Math Digests May-June 2026

May & June Digests:

OpenAI makes breakthrough on 80-year-old maths problem

The Guardian, May 21, 2026

In 1946, mathematician Paul Erdős posed a question about dots scattered across a plane. If you connect each pair of dots with a line, how many lines—each representing the distance between a pair of points—can have the same length? “Erdős proposed the number would rise only slightly faster than the number of dots themselves,” according to this article from The Guardian. The puzzle has challenged mathematicians for 80 years. Recently, a deep learning model from OpenAI made surprising progress. “The AI did not come up with a new answer for how fast the pairs of dots rise, but merely showed that the limit Erdős proposed was too low,” write Dan Milmo and Ian Sample.

Classroom Activities: geometry

  • (Mid-level) Draw a grid of dots, with points at integer pairs: (0,0), (0,1), (1,0), and so on.
    • Draw two squares: One with its bottom left corner at (0,0) and its top right corner at (0,1); the other with its bottom left corner at (2,0) and its top right corner at (5,3). Within each square, connect every pair of dots.
    • Write the length of every line connecting a pair of points in the 1 x 1 square. (There should be 6 lines.) How many pairs are equidistant?
    • How many lines (pairs of points) does the 3 x 3 square have? (Note: some lines go through one point on their way to another.)
    • Write the length for each line you drew.
    • What are the most frequent line lengths in the 1 x 1 square? And the 3 x 3 square?
    • As a class, generate a random arrangement of points by throwing darts (one per student). Work together to measure the distances between darts and plot the results in a histogram.
    • (High level) Is the activity more difficult with random points? Explain your answer.
    • (High level) The unit distance problem is often framed in terms of making as many pairs as possible distance 1 from one another. Why is this equivalent to the version described above?
  • (All levels) One mathematician in the story claims that the AI algorithm used paths of reasoning that humans have not yet explored, but that humans at OpenAI were very involved in finalizing the solution. What do you think he means by that, and why might he think it was important to say?

—Max Levy

Stonehenge and the Geometry of the Sky

Sky & Telescope, May 20, 2026

Stonehenge, in southern England, was built 4,000 years ago to track cycles in the sky. Its central axis aligns perfectly with both the summer solstice sunrise and the winter solstice sunset. It also marks the 18.6-year lunar cycle, which is more difficult to predict. “We are living at a time of extraordinary scientific understanding, yet also one in which many people feel increasingly disconnected from both the natural world and the institutions that study it,” writes Shanil Virani for Sky & Telescope.

Classroom Activities: probability, geometry, error

  • (Mid-level) The dual alignment with the two solstices is “not accidental—it is correct to within a few degrees,” Virani writes. “Statistical analyses suggest that such precise orientation is highly unlikely to arise by chance alone.” Confirm this statement for yourself with the following exercise.
    • Imagine the summer sunrise and winter sunset as two points opposite each other on a circle of radius 1. Draw a diagram that shows the line $\ell$ that passes between both points through the center of the circle.
    • Draw the two lines that pass through the center of the circle at angles of $\pm 2$ degrees from $\ell$. Shade in the region of the circle contained inside these lines.
    • What does the shaded region represent?
    • What fraction of the circle is covered by the shaded region?
    • What is the connection to Virani’s statement?
  • (High level) Stonehenge includes 56 chalk pits called Aubrey Holes. Scientists believe these holes track the moon’s 18.6-year cycle. In this activity, figure out why.
    • Suppose the ancient builders had estimated the moon’s cycle as being 18 years long. What would be their percent error, based on a true value of 18.6 years?
    • What would the error be of a 19-year estimate?
    • If each position in the Aubrey Holes represents one year, how many complete 18.6 year lunar cycles $N$ will it span? (Your answer should be a whole number.)
    • What is the true time it takes to complete $N$ lunar cycles?
    • What is the percent error represented by the Aubrey Holes? Justify your answer.

—Max Levy

Red & Black Knights

Numberphile, May 12, 2026

Complex patterns can emerge out of simple mathematical rules. In this video from Numberphile, mathematician Neil Sloane demonstrates a math experiment on a chessboard with numbered squares. He begins with a knight in the center of the board and adds more knights in a spiral pattern. Eventually, a symmetric pattern appears. But when the experiment features two types of knights and slightly different rules, the result is more puzzling.

Classroom Activities: patterns, complexity

  • (All levels) Watch the video. Describe in your own words how the rules of the experiment changed from the first example to the second.
    • Describe how the pattern changed.
    • Did both patterns eventually stabilize?
  • (High level) In the final experiment, red and black knights don’t interfere with each other’s path.
    • How would the final graphics change if the experiment began with a red knight instead of a black knight?
    • What events may lead to regions of only black or only red? Were you surprised by the large blocks of black and red in the final graphic? Why or why not?
    • Create a 30 x 30 grid numbered in the same spiral as shown in the video. Solve the experiment for three colors. (The rules remain the same, where knights can’t occupy squares already threatened by knights of a different color.)

—Max Levy

Physicist Richard Feynman’s forgotten notes on ‘the restaurant problem’ finally deciphered after 50 years

Live Science, June 9, 2026

In the 1970s, the physicist Richard Feynman watched a friend struggle to decide what to order for lunch. He responded in stereotypically physicist fashion—by deriving an equation for his friend to follow. “His equation showed exactly when Leighton—or any indecisive diner, for that matter—should stop taking risks and stick with what one knows is good,” writes Larissa G. Capella for Live Science. Capella covers new work that confirms Feynman’s solution is optimal, and tests whether people actually follow it. Spoiler: They don’t, but that’s not necessarily a mistake.

Classroom Activities: algebra, optimization, calculus

  • (Mid-level) The new paper reframes the restaurant problem as a choice among restaurants in a new city. Every restaurant has a quality score between 0 and 1.
    • If you have $n$ nights left in a new city, you should try a new restaurant if the best restaurant you’ve tried so far has a quality score less than $\sqrt{n}/(\sqrt{n} + 1)$. In which of the following scenarios should you try a new restaurant?
      • You have 9 days left in Paris, and the best restaurant you’ve visited so far scored 0.7.
      • You have 4 days left in Paris, and the best restaurant you’ve visited so far scored 0.7.
      • You are about to spend a year in Delhi, and on your first night, you try a restaurant that scores 0.95.
      • You are spending two nights in Santiago. On the first night, you try a restaurant that scores 0.6.
    • Give a qualitative justification for the rule. Would you follow it in real life? Why or why not?
    • The paper’s authors assume that quality scores are randomly distributed: You have an equal chance of choosing a restaurant that scores 0.2 as of choosing one that scores 0.8. How would the rule change if most restaurants in the city are really good? Not so good?
    • Feynman’s friend was torn between ordering a dish he’d eaten many times before and enjoyed, and between trying something new. What was the “right” decision? Justify your answer.
  • (High level) Feynman’s restaurant problem is closely related to another problem, called the optimal stopping problem. Try this lesson plan on the optimal stopping problem.

—Leila Sloman

Blue plaque honours pioneering female mathematician

BBC, June 12, 2026

Dame Mary Cartwright is being honored by the British government. During World War II, she proved that the radar used to detect German bombers could behave chaotically. Today, chaos theory is a field of mathematics in its own right. It describes the weather, ecology, financial markets, and mechanics, among other systems.

Classroom Activities: chaos, prediction, women in math

  • (Mid-High level) Try these lessons on chaos and fractals from the Cornell Math Explorers’ Club.
  • (High level) Rachael McMenemy describes the results of Cartwright’s research on radar systems: “Alongside leading mathematician J. E. Littlewood, they showed that the problem was not faulty equipment, but that some systems do not behave in predictable ways and that small changes can lead to different outcomes,” writes Rachael McMenemy for the BBC.
    • Research the definition of a chaotic system and describe how McMenemy’s description applies to radar.
    • Why did chaotic radar create a problem?
  • (High level) Several other British mathematicians are commemorated by blue plaques, including: Richard Price, Joan Clarke, Ada Lovelace, Arthur Eddington, Oliver Heaviside, Charles Babbage
    • Research one of these, or another mathematician with a blue plaque, and write a one-page report describing their mathematical contributions.
  • For more activities, see our previous digests on chaos theory from February 2025 and July 2023.

—Leila Sloman

More math headlines from May & June