Math Digests December 2025

December digests:

London Zoo: Dogs are natural mathematicians

Arizona Daily Sun, December 2, 2025.

My father once pointed out, watching our dog race through a grassy basin, that her sidelong route was surprisingly efficient. Recently, mathematician Tim Pennings confirmed this observation during a game of fetch on Lake Michigan. After 35 trials, Pennings discovered that his dog, Elvis, pursued routes that were “extremely close to the calculated optimal routes,” writes Karen London in this article. This, despite Elvis having to pass through both land and water. “Finding the solution to such problems requires entering the realm of calculus,” London writes. Unless you’re Elvis.

Classroom Activities: calculus, geometry, optimization

  • (Mid-level) Suppose Jane is rescuing Tarzan from quicksand. She can run 2.5 meters per second on solid ground, but she can only go 1 meter per second through quicksand.
    • Find a formula for Jane’s total travel time in terms of the distance she travels on ground $d_g$ and the distance she travels through quicksand $d_q$.
    • Assume Jane and Tarzan’s positions are recorded in the $xy$-plane, in units of meters. Jane’s starting point is $(0,0)$, Tarzan is at $(10,10)$, and the ground becomes quicksand when $x + \frac{1}{2} y > 10$. What is Jane’s travel time if she (1) travels in a straight line, (2) crosses into quicksand at the point $(5,10)$, (3) crosses into quicksand at the point $(7.5,5)$?
    • If Jane crosses at the point $(x,y)$, find a formula for her travel time in terms of $x$.
  • (High level) Using a calculator, find Jane’s fastest route.
    • Find an equation that tells you Jane’s fastest route if Tarzan is at an arbitrary point within the quicksand area.

—Leila Sloman

Law of ‘maximal randomness’ explains how broken objects shatter in the most annoying way possible

Live Science, December 2, 2025

Suppose you drop your parents’ favorite vase on the floor. The vase shatters into fragments, small and large. The mix of large fragments and small fragments seems random—and it is. “But that randomness has to obey certain limits,” writes Skyler Ware, for Live Science. Ware covers a new mathematical equation that describes how objects break, from dropped vases to exploding bubbles. Fragments tend to break in the “messiest” way possible, making it harder to glue them back together before your parents get home.

Classroom Activities: randomness, statistics

  • (All levels) Have each student in class draw a circle, as big or small as they like.
    • Measure each diameter with a ruler and tabulate the data on a spreadsheet. (Round to one decimal place for whatever unit you choose.)
    • Calculate the mean, median, mode, and range.
    • Graph the data in a histogram.
    • (High level) Randomly assign each student a number. Calculate the correlation coefficient for the data and infer whether the circles appear to have been drawn randomly. Explain your reasoning.
      • What if you assign numbers based on age, name, or in some other non-random way?
    • (High level) Evaluate the randomness of your circles with a gap analysis.
      • Sort diameters from smallest to largest.
      • Calculate gaps between consecutive values. Random selection tends to produce exponentially distributed gaps. Do your diameters appear to be random?

—Max Levy

How maths can help you wrap your presents better

BBC, December 13, 2025.

Christmas and Hanukkah are over, but Sarah Griffiths’ advice on wrapping presents is useful year-round. Griffiths discusses how to minimize the wrapping paper you use for gifts of various shapes, and how to get the most persnickety of gifts—like mugs and basketballs—tied up as neatly as possible.

Classroom Activities: geometry, surface area

  • (All levels) As a class, do a Secret Santa exchange with a twist: Each giver must wrap their gift in wrapping paper, with the gift’s surface area enclosed. After gifts are exchanged and unwrapped, measure how much wrapping paper was used to pack up the gift you received. Give out prizes for most creative wrapping and most economical wrapping.
  • (Mid-level) Read Sarah Santos’ procedure for wrapping cubes, in the section “Thinking outside the box.”
    • Calculate the surface area of a cube with side length $\ell$.
    • Calculate the area of the wrapping paper you would use to wrap that cube.
    • How much excess paper is there?
    • Answer the above questions, now using the recommended method for a triangular prism or cylindrical gift (section “Acute solution”).
    • In your own words, explain what happens if you try to wrap a cube according to Santos’ method, but you don’t place your cube diagonally in the center of the paper. What changes when you rotate it to sit diagonally?

—Leila Sloman

It’s time to unpack Spotify Wrapped. Here’s how the music streamer compiled your 2025 recap

Associated Press, December 3, 2025

Spotify’s annual streaming recap is notoriously confusing. This year, people learned about their “Listening Age” based on their musical habits and taste. “Spotify is billing the 2025 edition to be its biggest yet, with a host of new features it hopes may address some disappointments,” writes Wyatte Grantham-Philips for the Associated Press. The company claims that all of its metrics are based on simple rules and listening data, and this article unpacks the math behind your Wrapped.

Classroom Activities: data analysis, unit conversion

  • (Mid-level) Read this guide from Spotify with more details on calculating Wrapped results.
    • If a person ranked in the top 0.5% of listeners who streamed Chappell Roan, and Chappell Roan had 40 million monthly listeners, then how many people streamed Chappell Roan songs for more minutes than that person?
    • If a person listened to Lorde for 4,500 minutes this year, then how many months did they spend listening to Lorde?
    • (Mid-level) Based on this table of listening data and the Spotify guide, calculate each person’s “Listening Age.” (Assume that Alex, Beatrice, Cristina, Devin, and Evan are all 18 years old, and the “reminiscence bump” occurs when a person is 15 years old.)
  •  (All levels) If you use Spotify, discuss whether you were initially surprised by your Wrapped. Does the article help you better understand your results?

—Max Levy

Mathematicians discover a strange new infinity

New Scientist, December 3, 2025

Infinity is one of the strangest values in mathematics. We can use it in calculations, which suggests it’s a number. But unlike other numbers, you can’t say what number comes before or after infinity. Nearly 200 years ago, mathematician Georg Cantor came up with a way to organize the different versions in an “infinite ladder,” says physicist Abi James in this video from New Scientist. “We may have finally reached the end of infinity.”

Classroom Activities: infinity, real numbers

  • (Mid-level) Watch the video (at least until 6:16) and answer the following
    • What is 1 + infinity?
    • What is infinity + infinity?
    • Which of the two is larger? Explain.
  • (Mid-level) Explain the differences between the following terms: digit, number, integer, real number.
  • (High level) Explain in your own words the difference between a countable infinity and an uncountable infinity. Label the following as countable infinity or uncountable infinity.
    • Every prime number
    • Every odd number below 10
    • Every point on the circumference of a circle
    • Every integer below 10
    • Every real number between 1 and 3
  • (All levels) Watch the rest of the video and explain what the “Ultrafinitists” believe.
  • (Mid-level) For more infinity activities, refer to this Math Digest from 2022.

—Max Levy

More of this month’s math headlines: