like a preheated oven. In this TikTok, mathematician Hannah Fry teaches her audience a simple trick to quickly cool a hot car. “Don’t bother putting on the [air conditioning]. That would take about five minutes, it’s not worth it,” Fry said. Instead, she suggests opening a back window and flapping the opposing front door open and closed. It looks silly, she admits, but it’s based on mathematics.

Classroom Activities: algebra, heat transfer

  • (Mid-level) Suppose every time you shut the front door, 10% of the inside air is replaced by outside air at 80°F, so that the temperature within the car changes according to the formula$$\text{new inside temperature} = 0.9 \times \text{(old inside temperature)} + 0.1 \times \text{(outside temperature)}.$$If the air inside your car measures 100°F, the air outside is 80°F, and you want to cool the car to 85°F with Fry’s trick, how many times should you open and close the door?
  • (High level) Now consider the effects of heat radiating from the car’s interior and sunlight. If this “radiative heat transfer” adds 1°F to indoor air every 10 seconds and each open-close cycle takes 5 seconds, what will be the interior temperature after 12 open-close cycles?
    • Using spreadsheet software, calculate how many times you must open and close the door to reach 85°F inside.
    • Plot the temperature every five seconds.

—Max Levy

Math Puzzle: Discern the Door

Scientific American, July 5, 2025

In this puzzle, Emma R. Hasson challenges you to find a hidden prize as quickly as possible. The principles behind the puzzle solution connect to real-world applications in computer science, which you can explore in the activities below.

Classroom Activities: computer science, logarithms

  • (All levels) Reenact the “Discern the Door” puzzle in class. Choose a student volunteer to act as the “guard” and hide the prize. The rest of the class should brainstorm questions and discuss each possibility, with the teacher moderating the discussion. Students will vote on which questions they want to ask.
    • At the end of the round, discuss whether the class found the best strategy or not. If not, play again with a new guard.
  • (All levels) Play this guessing game from Khan Academy. Answer the following questions in your own words.
    • What is binary search?
    • What are the advantages of binary search?
    • What does binary search have to do with the “Discern the Door” puzzle?
    • When guessing a number from 1 to 300, Khan Academy says “you should need no more than 9 guesses.” Why?
  • (Mid-level) Brainstorm some real-world situations in which a computer scientist might want to use binary search. Describe in detail how you would apply binary search.
  • (High level) How would you apply or modify binary search to calculate the following?
    • The zeros of a linear function
    • The zeros of a quadratic function
    • The maximum of the function $-x^2 + \log(x)$

—Leila Sloman

Experiment: M&M’S geometry

Science News Explores, July 30, 2025

In this article from Science News Explores, Sara Agee and Teisha Rowland of Science Buddies present multiple ways to calculate the geometry of an M&M. You can measure it directly, or you can use a number of formulas which approximate the M&M as a sphere, cylinder, or ellipsoid. For the details on the experiment, head to the article; for some ideas to get students thinking more deeply about the concepts, check out the classroom activities below.

Classroom Activities: geometry, experiment

  • (All levels) Read the overview of the experiment. Order the four volume formulas (“Sphere – Short radius,” “Sphere – Long radius,” “Cylinder,” “Ellipsoid”) according to which you think will be most accurate. For each, note whether you expect the formula to over- or underestimate the true volume.
    • After doing the experiment, compare your results with your predictions. What did you get wrong? What did you get right?
  • (All levels) Repeat the experiment, and the predictions activity, using your personal favorite candy. (Besides M&Ms.)
  • (Mid-level) Partner up. As homework, find an item whose volume is difficult to calculate. At the next class, trade items with your partner. Use the formulas to compute the volume as accurately and quickly as you can.

—Leila Sloman

Can you solve it? The world’s most fascinating number – revealed!

The Guardian, July 7, 2025

Alex Bellos believes he knows the most interesting number in the world: 108. It’s a number of cultural significance from Buddhism to UNO decks. “Mathematically, 108 is also a superstar,” Bellos writes. It’s the number of degrees in each angle of a pentagon; it’s the product $1^1 \times 2^2 \times 3^3$; $2^{108}$ is the largest known power of 2 with no digits that are “9” (that’s no 9 among its 33 digits — an event with only a 3% probability). Bellos includes many more justifications in this article for The Guardian, writing: “I’ve only scratched the surface.”

Classroom Activities: number theory, puzzles

  • (All levels) Explain in your own words why Bellos believes 108 is the most interesting number.
    • What does Bellos seem to value in numbers?
  • (All levels) Solve each of the three puzzles at the bottom of the article. (Hint: For problem 3, what happens if the first digit is 9? If it’s 8?)
    • Write up your solution in your own words, and then compare solutions with a partner.
  • (All levels) What do you think is the most interesting number? Come up with at least three reasons to support your case. In small groups, present your case and vote for the most interesting number.

—Max Levy

More of this month’s math headlines: