Math Digests March 2025

The Uncertain Science of Certainty — the case for better evidence in post-evidence times

Financial Times, March 17, 2025

Is anything you hear ever truly certain? Mathematician Adam Kucharski might say no, according to his new book “Proof: The Uncertain Science of Certainty.” We live in a time of dubious scientific claims that distort evidence and scientific principles, such as the myth that vaccines cause autism. Kucharski is an expert in probability who teaches at the London School of Hygiene and Tropical Medicine. His book tells stories about the nature of evidence and how to best communicate it. This Financial Times review highlights a few relevant anecdotes and explanations from the book.

Classroom Activities: probability, statistics

  • (All levels) Complete this interactive activity representing the Monty Hall Problem mentioned in the article.
    • Why does this problem feel counterintuitive?
    • Why is it important to understand that probability is not always intuitive?
  • (All levels) Play with the “Expectation” portion of Brown University’s Seeing Theory lesson on probability and statistics.
    • Discuss and list real-world examples of where estimating “expectations” or “expected values” might be important.
  • (Mid level) Refer to the activities in this 2022 Math Digest about “The Math Behind False Positives and False Negatives.”
  • (High level) Read this article about the ethical consequences currently faced by using generative AI apps.
    • Discuss how using tools like ChatGPT can affect critical thinking skills.
    • How might misconceptions about math and computer science be informing how people use generative AI apps?
    • If more people rely on AI apps for answers, what effect might that have on the spread of misinformation and flimsy science, and why?

—Max Levy

‘Amazing’ spinning needle proof unlocks a whole new world of maths

New Scientist, March 10, 2025

One of math’s biggest open problems, the 3D Kakeya conjecture, was solved in February. The conjecture imagines a razor-thin needle rotating 360 degrees. As it rotates, it slides back and forth. If you color in the region that the needle moves through as it rotates, how big is it? The Kakeya conjecture guesses at the answer to this question. The new proof “has been hailed as one of the most important mathematical results in recent times,” writes Alex Wilkins for New Scientist.

Classroom Activities: geometry, fractals

  • (All levels) Play around with rotating a needle 360 degrees on a sheet of graph paper, using graphite sticks or unwrapped crayons as your “needle.” Have fun with this activity, and try to trace out a variety of different shapes!
    • Afterwards, estimate the area of your shapes by counting how many squares fall inside.
    • What types of motions produce the smallest shapes? The largest ones? Who managed to produce the smallest shape?
  • The precise statement of the Kakeya conjecture involves a concept called fractal dimension.
    • (All levels) Complete this worksheet from Fractal Foundation on Koch curves.
    • (High level) Read this chapter from Fractal Foundation on fractal dimension.
      • Explain in your own words the meaning of dimension, according to the chapter.
      • Take a copy of the Koch curve, and try to cover it with paper disks with radius: 1 cm, 0.5 cm, 0.2 cm, 0.1 cm. Do your results match the fractal dimension calculated in the chapter?
      • Calculate the fractal dimension of: A circle, $x^2 + y^2 = 1$; a disk, $x^2 + y^2 \leq 1$; the lines on your graph paper.
      • Following this Math Digests activity from November, estimate the fractal dimension of the US East Coast.

—Leila Sloman

An Internet Commenter Asked One Innocent Question—and Accidentally Caused a Major Math Conundrum

Popular Mechanics, March 8, 2025

“A curious situation,” tweeted mathematician Robin Houston in 2018, according to Popular Mechanics. “The best known lower bound for the minimal length of superpermutations was proven by an anonymous user of a wiki mainly devoted to anime.” Superpermutations are ways of stringing together scrambled sets of objects. Any way of ordering a set of objects is called a permutation. A superpermutation is a string of permutations where every possible permutation appears. (The permutations can overlap with each other.) In this article, writer Elizabeth Rayne describes a 4chan user who, in 2011, wanted to watch all the episodes of the anime series The Melancholy of Haruhi Suzumiya in every possible order, while minimizing the total amount of watching time. In thinking about how to do so, he unwittingly made a mathematical breakthrough — finding the shortest known superpermutation.

Classroom Activities: combinatorics, permutations

  • (All levels) Find all the permutations of the following sets: $\{1, 2, 3\}$, $\{\text{apples},\text{oranges},\text{pineapples}\}$, $\{\cdot, \star, \Delta, \Box\}$
  • (Mid level) How many permutations are there of
    • 3 elements?
    • 4 elements?
    • (Hard) N elements? (Hint: How many choices do you have for the first element? For the second element?)
  • (Mid level) Answer the following questions about superpermutations.
    • Find a superpermutation of the set $\{1, 2, 3\}$.
    • Find a superpermutation of the set $\{1, 2\}$.
    • What is the smallest superpermutation of the set $\{1, 2\}$? Why?
    • Find the smallest superpermutation you can of the set $\{1, 2, 3\}$. Explain your reasoning.
  • (Mid level) Read the article. Based on the given formula, how many superpermutations are there of 3, 4, and 5-element sets?

—Leila Sloman

Which knot is the strongest? (Instagram)

Scientific American, February 25, 2025

Can you recognize the strongest knot just by looking at it? In this Instagram Reel from Scientific American, Carin Leong shows four different knots with subtle differences. Two knots have more twists, and two have ropes that exit toward the side they entered from. Participants in a neuroscience study were shown these examples and asked to rank them in order of strength. “They consistently misjudged them,” according to the post, “pointing to a strange gap in our physical intuition.”

Classroom Activities: knots, systems of equations

  • (All levels) Did you rank the knots correctly? Explain the correct ranking in your own words. (Refer to this accompanying Scientific American article for more information.)
  • (All levels) Refer to this digest from 2023 for more activities. After completing the second activity in the link above, test out the strength of all the knots. Answer:
    • Which knots seem to be the strongest and why?
    • Is a figure 8 stronger than an overhand knot?

—Max Levy

A Mathematical ‘Fever Dream’ Hits the Road

The New York Times, March 14, 2025

Imagine a math lesson where you see math in a near endless list of artforms, rather than reading a dry textbook. “Beadwork, ceramics, crochet, embroidery, knitting, leatherwork, needle felting, origami, painting, polymer clay, 3-D printing, quilting, sewing, stained glass, steel welding, light, temari, weaving, wire bending and woodworking” — each of these crafts plays a part in a traveling exhibition called “Mathemalchemy,” according to writer Siobhan Roberts. Writing for The New York Times, Roberts describes the exhibition and one mathematician, Ingrid Daubechies, who has helped bring it to life.

Classroom Activities: tiling, artistic math

  • (All levels) Review the example artworks from the exhibitions. Discuss with a partner which is your favorite and why. Explain in your own words the mathematical concept being communicated.
  • (Mid level) The article says that Ingrid Daubechies’ $\pi$-shaped cookie cutter “tiles the plane.” Explain what this means in your own words.
    • Name and sketch 3 other shapes that tile the plane and 3 shapes that don’t.
  • (High level) The temari balls show the pairs 3 & 5, 5 & 7, and 11 & 13. Answer the following about twin primes.
    • What are the next 3 pairs of twin primes?
    • Read this article and look at the code for a program to check twin primes. Control the code by changing the values of n1 and n2 under the “Driver code” comment.
      • Are 137 and 139 twin primes?
      • Are 181 and 183 twin primes?
      • Can you explain in your words how the code tests whether two numbers are twin primes?

—Max Levy

More of this month’s math headlines: