Diana Liao and her colleagues in Tübingen were able to train crows to caw a specific number of times in response to visual and auditory cues. Their article in *Science* was picked up by *Audubon* on May 28th and by *Smithsonian Magazine* on the 29th.

Three male crows, borrowed from the University aviary, participated in this study. They were of the species *Corvus corone* (Carrion crow), common in Western Europe.

The crows were trained over a year to respond to a cue by cawing one, two, three or four times. They were then evaluated over a period of ten days. In a typical trial, in visual mode, a crow would be shown a colored number: purple 1, orange 2, blue-green 3 or pink 4. (Presumably in this context the Indo-Arabic numerals just served as four different shapes). It then had 10 seconds to caw a certain number of times and peck at an “enter key” to signal it was done. If the number of caws matched the cue, the crow was rewarded with a meal worm or a birdseed pellet. Otherwise, time out.

The distribution of responses is compatible with a real engagement of the subjects with the underlying number concepts. For example, as the authors observe, the errors were usually off by plus or minus one unit and tended to be larger when the target number increased. Another significant observation the authors report is that the *reaction time—*the delay between the stimulus and the first vocalization—was systematically longer for larger target numbers; as they remark, this “suggests that crows plan the entire number of impending vocalizations before motor production.” Furthermore, as they report, the acoustic quality of the first “caw” allows fairly accurate prediction of what the total number of vocalizations will be.

This item was covered by Ari Daniel for NPR on July 18. The four-minute broadcast includes the four auditory cues and a recording of one of the auditory-cue trials, and ends with a quote from Chris Templeton, a biologist unaffiliated with this project: “Animals are smart in a whole bunch of different ways, and those may or may not be the same things that we do.” To which Daniel adds: “Meaning that if crows were to give *us* an intelligence test, we may not pass.”

Two Tufts University professors, Barbara Brizuela and Susanne Strachota, published an article in the journal *Learning and Instruction* (May 17, 2024) with the title “When algebra makes you smile: Playful engagement with early algebraic practices.”

Brizuela and Strachota followed the mathematics instruction of 69 children (four classrooms) for two and a half years. The level is *early algebra,* grades 2, 3 and 4, with students aged 7-10. The authors wanted to track “the joy that students can experience when engaging with mathematics practices.” As they explain, most studies about students’ emotional responses to mathematics instruction have focused on the negative side. Even studies on how best to support self-driven, “playful” learning were aimed at preventing boredom and “math phobia.” No one considered the possibility that children could actually *enjoy* math, or experience what the authors call “positive epistemic affect” (essentially, the pleasure of understanding).

The authors set out to characterize instances in which students demonstrated positive epistemic affect in response to the four *practices* the late James Kaput identified as characteristic of students at this stage:

*generalizing*mathematical structures;*representing,**reasoning*with and*justifying*these generalizations.

The classes were recorded, and Brizuela and Strachota identified observable behaviors that could be correlated with students’ enjoyment: overt utterances, body postures and movement, facial expressions (e.g., smiles or smirks) and gaze. As they put it: “we identify markers of joy and describe how those co-occur with students’ engagement with the specific mathematics practices within early algebra.” In this paper they single out three cases involving different mathematical tasks, analyzing how they were taught and how the students engaged with the work.

One nice example comes from a class on the number line. The teacher-researcher had asked an open question about how far the number line would extend. The authors reproduce the transcript of the ensuing conversation, annotated to record the practices represented and the markers of affect manifested. At one point, one of the students, Talik, mentions “infinity.” The teacher asks him, “What’s that, Talik?” He explains with a smile: “The number line just goes on and on.” A bit later the teacher repeats what Talik said: “It should go on and on and on,” and asks: “When would we ever stop?” An unidentified student says: “It’s infinity. We would never stop.” The teacher notices Felipe wants to speak and calls on him. He volunteers: “We would stop at infinity and beyond.” The teacher repeats what he said and asks, “What does that mean?” Felipe answers, in statement form and without smiling: “Buzz Lightyear.”

The authors note that Buzz Lightyear, from *Toy Story 2*, counts as “relevant and familiar context,” part of the criteria for “playful stances to learning.” (Lightyear’s catch-phrase, “To infinity … and beyond!” was once voted the best film quote of all time).

An ArXiV posting by Kevin Buzzard (May 16) was brought to the wider public on June 16 by Clare Watson, on *Science Alert*: “Mathematician Reveals ‘Equals’ Has More Than One Meaning in Math.” The title may call to mind the Monty Python sketch where Peter Cook asks John Cleese: “Are you using ‘yes’ in the affirmative sense?” or Bill Clinton’s Grand Jury testimony where he states: “It depends on what the meaning of the word ‘is’ is.” But this is serious. The problem arises when we attempt to enlist computers into processing mathematical arguments, for example checking if a proof is correct or not. It turns out that our use of “equals” follows human conventions, often tacit, which lead to statements that are indigestible to a machine.

Buzzard’s posting focuses on fairly arcane examples from the work of the celebrated algebraic geometer Alexander Grothendieck, but there are more elementary ones. The group of symmetries of a polygon is the set of rigid motions (rotations and reflections) taking that polygon to itself. For example, if the polygon is an equilateral triangle, a 120$^{\circ}$ rotation clockwise or counterclockwise brings it back onto itself. This is a symmetry. Likewise, reflecting the triangle across any one of its side bisectors brings it back onto itself. This gives three more symmetries. Calling a set a *group* means that there is a multiplication combining any two elements to give a third. For symmetries of a polygon the “multiplication” is *composition*: performing the first motion, then the second. The top table below shows how this works for an equilateral triangle. For example, rotating 120$^{\circ}$ and then rotating 240$^{\circ}$ takes us back to where we started. (The symmetry that leaves everything fixed is labeled ${\bf I}$, the identity).

Googling “group of symmetries of equilateral triangle” leads to this page where you can read that “the dihedral group ${\bf D_3}$ is the symmetry group of an equilateral triangle” and “in mathematics, ${\bf D_3}$ … equals the symmetric group ${\bf S_3}$.” On another page you will find that ${\bf S_3}$ is the group of permutations that can be performed on 3 symbols. What does it mean to say that these two groups are “equal”?

When we say that two *sets* are equal, we mean that they have the same elements. The groups ${\bf D_3}$ and ${\bf S_3}$ obviously don’t have the same elements, but a comparison of the two multiplication tables shows that the correspondence that matches the 120$^{\circ}$ rotation with the permutation $\lt \mathsf{abc}\gt$, the 240$^{\circ}$ rotation with the permutation $\lt \mathsf{acb}\gt$, etc., going down the two lists, takes products to products and shows that the groups are really “the same” up to notation. Could this be the meaning of “${\bf D_3}$ equals ${\bf S_3}$”? The problem is that a machine would need to know which element goes with which, and the identification given here is *not* automatic.

In fact, our matching amounted to labeling the vertices of the triangle with the symbols $\mathsf{a, b, c}$, reading counterclockwise. But we could just as well have labeled them clockwise; then the 120$^{\circ}$ rotation would correspond to the permutation $\lt\mathsf{acb}\gt$, etc.; the two group structures would still match perfectly, *but in a different way*.

Buzzard explains that this imprecision in talking about structures and “equality” has not led to any errors in mathematics, but that success in formalizing mathematics to the point where computers can do it usefully depends on clearing it up.

*—Tony Phillips, Stony Brook University*

*Politico*, June 2, 2024.

Much is made of the conservative-liberal divide on the Supreme Court. But in this article, legal commentator Sarah Isgur and economist Dean Jens argue — using math! — that the conservative group (currently comprising six justices) should be viewed as two distinct groups. Isgur and Jens argue that the main difference between these groups is institutionalism and show some of the numbers that they used in their analysis.

**Classroom Activities: ***linear algebra, statistics*

- (Mid level) Read the article, then navigate to the graphic titled “Three Groupings on the Supreme Court Show Through in the 2023 Term.”
- Describe, in your own words, the meaning of the number in top left corner of the graphic.
- Examine the row representing Justice Sotomayor. Calculate the mean and variance of the data in this row. Calculate the mean and variance of the data in Sotomayor’s row
*within*each of the clusters proposed by Isgur and Jens (this will give a different mean and variance for each of the three clusters.) - Randomly select two other justices besides Sotomayor and repeat the exercise on the data from their rows. Do your results support the clustering proposed by Isgur and Jens? Why or why not? As a class, vote on whether you agree with the 3-3-3 clustering.

- (Mid level, Linear algebra) Isgur and Jens used a technique called
*principal component analysis*. Principal component analysis involves finding eigenvalues and eigenvectors of a matrix of data.- Consider a hypothetical court that has three justices. Let $\mathbf{j}_i$ be the vector of how Justice $i$ voted on five separate cases, with each vote being either “yes” or “no.” Suppose

$$\mathbf{j}_1 = \begin{bmatrix} \text{yes} \\ \text{yes} \\ \text{no} \\ \text{no} \\ \text{yes} \end{bmatrix},\; \mathbf{j}_2 = \begin{bmatrix} \text{yes} \\ \text{no} \\ \text{no} \\ \text{no} \\ \text{yes} \end{bmatrix},\; \mathbf{j}_3 = \begin{bmatrix} \text{no} \\ \text{no} \\ \text{yes} \\ \text{yes} \\ \text{no} \end{bmatrix} $$

Find the matrix whose $(i,j)$-th entry is the percentage of the time that Justice $i$ and Justice $j$ agree with one another. (The analog to the plot shown in the article.) Find the eigenvalues and eigenvectors of $M$.

- Consider a hypothetical court that has three justices. Let $\mathbf{j}_i$ be the vector of how Justice $i$ voted on five separate cases, with each vote being either “yes” or “no.” Suppose

*—Leila Sloman** *

*Numberphile,* June 13, 2024.

Suppose that if you roll the number $N$ on a six-sided die, you win $N$ dollars. How much money would you expect to earn in a single roll? This scenario is an example of “expected value” in statistics. In this *Numberphile *video, we learn that expected value equals the sum of each outcome multiplied by its probability of occurring. This statistical method is useful in games such as Pokémon, where we can calculate the expected number of encounters it takes to “catch ‘em all” using geometric series.

**Classroom Activities:** *expected value, geometric distributions*

- (All levels) Based on the rules outlined above, would you expect to win or lose money with the dice-rolling game if playing the game cost $30 for 9 rolls?
- How much money would you win or lose? Show your work.

- (Mid level) Watch the video. Explain in your own words:
- Why does the mathematician in the video use an “infinite sum” when calculating expected value?
- What is a geometric distribution?

- (Mid level) How many encounters would it take to catch 5 different Pokémon? Assume that each Pokémon has equal likelihood of appearing. Show each step of your work.
- (High level) Write out each step of the generalized example from the video of expected encounters for any number n of Pokémon. Explain why the mathematician compares the harmonic sequence to $1/x$.
- Why does integrating $1/x$ give a useful approximation for expected value?
- How many encounters do you expect it would take to catch 500 Pokémon?

*—Max Levy*

*Astronomy*, June 1, 2024.

In the early 20th century, Albert Einstein presented a new theory of gravity that applied not just to our planet or our solar system, but to the universe. Einstein’s theory included mathematical terms for the familiar attractive forces of gravity, as well as for repulsive forces that prevented his mathematical universe from imploding due to gravity. This repulsion was represented by a cosmological constant, lambda. Scientists didn’t know what lambda represented in the physical world until many years later. “For physicists and mathematicians who work with these equations today, lambda represents dark energy,” wrote Steve Nadis and Shing-Tung Yau for *Astronomy.* In this article, Nadis and Yau describe how Einstein’s theory was connected to dark energy, as well as how mathematics predicted other important physics discoveries like black holes.

**Classroom Activities:** *mathematical physics*

- (Mid level) Read the overview of this “Galaxies and the History of the Universe” lesson from PBS.
- Complete the problems 1-6 in the worksheet on your own.
- For problem 7, discuss each response as a class.

- (All levels) Watch this video introducing the math of general relativity.
- Define in your own words: spacetime; worldlines; coordinate system
- Propose three different 2D coordinate systems to track the progress of an airplane arriving in New York from Los Angeles. Explain the differences of each system.

*—Max Levy*

*AP News,* June 5, 2024.

Japan’s population is shrinking. As the population of Japan ages, young people are starting fewer and smaller families. Women in Japan average about 1.2 babies each in their lifetime, based on recent data — about 25% lower than the U.S. birth rate and about 45% below the global average. The new data makes 2023 the eighth consecutive year that Japan’s birth rate reached a new low. In this article for *AP News*, Mari Yamaguchi writes about the implications of low birth-rates on future economic predictions.

**Classroom Activities:** *data analysis, compounding*

- (All levels) Based only on the numbers above, calculate the U.S. and global birth rates.
- (Mid level) Read this MathWorld resource to learn more about compounding. Answer the following questions:
- If Japan’s birth rate continues to fall 5.6% annually, when will the birth rate fall below 1?
- By what percent would Japan’s birth rate need to increase annually to catch up to the global average within:
- 10 years?
- 5 years?

*—Max Levy*

*STV News*, June 24, 2024.

June 26 was Lord Kelvin’s 200^{th} birthday, and in celebration, the university where he spent most of his career launched a two-week exhibit titled “Lord Kelvin: Beyond Absolute Zero.” Two paintings made as part of the bicentennial highlight a project of Kelvin’s that was decidedly mathematical. In 1887, Kelvin proposed a shape which tiles all of three-dimensional space while keeping its surface area as small as possible. His project was misguided — he wanted to model an “ether” through which light could travel, when in fact no such ether exists — but he made long-lasting progress on the underlying math problem. It took 106 years to find a shape that both fills 3D space and has a smaller surface area than Kelvin’s.

**Classroom Activities: ***tessellation, geometry*

- (All levels) A shape is called
*space-filling*if copies of it can be packed together with no gaps or spaces in between the copies. For example, cubes are space-filling: You can stack them on top of and next to one another with no gaps. Are the following shapes space-filling?- Rectangular prisms
- Spheres
- Equilateral tetrahedrons (shapes where each of 4 sides is an equilateral triangle).
(Note for teachers: This example may be hard to visualize without physical tetrahedrons for students to play with.)

- (All levels) Kelvin thought his shape had the smallest surface area possible for a space-filling shape. Here is a list of space-filling polyhedra. Calculate the surface area of:
- A regular hexagonal prism (all edge lengths are the same) of volume 1.
- A cube of volume 1.
- Using formulas (9) and (10) here, calculate the surface area of a truncated octahedron whose volume is 1. Kelvin’s shape was a curved version of this shape.

*—Leila Sloman** *

- Can you solve it? Try this triple Tetris teaser

*The Guardian*, June 24, 2024. - Texas team uses AI to prevent power outages

*The Engineer*, June 24, 2024. - Mathematicians discover impossible problem in Super Mario games

*New Scientist*, June 13, 2024. - Mathematicians Are Suddenly Rethinking the Equal Sign

*Popular Mechanics*, June 12, 2024. - AI Will Become Mathematicians’ ‘Co-Pilot’

*Scientific American*, June 8, 2024. - Prime Number Puzzle Has Stumped Mathematicians for More Than a Century

*Scientific American*, June 7, 2024.

During the Edo period (1603-1868) there was a custom in Japan of individuals hanging mathematical tablets in shrines or temples. On these wooden *sangaku* (“calculation tablets” measuring typically around 80 $\times$ 170cm) were written problems or solutions to other problems, often with elaborately colored diagrams. In a paper from the 2014 Bridges conference, Hidetoshi Fukagawa and Kazunori Horibe explain that, in the Edo era, “ordinary people enjoyed mathematics in daily life, not as a professional study but rather as an intellectual popular game and a recreational activity.” However, the careful display of *sangaku* in sacred precincts suggests that it was also taken quite seriously. These tablets were also surveyed by Tony Rothman in the (paywall protected) May 1998 issue of *Scientific American*.

A May 30, 2024 posting on Nippon.com, a site whose mission is to “share Japan with the world,” highlights three instances in which the mathematics found during the Edo period anticipated its discovery in the West, often by many years. One of these comes from a *sangaku*. The item in question is “Soddy’s hexlet,” named for Frederick Soddy, FRS. Soddy first announced the hexlet in verse (!) (*Nature*, December 5, 1936). A proof by Frank Morley was published in *Nature* in January of the next year. That issue also included Soddy’s elaboration of a special case of the hexlet to construct his “Bowl of Integers”.

In Soddy’s hexlet, we start with three spheres $S_1, S_2, S_3$. One of the spheres, say $S_1$, encloses both of the others. Each sphere is tangent to the other two. The statement is as follows: let $S_A$ be any sphere tangent to all three. Then $S_A$ is part of a ring of six spheres, $S_A, S_B, \dots, S_F$, which encircle $S_2$ and $S_3$ inside of $S_1$. Each sphere in the ring of six is tangent to its two neighbors in the ring, as well as to all three of $S_1, S_2, S_3$. This ring is the “hexlet.”

Morley also proved that the radii of the nine spheres are related by the identity

$$ \frac{1}{r_A} + \frac{1}{r_D} = \frac{1}{r_B} + \frac{1}{r_E} = \frac{1}{r_C} + \frac{1}{r_F} = 2\left(-\frac{1}{r_1} +\frac{1}{r_2} + \frac{1}{r_3}\right)$$

As Soddy put it:

Now these beads without flaw obey this first law

For the aggregate sum of their bends.

As each in the tunnel slims through the funnel

Itsvis-à-visgrossly distends.

But the mean of the bends of each opposite pair

Is the sum of the three of the thoroughfare.

(The *bend* of a sphere is the inverse of its radius, except that for $S_1$, inside-out with respect to the others, the bend counts as negative.)

The *sangaku* in question was hung in a shrine in 1822, more than a hundred years before Soddy’s discovery, by one Irisawa Hiroatsu. Irisawa poses the problem: suppose $S_1, S_2, S_3$ and $S_A$ have diameters respectively 30, 10, 6 and 5. What are the diameters of the other spheres? He works through the calculation, obtaining for $S_B, \dots, S_F$ the diameters 15, 10, 3.75, 2.5, 2$\frac{8}{11}.$ You can check that these numbers satisfy Morley’s identity.

The original *sangaku* is lost, but a copy was made and appears in a book published in 1832. Details are in the report by Fukagawa and Horibe mentioned above.

Thomas Fink, the director of the London Institute for Mathematical Sciences, argues in a May 14 *Nature* article that *conjectures* are the part of math research where AI (artificial intelligence) will have the greatest impact.

In mathematics, progress comes when a mathematician has an idea (perceives a pattern, feels a correspondence, intuits a structure) and then sets out to check if that idea is correct. An idea is only dignified with the name “conjecture” if it becomes known before a proof has been established and if it has interesting implications. This is what Fink calls a “good conjecture.” Good conjectures are the way forward in mathematics, and Fink gives several examples to show how surprisingly good conjectures can be generated by AI. Nevertheless, as he states in his conclusion, for now it will take human judgment and experience to pick out the good ones: “AI will act only as a catalyst of human ingenuity, rather than a substitute for it.”

One set of examples Fink gives comes from a 2021 paper which harnessed AI to systematically look for mathematical characterizations of fundamental constants like $e$ and $\pi$. These turned out to be mostly in the form of *continued fractions*.

We’ll define continued fractions by example: The “golden mean” $\phi$ can be written as the continued fraction

$$\phi = 1 + \frac{1}{1+{\displaystyle\frac{1}{1+{\displaystyle \frac{1}{1+\cdots}}}}}$$

which means (enough terms are given for the pattern to be apparent) that $\phi$ is the limit of the sequence

$$1,\; 1 + \frac{1}{1+1}=\frac{3}{2},\; 1 +\frac{1}{1+{\displaystyle\frac{1}{1+1}}}=\frac{5}{3},\; 1 + \frac{1}{1+{\displaystyle \frac{1}{1+{\displaystyle \frac{1}{1+1}}}}}= \frac{8}{5},\; \cdots .$$

Among other formulas, the “Ramanujan machine” came up with

$$\frac{8}{\pi^2} = 1 – \frac{2 \times 1^4-1^3}{7-{\displaystyle\frac{2\times 2^4-2^3}{19-{\displaystyle \frac{2 \times 3^4-3^3}{37-{\displaystyle \frac{2 \times 4^4-4^3}{\cdots}}}}}}}$$

The numbers 1, 7, 19, 37 are the first hexagonal numbers. This formula, according to the authors, isn’t yet proved. Whether it is a “good” conjecture or not remains to be seen.

Fink also gives examples from the preprint Murmurations of Elliptic Curves. An elliptic curve $E$ consists of all the solutions to an equation of the form $y^2=x^3+ax+b$, just as a circle of radius 1 corresponds to the equation $x^2 + y^2 = 1$. But an elliptic curve is not only a curve corresponding to a polynomial equation. It is also, in a natural way, an abelian group (for details see here or here), and as an abelian group $E$ has a *rank* (analogous to the dimension of a vector space), a non-negative integer. Rank is an elusive invariant. As the “Murmurations” authors remark, “there is no general algorithm to compute the rank of an elliptic curve.” In fact the *Birch and Swinnerton-Dyer conjecture*, in which the rank of an elliptic curve appears, is a central open problem in the field.

In an earlier paper, the authors had used machine learning to train a computer to predict the rank of an elliptic curve. According to Lyndie Chiou, writing in *Quanta*, the authors were led to the discovery of murmurations by their efforts to explain why that earlier program worked so well.

Elliptic curves are important in number theory (and in cryptography). Often for these applications, we look for solutions modulo a prime $p$. That is, we want pairs of numbers $x,y$ in the range $0 \dots p-1$ where $y^2 – (x^3 + ax + b)$ is a multiple of $p$. For an elliptic curve $E$ the number of mod $p$ solutions, $N(E,p)$, is always finite. Murmurations turn up when the authors investigate the way $N(E,p)$ varies with $p$, and how this variation depends on the rank of $E$. They calculate the average value of the quantity $p + 1 – N(E,p)$ (averaged over all the elliptic curves whose coefficients satisfy certain number-theoretic inequalities) for the first 10,000 primes (from 2 to 104,729). When they plot the average values for odd-rank and even-rank curves separately, as shown in the figure below, the points form two clouds that *oscillate* up and down as $p$ increases (the term “murmuration” usually describes the coordinated behavior of a huge flock of starlings), with the direction of the oscillation depending on the parity of the rank. As the authors remark, this oscillation is still unexplained.

*Nature News*, May 22, 2024.

In this article for *Nature*, Rachel Crowell discusses the work of several mathematicians who have brought their quantitative skills to bear on social justice issues — from developing a tool that helps residents of Rhode Island’s Woonasquatucket River Watershed find healthcare resources to exploring the dependencies among Sustainable Development Goals set by the United Nations. “Mathematicians can experience first-hand the messiness and complexity — and satisfaction — of applying maths to problems that affect people and their communities,” writes Crowell.

**Classroom Activities: ***statistics, data science*

- (All levels) Read the article. Choose one of the featured projects and describe how mathematics helps the relevant social justice cause.
- (Mid level) Identify a social justice cause you care about. How might mathematics be helpful in studying it?

- (Mid level) Navigate to the CDC’s interactive CovidVaxView page for adults. View Figure 3A. This figure shows the percent of a population over time that has received the latest Covid-19 booster. Under “Jurisdiction,” select “National.” For each of the following “Demographics” selections, plot the 95% confidence intervals of the most recent figures. Then, interpret the results. Which of the demographic differences are statistically significant?
- “Disability Status”
- “Gender Identity”
- “Health Insurance”
- “Poverty Status”
- “Race/Ethnicity”

- (Mid level) Try these statistics activities on the relationship between earnings, education, and gender from the US Census Bureau’s Statistics in Schools program: One on plotting and analyzing scatterplots for 8
^{th}graders, and one interpreting box plots for 9^{th}graders.

*—Leila Sloman*

*60 Minutes*, May 5, 2024.

Last year, two teenagers from New Orleans announced a rare trigonometric proof of the Pythagorean theorem, which states that a right triangle’s hypotenuse squared equals the sum of its shorter sides squared ($a^2 + b^2 = c^2$). Many trigonometric principles depend on the Pythagorean theorem, so some mathematicians once thought using these principles was self-referential. In this *60 Minutes* segment, the two students describe their work and achievement in discovering up to five new proofs.

**Classroom Activities:** *trigonometry, algebra*

- (All levels) Read more about the teens’ proof presented last year in this
*Scientific American*article. (Note: The*Scientific American*article was written by the editor of this column.)- Describe in your own words the flaw in using the equation $\sin^2(\theta) + \cos^2(\theta) = 1$ to prove the Pythagorean theorem.
- Complete the AMS activities based on the initial news from last year.

- (High level) Follow along with the students’ “waffle cone” proof in the 60 Minutes video and go deeper with this step-by-step video from Polymathematic.
- Why does the proof require reflecting the initial right triangle? (Hint: Can you create the “waffle cone” without doing so?)
- What is the law of sines, and how does it enable this proof?
- Explain why using “similar” right triangles allows Johnson to draw a general conclusion for her proof. (Hint: think of convergent infinite series.)

*—Max Levy*

*RNZ*, May 23, 2024.

In the famous Monty Hall Problem, you are a contestant on a fictitious game show. In front of you are three closed doors, labeled 1, 2, and 3. One of the doors — you must guess which — conceals a valuable prize. To start, you choose a door. The game show host responds by opening one of the other two doors to reveal there is no prize behind it. You now get a chance to make your final guess: Stay with your first guess, or switch? On May 23, mathematician Chris Tuffley joined Radio New Zealand to explain what the right choice is, and why.

**Classroom Activities: ***probability*

- (All levels) Before learning the solution to the Monty Hall Problem, answer the following questions individually. Then, discuss answers as a class. After a brief discussion, vote on the right answers.
- Should you switch your choice? Why or why not?
- At the beginning of the game, what is the probability that your first choice is the right answer?
- Does this change after the host makes the reveal? Why or why not?

- (All levels) Split into pairs and play 6 rounds of the Monty Hall Problem with your partner, using a quarter hidden underneath one of three plastic cups. Take turns playing the host. Jot down anything you notice while playing. (You can also use this lesson plan, in which students play the game on a simulator such as this one and track the results.)
- Revisit the questions from the previous exercise and vote again if it seems like the class’s perspective has changed.

- (Mid level) Read Keith Ellis’s explanation of the solution. Write out the reasoning in your own words. Read this article about how mathematicians and scientists reacted to this counterintuitive problem in 1990.
- (Mid level) Consider a variant of the Monty Hall Problem in which the placement of the prize is not totally random. How does the optimal strategy change if everything stays the same, but at the beginning of the game:
- There is a 50% chance the prize is behind Door 1, a 25% chance it’s behind Door 2, and a 25% chance it’s behind Door 3
- There is a 40% chance the prize is behind Door 1, a 30% chance it’s behind Door 2, and a 30% chance it’s behind Door 3
- There is a 20% chance the prize is behind Door 1, a 30% chance it’s behind Door 2, and a 50% chance it’s behind Door 3
- There is a 20% chance the prize is behind Door 1, a 40% chance it’s behind Door 2, and a 40% chance it’s behind Door 3

*—Leila Sloman** *

*Numberphile*, May 1, 2024.

Is any single number more special than the rest? It depends on what question you ask. For mathematician Ben Sparks, the number 276 stands out. It’s not because 276 is an even number or a “triangular” number. It’s because of a special application of number theory called an aliquot sequence, in which you sum a number’s divisors (excluding itself), then sum *that* number’s divisors, and so on. Under this procedure, Spark’s number packs a strange surprise. In this *Numberphile* video, Sparks explains the aliquot sequence process and plots the results to reveal surprising mathematical behaviors.

**Classroom Activities:** *aliquot sequences*

- (All levels) Using only pen and paper, write the aliquot sequences for the following numbers: 10, 33, 15, 100.
- Which number has the longest sequence?
- Was this surprising? Describe in your own words what factor(s) allow some numbers to have longer or shorter sequences than others.

- (Mid level) Use this GeoGebra program to generate and plot the aliquot sequences for the numbers below.
- 79
- 30
- 220
- 1264460
- 119
- Find the definitions of
**perfect**,**abundant**,**amicable**, and**sociable**numbers in this resource from Dartmouth University, and say which word applies to each number above.

*—Max Levy*

*Quanta Magazine*, May 6, 2024.

More than 80 years ago, French mathematician André Weil first wrote about a surprising connection between two disparate fields of mathematics: geometry and number theory. Weil’s “Rosetta stone” connected the two topics with a third, the study of finite fields: “Finite fields are a place where number theory and geometry begin to blend,” writes Kevin Hartnett. In this *Quanta Magazine *article, Hartnett tells the story of Weil’s discovery and how it laid the groundwork for the Langlands program, a “grand project to unify disparate fields of mathematics.”

**Classroom Activities:** *number theory, geometry, finite fields*

- (Mid level) Based on the reading and your own online searching, describe each of these areas of math in your own words, and write a simple example problem related to each.
- Geometry
- Number theory
- Finite fields

- (Mid level) Write three different polynomial expressions that can be written in a finite field with the elements
**0**and**1**.- Write the binary form of each polynomial, as described in the article. What whole number does each binary form correspond to?
- What polynomial expression in this field would correspond to the whole number
**121**? - Are the polynomials you listed irreducible or reducible? Explain.

*—Max Levy*

- Five times numbers helped us make sense of the world

*BBC Teach*. - Cambridge code-breaker played by Keira Knightley in The Imitation Game gets blue plaque

*Cambridge News*, May 29, 2024. - How mathematician Freeman Hrabowski opened doors for Black scientists

*Nature Podcast*, May 28, 2024. - What are fractals and how can they help us understand the world?

*New Scientist*, May 21, 2024. - Incredible maths proof is so complex that almost no one can explain it

*New Scientist*, May 20, 2024. - Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture

*Quanta Magazine*, May 14, 2024. - Why mathematics is set to be revolutionized by AI

*Nature*, May 14, 2024. - Scientists Say: Correlation and Causation

*Science News Explores*, May 13, 2024. - The Mathematical Case for Monkeys Producing Shakespeare—Eventually

*Scientific American*, May 7, 2024. - Why are algorithms called algorithms? A brief history of the Persian polymath you’ve likely never heard of

*The Conversation*, May 7, 2024. - Cake-cutting math offers lessons that go far beyond dessert plates

*Science News Explores*, May 2, 2024.

- Galileo was wrong about mathematics?
- Updates on the shape of the universe
- Topology and schizophrenia

In 1623 Galileo published *Il Saggiatore* (“The Assayer”), a salvo in his polemic war against the Jesuit astronomer Orazio Grassi, an apologist for a mixed geo/helio-centric solar system model. In *Il Saggiatore*, he compares the universe to a huge book which one can’t understand without first understanding the language and the letters in which it is written: “[That book] is written in the language of mathematics, and the letters are triangles, circles and other geometric figures.” The prize-winning British science writer Philip Ball disagrees. “Galileo was, to put it bluntly, wrong: maths is not the language of nature, but a tool with which we are able to make quantitative predictions about some aspects of nature,” Ball wrote in an April 24 opinion piece in *ChemistryWorld*.

Ball, who writes mostly about the life sciences, is actually taking Galileo out of context. Galileo was talking about astronomy and physics (“the universe”) without addressing medicine or the life sciences of his day, as the word “nature” might suggest. On the other hand, leaving Galileo out of the picture, Ball has a point. As he puts it, “There is a profound difference between using the maths and understanding the science.” That is, we need pictures and stories (“causal narrative”) for the equations to make sense; too often, Ball says, “maths can even be something to hide behind: if you can crank the handle and get results, you can disguise (for a while) the fact that you don’t quite grasp the underlying science.”

But why does science have to make sense? Ball starts his piece, in fact, with a 1930 quotation from Werner Heisenberg, the discoverer of quantum mechanics, who wrote: “It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms.” More generally, why should our minds be able to comprehend the fundamental structure of the universe or, for that matter, of the human brain itself? Luckily, as Heisenberg also remarked, we have mathematics at our disposal, which can lead us safely beyond what we can comfortably “understand.” There may be parallel universes where the speed of light is different, but there can’t be one in which 47 is not a prime. We should be grateful that we have this reliable guide, and not complain when it sees better than we can.

If our three-dimensional universe is not infinite, then what does it look like? Presumably, it doesn’t stop at a wall, so it goes on being 3-dimensional everywhere; it’s what we call a 3-dimensional manifold. But just as a 2-dimensional finite surface with no boundary can have different *topologies—*it can be a sphere, a torus, a 2-holed torus, etc.—there are many possible topologies in dimension 3.

Some 3-dimensional topologies have been excluded as possibilities for our universe because they don’t match with current observations. For example, by current consensus the universe is geometrically *flat*, so that in any region we can examine the overall geometry is essentially like 3-dimensional Euclidean space. This rules out the simplest model for a finite topology, which would be the 3-dimensional sphere since the sphere requires overall positive curvature. “Promise of Future Searches for Cosmic Topology”, published in *Physical Review Letters* on April 26, 2024, reviews the remaining options. One possibility is the 3-dimensional torus, the space obtained by identifying opposite sides of a cube. This image of what that space would look like inside was used in a *Physics Magazine* commentary on the *PRL* article.

The authors of the *PRL* article (the 15 members of the international “COMPACT collaboration”) explain that the place to look for information on the topology of the universe is in the map of the cosmic microwave background (CMB). If topology is the explanation for anomalous features of this map, then it must contain detectable topological information. The most obvious sign of a topology like the 3-torus would be recurring images of features like galaxies, or, more generally, what the authors describe as “pairs of circles with matched temperature (and polarization) … visible in different parts of the sky,” but there is no sign of these.

To show that “no matched circles” does not rule out an interesting topology for the universe, the authors examine several flat models and conditions under which they would not be expected to show such circles. They remind us that there are 17 non-equivalent flat possibilities $E_1, \dots, E_{17}$ and concentrate on the first three: $E_1$, the 3-torus, where the faces to be identified can be parallelograms, or *rhombi*; $E_2$ (one of the rhombic faces has been rotated by $\pi$ before the identification) and $E_3$ (a square face has been rotated by $\pi/2$ before the identification). In these examples they work out a range of parameters (e.g. relative sizes of faces) for which the topology would not manifest matched circles, and so which are still viable candidates as models of a finite universe.

An application of graph theory to the diagnosis of mental disease is reported in the *Schizophrenia Bulletin*, April 26, 2024. In the article, Topological Perturbations in the Functional Connectome Support the Deficit/Non-deficit Distinction in Antipsychotic Medication-Naïve First Episode Psychosis Patients, by Matheus Teles, Jose Omar Maximo, Adrienne Carol Lahti (all at the University of Alabama, Birmingham) and Nina Vanessa Kraguljac (Ohio State), the authors seek physiological support for the understanding that deficit syndrome (DS, also called Deficit Schizophrenia) is actually a distinct disease, and not just a peculiarly severe manifestation of schizophrenia. Deficit Syndrome, as its name implies, is marked by enduring negative symptoms, e.g., restricted affect, poverty of speech, diminished sense of purpose.

The team worked with 61 patients, 18 DS and the others diagnosed with non-deficit schizophrenia (NDS), comparing them with 72 healthy controls (HC). Their hypothesis was that DS is special in that it involves disruptions in the connectivity of the workings of the brain. To test it, they defined 246 regions of interest in the brain, and considered these regions as the nodes of a graph. They used functional magnetic resonance imaging to infer the strength of connections between the regions by correlating their activity, and based on these strengths, assigned weights to edges between the corresponding nodes. This yielded a weighted graph for each subject. The graphs were evaluated with graph-theoretic criteria including:

**Global efficiency**. For an unweighted graph, global efficiency is defined as the average of the reciprocal of the length of the shortest path between any two different nodes (so longer paths contribute less). For a connected graph $G$ with $N$ nodes, with $d_{ij}$ the length of the shortest path between node $i$ and node $j$, the formula for the global efficiency $E(G)$ is $$E(G)=\frac{1}{N(N-1)}\sum_{i\neq j}\frac{1}{d_{ij}}.$$**Global shortest path length**here refers to the average length of the shortest path between any two nodes.**Local efficiency**at a node $i$ is the global efficiency $E(G_i)$ of the graph $G_i$ made up of node $i$, all the nodes directly connected to $i$, and all the edges between them.**Global local efficiency**of $G$ is the average of $E(G_i)$ taken over all the nodes of $G$.

The findings show that the global efficiency, global shortest path length, and global local efficiency differed—in a statistically significant way—between the deficit syndrome groups and the others. (In their conclusion, the authors remark that the topological metrics showed no statistically significant difference between the controls and the NDS (schizophrenia without deficit syndrome) subjects.) Put together, this is evidence that deficit syndrome really is a distinct disease.

*—Tony Phillips, Stony Brook University*

*Vox*, April 10, 2024.

When people say that “a picture is worth a thousand words,” they usually mean that one image can relay a lot of information. However, you could also conclude that one word doesn’t say much. In this article, *Vox*’s Oshan Jarow writes about how mathematics might be better suited than words for describing our conscious experience. “Words could offer you a poem about the feeling of standing on a sidewalk when a car driving by splashes a puddle from last night’s rain directly into your face,” Jarow writes. “A mathematical structure, on the other hand, could create an interactive 3D model of that experience, showing how all the different sensations — the smell of wet concrete, the maddening sound of the car fleeing the scene of its crime, the viscous drip of dirty water down your face — relate to one another.”

**Classroom Activities:** *math in language, data analysis*

- (All levels) The limitation of language, according to the article, is that an adjective like delicious mainly exists in one dimension. A chocolate mousse can be delicious and so can a burrito. The word tells you
*something*about food, but not enough to distinguish between them. Answer the following questions.- Describe the last thing that you ate with one adjective.
- Now describe the last thing you ate with four adjectives.
- Find a partner and describe your food with the four adjectives you chose
*without saying what the food is.*Try to guess each other’s food. (Avoid words like “cheesy” which reveal the ingredients. A word like melty might be better, for example.)

- (Mid level) List five pairs of adjectives (e.g., “salty/unsalted,” “dry/moist,” and “fragrant/odorless”) that you can use to describe a variety of different foods by assigning a score of 1 to 10 for each pair. (For example, a saltine cracker might be a [8,1,3] meaning very salty, very dry, and fairly odorless).
- Assign scores to the following foods: apple, chocolate bar, raw broccoli, parmesan cheese, mozzarella cheese.
- Tabulate the numbers for each of the foods.
- Write five observations based on the data, such as “XX is similar to YY in ZZ way but different in VV way.”
- In what ways is this system useful for comparing and contrasting the experiences of eating different foods? In what ways is it insufficient?

- (Mid level) Explain how the above exercises relate to the mathematics of consciousness described in the article.
- Explain what “ineffability” refers to in your own words.
- Upload this image of Earth to this online tool for pixelating images. Pixelate the image with a block size of 1, 50, and then 100.
- Describe how you would apply the idea of ineffability to this example. What happens when ineffability is low versus high?
- Is there a block size limit where the image no longer looks like anything?

*—Max Levy*

*New Atlas*, April 15, 2024.

When you look closely at a fractal, patterns that are obvious in the zoomed-out view repeat infinitely on smaller and smaller scales. Fractals are present all over nature, from snowflakes and trees to mountains and coastlines. However, they’ve never been observed in proteins, the building blocks of life, until now. “We stumbled on this structure completely by accident,” Franziska Sendker, a microbiologist who recently discovered a fractal molecule, said in an interview with *New Atlas. *Sendker was studying a protein isolated from cyanobacteria when she noticed its strange shape. Its atoms were arranged in countless triangles with triangular holes in their center. The pattern repeated at different sizes, “totally unlike any protein assembly we’ve ever seen before,” according to Sendker. This article describes the research and the specific type of fractal observed.

**Classroom Activities:** *fractals, Sierpiński triangle *

- (All levels) Read the description of the Sierpiński triangle observed in Sendker’s research and the instructions on this worksheet from the National Oceanic and Atmospheric Administration.
- Complete the worksheet for 6 iterations of the Sierpiński triangle.
- How many white triangles do you count at steps 3, 5, and 6? Describe the pattern.

- (Mid level) Follow this activity from the Fractal Foundation, “Fractal Trees,” where you calculate the ratios of branch length.
- (All levels) For more, refer to previous Math in Media digests about fractals from 2021 and 2022.

*—Max Levy*

*Ars Technica*, April 10, 2024.

On April 10, the Association for Computing Machinery announced the winner of the 2023 Turing award, one of the most prestigious awards in computer science. It went to Avi Wigderson of the Institute for Advanced Study in Princeton, New Jersey. In this article for *Ars Technica*, Jennifer Ouellette describes some of Wigderson’s accomplishments, including work showing that the advantages of random algorithms could be reproduced without randomness and on zero-knowledge proofs.

**Classroom Activities: ***randomized algorithms, algebra*

- (High level) In Richard Karp’s 1991 survey, he gives several examples of randomized algorithms. One is for testing whether a polynomial equation is valid or not.
- Read Section 4.1 of the survey. Describe in your own words the randomized algorithm for testing whether a polynomial $f$ is zero or not.
- Graph the polynomial $f(x,y) = x^3-5x^2y + 7xy^2 -3y^3$ using an online 3D plotter like WolframAlpha. Is $f(x,y) = 0$?
- Evaluate $f(x,y)$ at the points $(0,0)$, $(3,1)$, and $(2,2)$. Then use a random number generator to choose 3 coordinate pairs from the set ${-3, -2, -1, 0, 1, 2, 3}^2$ and evaluate $f(x,y)$ again. Interpret the results.
- Based on Theorem 4.1, what are the chances that you got $0$ for all 3 of your choices in the last exercise? Why do you think it’s important that the variables $a_1,\dots,a_n$ are chosen randomly?
- Come up with your own secret polynomial, and have a classmate try to guess whether it’s zero using the randomized algorithm.

- (High level) Read the
*Ars Technica*article and state, in your own words, what Wigderson showed about randomized algorithms like the one you just used.- Come up with a non-random algorithm for figuring out whether a polynomial of the form $ax^2 + bx + c$ is zero, by testing it on various values of $x$.

- Check out some activities on zero-knowledge proofs from our October 2022 digests.

*—Leila Sloman*

*The Guardian*, April 29, 2024.

Every two weeks, Alex Bellos publishes a mathematical puzzle for readers of *The Guardian*. On April 29, Bellos had readers come up with a grid whose tiles were colored in black and white according to some constraints. The problem is reminiscent of mathematical concepts like map coloring.

**Classroom Activities: ***graph theory, coloring*

- (All levels) Solve Bellos’ puzzle. Write a short paragraph describing how you came up with your solution.
- (Mid level) Bellos’ puzzle, along with the map coloring problem, can be represented on mathematical objects called
**graphs**. Complete this lesson on graph theory from the Park School in Baltimore, up through Exercise 17.- (Mid level) Rewrite Bellos’ puzzle as a problem about graphs. Do you find it easier or harder to solve now? Why?

- (High level) Read about the map coloring problem in the first section titled “Map Colorings” here.
- Come up with a graph that cannot be colored with only 4 colors. Describe why your graph is not a planar graph.
- Read the proof of the simpler six-color theorem given in the notes. Does the proof help you understand the examples you’ve worked with?

*—Leila Sloman*

*Scientific American*, April 18, 2024.

In mathematics, the concept of **measure **is used to quantify the length of sets of real numbers. For example, the measure of the interval $[0,1]$ is 1; the measure of the interval $[0,0.01]$ is 0.01. More complicated sets can be measured, too. The measure of the natural numbers is 0, while if you take each natural number $n$ and surround it by an interval of length $10^{-n}$, the total measure of all those intervals is $\frac{1}{9}$. But in 1905, Giuseppe Vitali showed that there are sets of real numbers that can’t be measured. In this article for *Scientific American*, Manon Bischoff explains the concept of measure and Vitali’s proof. (Note: The proof given that Vitali’s set is non-measurable contains some minor errors. The reason $\mu(V^*) = \sum_p \mu(V_p)$ is because the sets $V_p$ are disjoint, and there are uncountable sets with measure zero.)

**Classroom Activities: ***measure*

- (Mid level) Read the first section of the article. Based on the information provided, guess the measure of the following sets, with a brief justification for your answer:
- The interval $[0,4]$
- The union $[0,1] \cup [2,3]$
- The union $[0,1] \cup [1/2,1]$
- The set of all real numbers bigger than 1
- The set containing only 0

- (All levels) In dimension 1, the measure described by Bischoff represents “length.” The concept of measure can be generalized to two-dimensional sets, three-dimensional sets, and higher. Brainstorm as a class how this generalization should work before moving on to the next activity.
- (Mid level) Usually, the simplest measures in two-dimensional space represent
**area**and the simplest measures in three-dimensional space represent**volume**. Guess the measure of the following sets, with a brief justification for your answer:- The inside of a $1 \times 1$ square in the $xy$-plane
- The inside of a circle of radius 1 in the $xy$-plane
- The line $y = 2x$ in the $xy$-plane
- A sphere of radius 2 in 3D space
- The surface of the sphere of radius 2 in 3D space

*—Leila Sloman** *

*Yahoo Finance*, April 13, 2024.

Usually, sellers are happy when the price of what they’re selling is abnormally high. Homes are different. Most homeowners who sell would need to purchase another home — either upgrading to a more expensive house or moving “laterally” to a similar one — and right now, the math is not in their favor. They’d wind up paying far more on their monthly loan than they currently do. “That gap is convincing many American homeowners to stay put, stunting the number of homes for sale and buoying prices on the paltry supply that exists,” writes Janna Herron in *Yahoo Finance*. In this article, Herron explains the math behind the mounting pressure on buyers and sellers in the housing market.

**Classroom Activities:** *mortgage math, data analysis *

- (All levels) Most people purchase homes with a long-term loan called a mortgage, which requires monthly payments based on the total loan amount, plus interest. Read this interactive guide about mortgage math, and test the mortgage calculator with different numbers to answer the following questions:
- What would be the monthly payment on a \$500,000 loan with 7% interest paid over 30 years? What total amount will be paid by the end of the 30 year period? How much of that is interest?
- How about a \$500,000 loan with 7% interest paid over 20 years?
- A \$600,000 loan with 5.5% interest paid over 30 years?

- (Mid level) Using a spreadsheet such as Google Sheets or Excel, create your own mortgage payment calculator.
- What would be the initial monthly payment on a \$700,000 loan with interest of 4.5% paid over 30 years? What total amount will be paid by the end of the 30 year period?

- (Mid level) Read the example given in the article comparing the “payment shock” of two homeowners A and B.
- Suppose you and your neighbor each own homes worth \$400,000. You have \$100,000 in home equity, and your neighbor has \$300,000 of equity. If each of you decides to sell and use your equities for a down payment on \$500,000 homes, what will each of your monthly payments be? (Assume 5% interest and a 30-year loan)
- Define the payment shock as the percent change in the mortgage balance after selling your home and buying a new one. Calculate the payment shock for you and your neighbor.

*—Max Levy*

- Female code-crackers get belated recognition

*Taipei Times*, April 24, 2024. - Mathematicians Gave a Billiard Ball a Brain—and It Led to Something Unbelievable

*Popular Mechanics*, April 22, 2024. - Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions

*Quanta Magazine*, April 22, 2024. - Particles move in beautiful patterns when they have ‘spatial memory’

*New Scientist*, April 18, 2024. - Why AIs that tackle complex maths could be the next big breakthrough

*New Scientist*, April 10, 2024. - Can You View a Round Solar Eclipse Through a Square Hole?

*Wired*, April 5, 2024.

Drawings of mathematical problems predict their resolution was a University of Geneva press release picked up by *Phys.org* on March 7, 2024. It concerns research published in *Memory & Cognition* on February 14. That article, by Hippolyte Gros, Jean-Pierre Thibaut, and Emmanuel Sander, focuses on the role mental representations play in how people solve word problems involving whole numbers.

As the authors encapsulated their findings, “*What* we count determines *how* we count.” According to them, there are two different mental models for arithmetic, with different problem-solving strategies, and the one we choose is often determined by the context of the problem. They call the first model the *cardinal representation*. In a cardinal representation, a number stands for how many elements are in a certain set. The other model is an *ordinal representation*, in which a number refers to a certain position on the number line. (This second usage is related to, but different from, the way the term “ordinal” usually occurs in mathematics.)

Here is an example of a cardinal problem, taken from the list used in their experiment:

A bag of pears weighs 8 kilograms. It is weighed with a whole cheese. In total, the weighing scale indicates 12 kilograms. The same cheese is weighed with a milk carton. The milk carton weighs 3 kilograms less than the bag of pears. How much does the weighing scale indicate?

In an earlier paper the authors had established that subjects faced with a cardinal problem of this type, and asked to use as few operations as possible, would get to the solution in three steps. Using the terms from this example,

- The cheese weighs 12 – 8 = 4kg.
- The milk carton weighs 8 – 3 = 5kg.
- The weighing scale will indicate 4 + 5 = 9 kg.

Now an example of an ordinal problem, *with the same arithmetic structure:*

Obelix’s statue is 8 meters tall. It is placed on a pedestal. Once on the pedestal, it reaches 12 meters. Asterix’s statue is placed on the same pedestal as Obelix’s. Asterix’s statue is 3 meters shorter than Obelix’s. What height does Asterix’s statue reach when placed on the pedestal?

(Asterix and Obelix are cartoon characters, as familiar to the French as Tom and Jerry are here.)

Subjects with the ordinal problem and the same instructions would not be distracted by the equivalent of calculating the weight of the cheese (which would be the 4m height of the pedestal), and solve the problem in one step:

- Asterix’s statue reaches 12 – 3 = 9 meters.

As they describe it, problems involving weight, price and collection tend to lead to a cardinal analysis, “due to these quantities usually describing unordered entities.” On the other hand, problems involving height and duration are more often treated as ordinal problems “due to daily-life knowledge underlining the intrinsic order of the entities they mention.”

In this new paper, the authors investigate what mental activity leads to these different strategies. To this end, they asked subjects to give a solution to the problem, as well as a *diagram *of the problem. The authors worked with 111 subjects (59 adults and 52 fifth graders). Each was given a booklet of 12 problems, a mix of 6 cardinal and 6 ordinal (the $x, y, z$ in these tables were substituted with numbers in the range $z < 4 < x < y < 15$ in the tests).

The authors then developed a protocol for evaluating the extent to which a diagram represented cardinal or ordinal thought. Identifiable clusters of objects and graphically rendered set inclusion were cues to the presence of cardinal thought, while ordinal thought would manifest itself as axes, graduations, and intervals. Analyzing the results, the authors found that the cardinality of the diagrams predicted the use of the three-step strategy, and the ordinality of the diagrams predicted the one-step, regardless of whether the problem itself was of cardinal or ordinal type. This held for both children and adults.

The awarding of the 2024 Abel Prize (sometimes described as the Nobel Prize of mathematics) to the French probability theorist Michel Talagrand was reported by Kenneth Chang in the *New York Times*, March 20, 2024. Chang largely followed the Abel Committee announcement in citing three different areas of Talagrand’s work.

**Stochastic processes.**These are phenomena subject to random variation. Chang mentions the water level in a river—let’s call it $L$. Characteristics of such a process are its average value $\mu$ and its standard deviation $\sigma$. The standard deviation is a measurement of how often and how much $L$ deviates from $\mu$. A simple example of Talagrand’s work is an estimate for the*maximum*of the water level $L$ over a long period of time. The estimate calculates the probability $P$ of $L$ exceeding its mean by a certain amount; it has the form (I’m following Bodhisattva Sen’s 2022 lecture notes at Columbia, p. 101)

$$P[L\geq \mu + f(\mu, \sigma, x)] \leq e^{-x}$$

where $f$ is a specific function devised by Talagrand.**Concentration of measure.**As Chang describes it, in this second area Talagrand “helped show that there is a measure of predictability within random processes.” Here we have an expository lecture, “A new look at independence,” by the master himself. He describes the phenomenon as follows: “A random variable that depends (in a ‘smooth’ way) on the influence of many independent variables (but not too much on any of them) is essentially constant.” The setting for his estimate is much more general, but he starts his lecture with tossing a coin $N$ times, counting $+1$ for heads and $-1$ for tails and tallying the sum $S_N$. He estimates, for any $t\geq 0$, the probability that $S_N$ is greater than $t$ in absolute value:

$$P[|S_N| \geq t] \leq 2\exp\left(-\frac{t^2}{2N}\right)$$

for any $t\geq 0$. So the probability of $t = 550$ or more heads or tails in $N=1000$ tosses is $P[|S_{1000}|\geq 100]\leq .014$, and the probability that the number of heads is between 450 and 550 is at least $1-.014=.986$. (Besides the point for this exposition, a much sharper estimate is possible using the Chernoff inequality. This number, $.997$, is the one reported by Chang, presumably working from Talagrand’s 2019 Shaw Prize citation.)**Spin glasses.**A spin glass is a mathematical model for a special kind of matter in which atoms are arranged “amorphically” (not regularly as in a crystal) and in which their interactions are not confined to nearest-neighbors (unlike, for example, the Ising Model.) To investigate what configurations give local minima for energy, physicists have made mathematical models such as the Sherrington-Kirkpatrick (SK) model. This model dates back to 1975 and is described allegorically in this 2014 article by Dmitry Panchenko, in terms of the*Dean’s problem.*With this notation, in a perfect solution the product $g_{ij}\sigma_i\sigma_j$ is positive for every pair $i,j$: if students $i$ and $j$ like each other ($g_{ij}>0$), they get assigned to the same dorm ($\sigma_i\sigma_j =1$) and if they don’t ($g_{ij}<0$) they will be in different dorms ($\sigma_i\sigma_j =-1$). This is often too much to hope for—say 2 is BFFs with 4 and 5 while 4 and 5 cannot stand each other. But the Dean can make everybody as happy as possible by

*maximizing*the sum $\sum_{i,j}g_{ij}\sigma_i\sigma_j$. Formally, this is the same problem as minimizing the energy in the SK model of a spin glass.When $N$ is large, solving the problem is very hard. Around 1982, the Italian physicist Giorgio Parisi found a way to package the configurations to make the computation possible. He derived his equation using his powerful physical intuition. This work, in part, earned him the 2021 Nobel Prize. But mathematically speaking, it remained a conjecture. Chang quotes Talagrand: “For a mathematician, this doesn’t make any sense whatsoever.” Nevertheless, Talagrand developed the mathematical machinery necessary for a rigorous proof in 2006.

Michel Talagrand’s Abel Prize was also covered by Davide Castelvecchi in *Nature* on March 20. Both reporters mention Talagrand’s “unconventional” career among French mathematicians: he did not attend the *École Normale Supérieure* in Paris, their traditional spawning ground, and went instead through the regular university system, in Lyon. But he was not some nobody from the provinces. At age 22 he took the *Agrégation* examination in mathematics, the same national test that all the math *normaliens* have to pass, and came in first.

*—Tony Phillips, **Stony Brook University*

*WMDT*, March 14, 2024.

The concept of $\pi$ is essential. How could you study geometry without marveling at how the circumference of a circle is always a factor of $\pi$ times larger than its diameter exactly? To approximate its value, students usually prefer 3.14, while others prefer 22/7, said mathematician Dawn Lott. “I tend to use ten digits, which is 3.14159265358.” This story from Leila Weah at WMDT shares the history of calculating $\pi$ and the differences in how people use and represent it.

**Classroom Activities:** *pi*,* algorithms*

- (All levels) Read this Twitter/X thread from mathematician Alex Kontorovich about how to demonstrate $\pi$ using pizza crusts.
- Identify and explain at least 3 limitations of this exercise that would lead to falsely high or low estimations of $\pi$.

- (Mid level) Design your own pizza-based demonstration of $\pi$ that only uses pizzas but involves no cutting. (Think about other ways you might measure the circumference!)
- Identify and explain at least three limitations of your exercise.
- Write a concise step-by-step protocol so that someone not in this class could carry out the demonstration and understand what it says about $\pi$.
- Compare and discuss your idea with other students.

*NOTE: The article states that $\pi$ contains trillions of digits. The decimals of $\pi$ extend infinitely. The record for most decimal digits calculated by a human is 100 trillion.*

*—Max Levy*

*CBS News*, March 20, 2024.

*ESPN*, March 19, 2024.

During the yearly college basketball tournament known as March Madness, fans often fill out a “bracket” to try to predict the outcomes of each of the 63 games. What are the odds of getting the entire bracket right? Around 1 in 120 billion, according to *CBS News*. But 5 years ago, Gregg Nigl of Columbus, Ohio made history by predicting the outcomes of 49 March Madness games in a row. An *ESPN *article by Ryan Hockensmith tells Nigl’s story.

**Classroom Activities: ***probability, statistics*

- (Mid level) There are 63 games in the main March Madness bracket. Assuming every team has an equal chance at winning each game, calculate the following:
- The number of possible brackets.
- The probability of guessing a bracket perfectly.
- Nigl’s odds of finishing the bracket perfectly, once he had correctly predicted the first 48 games.
- The probability of correctly guessing 31 out of the 32 games in the first round.
- According to this article, a top-of-the-line prediction model is accurate 75% of the time. Repeat your calculations from above with 75% as the chance of guessing a game’s winner correctly.

- (All levels) Simulate a March Madness competition in class by breaking into 8 teams and running a tournament. The game you choose for the tournament should have outcomes that are somewhat, but not totally, random.
- Ahead of the tournament, have students fill out brackets. Did anyone guess the entire tournament correctly?
- By studying the class predictions, estimate the chances of correctly predicting the winner of each game.

*—Leila Sloman*

*Forbes*, March 13, 2024.

More than a century ago, a brewer from the Guinness company named William Sealy Gosset invented a far-reaching statistical method known as the $t$-distribution. Gosset was helping a fellow brewer who suspected hops with higher levels of a chemical called resin would improve the beer. Gosset’s colleague sought to prove his hunch with the scientific method: collect samples, measure resin concentration, and record his observations. But where was the line between “more resin in these samples” and “*significantly* more resin?” Gosset came up with the t-distribution to delineate between these two scenarios. “There is no doubt as to the significance of the method. The ‘$t$-distribution’ is featured in statistics textbooks and used in all fields utilizing stats from medicine to agriculture and much more,” writes Erik Ofgang for *Forbes*. In this article, Ofgang tells the story of Gosset’s work and influence.

**Classroom Activities:** *T-distribution; scientific analysis *

- (Mid level) Read this guide on the Student’s $t$-distribution from
*Wolfram MathWorld*. Then read Sections 3, 3.1, and 3.2 of Penn State’s “Statistics Online” textbook. Now suppose that the brewers collected the following data:

Batch |
Resin level |
Beer Quality |

1 | 3 | 20 |

2 | 3 | 13 |

3 | 10 | 20 |

4 | 12 | 20 |

5 | 15 | 29 |

6 | 16 | 31 |

7 | 17 | 25 |

8 | 19 | 21 |

9 | 23 | 27 |

10 | 24 | 32 |

11 | 32 | 16 |

- Calculate the $t$-score by hand.
- Calculate the degrees of freedom and determine a $p$-value for the hypothesis that more resin improves the beer.
- (High level) Read this
*WIRED*article about a potential new treatment for Lyme disease and follow the link to the press release.- Describe what the scientists were testing specifically in 2 or 3 sentences, using your own words.
- By looking through the technical report, describe how the scientists used the $t$-distribution test, also known as the “$t$ test” or “$p$-value test.”
- What $p$-values does the scientific team report in their study? Does this seem relatively more conservative or less conservative as a threshold for significance?

- (High level) Complete this calculator-based t-distribution activity and worksheet from TI Instruments.

*—Max Levy*

*BBC*, March 30, 2024.

Last month, Netflix premiered a new show featuring an old and famous scientific problem: The three-body problem. In the show, a group of aliens lives on a planet which orbits three suns at once. As physicists and mathematicians know, the movement of three objects acting on one another via gravity is nearly impossible to predict. Thus, the future of the planet is uncertain, and its inhabitants want to take over Earth instead. “The three-body problem is, then, the root cause of all the drama that plays out throughout the rest of the series,” writes Kit Yates for *BBC*.

**Classroom Activities: ***physics, algebra*

- (All levels) Read the article. Devise your own premise for a book, movie, or TV show centered around a mathematical or scientific problem you have studied in class. Describe how the details of the problem and its solution lead to a high-stakes plot. For more activities on math-related stories, check out our September digest on graphic novels.
- (High level) A two-body system follows paths described by conic sections. The equation $x^2 + y^2 = z^2$ describes a cone.
- A conic section is the curve you get when you intersect a cone with a plane of the form $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Find and draw the following conic sections:
- $x^2 + y^2 = z^2$ intersected with the plane $y = ½$
- $x^2 + y^2 = z^2$ intersected with the plane $z = 1$
- $x^2 + y^2 = z^2$ intersected with the plane $x – z = 1$

- On your own, come up with a few more planes to intersect the cone with, and draw the results. Write down any observations you have.
- A conic section is one of the following types of curves: An ellipse, a hyperbola, or a parabola. Derive the equations of an ellipse, hyperbola, and parabola in polar coordinates.
- For more on conic sections, read this section from Lumen Learning and try the Section Review Exercises.

- A conic section is the curve you get when you intersect a cone with a plane of the form $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Find and draw the following conic sections:

*—Leila Sloman*

*Quanta Magazine*, March 13, 2024.

To Claire Voisin, mathematics is something you feel, imagine, and meditate on while walking through Paris. Voisin is an award-winning French mathematician. Voisin appreciated math from a young age, thanks to its elegant proofs and definitions, but lost interest as a teen when math instruction felt more like a game. She turned to the creativity and structure of poetry, philosophy, and painting. Eventually, however, Voisin found the same depth in math that she loved in her creative pursuits and began a four-decade-long career. In this article, Voisin spoke with *Quanta Magazine*’s Jordana Cepelewicz about her work and inspiration.

**Classroom Activities:** *creativity in math*

- (Mid level) Based on the reading, describe what “deeper” things that Voisin eventually found in math. What do you think it has in common with the depth she found in philosophy, painting, and painting?
- (All levels) Use this mathematical art resource to draw the “three bugs problem.”
- How does the problem compare and contrast when you begin with a square versus a hexagon?
- Now, draw a shape with a concave region and repeat the exercise. Discuss how this compares to the square and hexagon versions.
- Next, follow the directions for “curves of pursuit” and sketch one that begins with a pentagon.

- (High level) Voisin works in Hodge theory, a subfield of topology. Complete the Day 1 and Day 2 exercises from this topology introduction for high school students, which cover differences between topology and geometry. (Note for teachers: An introduction and more difficult lessons are available within the document. #6 of the Day 1 exit slip is incorrect.)

*—Max Levy*

- Why 2024 Abel prize winner Michel Talagrand became a mathematician

*New Scientist*, March 21, 2024. - Mathematicians plan computer proof of Fermat’s last theorem

*New Scientist*, March 18, 2024. - Pi Day: How One Irrational Number Made Us Modern

*The New York Times*, March 15, 2024. - Pi calculated to 105 trillion digits, smashing world record

*Live Science*, March 15, 2024. - A Black mathematical history

*Nature*, March 14, 2024. - This 200-Year-Old Law Of Heat Has A Blind Spot. It Could Change Engineering.

*Yahoo! News*, March 14, 2024. - This Pi Day Let’s Celebrate The First Black Woman To Earn Her PhD In Mathematics

*Essence*, March 14, 2024. - Applied maths to the rescue: the Jack Powers story

*Cosmos*, March 8, 2024. - Have You Heard The One About The Mathematician…

*Discover*, March 7, 2024. - New Breakthrough Brings Matrix Multiplication Closer to Ideal

*Quanta Magazine*, March 7, 2024. - When a Math Museum Moves, Geometry Helps

*The New York Times*, March 4, 2024.

“Decimal fractional numeration and the decimal point in 15th-century Italy”, by Glen Van Brummelen (Trinity Western University) was posted online by *Historia Mathematica* on February 17, 2024, and picked up two days later by Jo Marchant in a *Nature* news item whose headline proclaimed “The decimal point is 150 years older than historians thought.”

Before Van Brummelen’s paper, historians thought the first appearance of the decimal point was in a table published by the Jesuit German astronomer Christopher Clavius in 1593. Now, we’ve learned from Van Brummelen that it can be traced back to trigonometric tables written by Giovanni Bianchini, a Venetian mathematician, around 1440. Bianchini’s tables have been digitized by the Jagellonian Library in Kraków; the decimal points Van Brummelen noticed are on the page numbered 108.

For some perspective, the “Treviso Arithmetic,” published in 1478 (the first known printed, dated arithmetic book) has no trace of this notation.

Below, you can see an excerpt from Bianchini’s table. The first column gives angles in degrees and minutes. The second gives corresponding values of the tangent function multiplied by 10,000 [my calculator gives $\tan$(68$^\circ$) = 2.47508]. The third column, where the decimal points occur, gives the increment corresponding to an additional minute of arc. [Since 21.2 = (25,387 – 24,751)/30, etc., this sets up linear interpolation between the whole-degree and half-degree values]. So for 68$^\circ$5′ the table would give 24,751 + 5 $\times$ 21.2 = 24,857. My calculator ($\times$ 10,000) gives 24,855.

Besides illustrating this early occurrence of the decimal point, Van Brummelen’s article points to the paradoxical situation of calculation in Europe in the early 15th century. While merchants and bankers were struggling with the transition from Roman numerals to Indo-Arabic notation, and choosing the best way to do long division, astronomers and mathematicians were computing trigonometric tables with four or five significant digits of accuracy.

A team in Japan has used concepts from graph topology to probe the way in which our brains synchronize with those of people we are interacting with. The experiment, reported in “The topology of interpersonal neural network in weak social ties” (*Nature Scientific Reports*, February 29, 2024), focused on the difference between interaction with a stranger and with an acquaintance. The authors, Yuto Kurihara, Toru Takahashi and Rieko Osu of Waseda University, explain that each run of the experiment involved a pair of participants. In total, there were 14 pairs of strangers and 13 pairs of acquaintances. Each participant wore headphones and a wireless 29-channel electroencephalogram (EEG) headset; they faced away from each other; each controlled a computer mouse which when clicked produced a tone audible to both of them. They were instructed to listen to a sequence of eight equally-spaced tones produced by a metronome and then to continue the sequence in alternation: participant 1, then participant 2, then participant 1, up to 300 clicks (150 each), matching the metronome’s inter-tap interval (ITI) as closely as possible.

The experiment was run for each pair under four conditions, including slow tap (ITI = 0.5s) and fast tap (ITI = 0.25s). The EEG records were filtered into three bands: waves of low (theta), medium (alpha) and high (beta) frequency, which were analyzed separately. In each case a 58$\times$58 matrix of inter/intra-brain synchronization was set up with an entry for each pair of monitored EEG channels (there are 29 + 29 = 58 channels in all), including pairs from the same participant. After processing to remove background effects, the matrix entries were reduced to 1 if the synchrony between the channels was significantly higher than the background and 0 otherwise. These data were used to construct a graph with one vertex for each of the 58 channels and links between the pairs corresponding to 1s in the matrix.

The team report that for the “fast tap” condition the theta band graph shows a significant difference between the performance of stranger pairs and acquaintance pairs of subjects. The graph-theoretic criterion used for this distinction is the *local efficiency*, first introduced in a *Physical Review Letters* paper in 2001. This is the average of the local efficiency at a particular vertex $v$, which is calculated as follows: Look at the neighbors of $v$, which form a subgraph $G_v$. For each pair of vertices in $G_v$, calculate one over the length of the shortest path in $G_v$. (Notice that when it is easy to get from one vertex to another, there will be short paths, making this number high.) The local efficiency at $v$ is the average of this number over all pairs in $G_v$. The local efficiency overall is the average local efficiency over all $v$.

In their Discussion section, the authors cite previous research showing a correlation between interpersonal interactions and activity at lower EEG frequencies. They further remark that interacting with a stranger, a more novel experience, will require more processing and more retention than interacting with an acquaintance. The question why exactly local efficiency should be the criterion that exhibits that difference is not, to my understanding, addressed.

Edward Frenkel (University of California, Berkeley) posted “Maths, like quantum physics, has observer problems” on the Institute for Art and Ideas website, February 6, 2024. Frenkel starts by reminding us how in the quantum world, different experimental setups can lead to seemingly contradictory results, with electrons sometimes looking like particles and at other times like waves.

How could something like this happen in mathematics? The answer is that the mathematics we do depends ultimately on the *axioms* we choose as a base for our logic, and that there is no way to check that we have a completely satisfactory set of axioms.

Frenkel focuses on *set theory*, the system that provides the language of mathematics. It has its axioms. The current set used by most of us is called ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) but a minority (Frenkel estimates 1%) reject one of the axioms: the *axiom of infinity*. The axiom of infinity states that the set 1, 2, 3, … of natural numbers exists; as Frenkel explains, this is much stronger than the statement that for every natural number there is a bigger number–there, you are always talking about some finite number. Accepting the natural numbers as a set means that there can be a set with infinitely many elements and some, the “finitists,” find this disturbing and refuse to do it. (It is disturbing, if you stop to think about it.) And as Frenkel explains, Gödel’s Second Incompleteness Theorem means it is impossible to know how the axiom of infinity affects the soundness of mathematics.

Why does this matter? It turns out that there are some theorems *about finite sets* that cannot be proved without the axiom of infinity. One of them (see below) is a strengthened general version of the “Cocktail Party theorem” we discussed last November. In 1977 Jeff Paris and Leo Harrington proved that this was a theorem, but that its proof made essential use of the axiom of infinity. Using the language of quantum physics, in one “setup” the statement is true while in the other it is unprovable. Whether the “observer” is a finitist or not makes a difference.

The theorem in question is the *strengthened finite Ramsey theorem*. To start working towards the statement, let’s go back to the “Cocktail Party theorem” for a moment. It is based on the following special case of the (unstrengthened) finite Ramsey theorem: suppose $N$ points are joined two by two by lines colored either red or blue. Then if $N$ is large enough any such configuration must include a red triangle or a blue triangle. (In this special case $N=6$ is large enough, and that’s the “Cocktail Party theorem.”) The finite Ramsey theorem is about sets, but it is convenient for us to continue to “visualize” it in terms of geometric objects, which here are $k$-*simplexes*. A $0$-simplex is a point, a $1$-simplex is a line segment, a $2$-simplex is a triangle, a $3$-simplex is a tetrahedron. In general a $k$-simplex consists of $k+1$ vertices and all the line segments, triangles, tetrahedra, etc. spanned by any pair, triple, 4-tuple, etc. of those points. Those sub-simplexes are called the *faces* of the $k$-simplex: 0-faces (vertices), 1-faces (edges), 2-faces (triangular faces), etc. For the general statement of the theorem we start with three numbers $n,m$ and $c$ with $m\geq n$, and an $N$-simplex we’ll call ${\bf K}$. We start by coloring every $n$-face of ${\bf K}$ with one of the $c$ colors. The finite Ramsey theorem states that if $N$ is large enough then ${\bf K}$ has a face ${\bf Y}$ of dimension at least $m$, all of whose $n$-faces are of the same color. (In the context of the “Cocktail Party theorem,” $n=1$ and $m=c=2$. If $\bf{K}$ has at least $6$ points, then there is a $2$-face all of whose $1$-faces—interpreted as edges of the triangle—are the same color.)

The finite Ramsey theorem can be proved without using the axiom of infinity. The *strengthened* version cannot. It involves identifying the $N+1$ vertices of ${\bf K}$ with the integers $1, 2, \dots, N+1$, and it reads the same as above except that ${\bf Y}$ can be required to have the additional property that its dimension is greater than the smallest of the integers corresponding to its vertices.

*The Conversation*, February 28, 2024.

In this article, education professor Leah McCoy argues that the game of Tetris is intrinsically linked to an area of mathematics known as *transformational geometry*. Introduced in middle school, transformational geometry involves linking two distinct mathematical objects through transformations like translation or reflection. What’s more, transformational geometry is used by a wide variety of professions, including animation and architecture. “There’s far more to Tetris than the elusive promise of winning,” McCoy writes.

**Classroom Activities: ***geometry, optimization*

- (All levels) Have each student play five minutes of Tetris on their own. Then, teach the definitions of the four basic geometric transformations: Translation, reflection, rotation and dilation. Ask each student to answer the following questions:
- Which transformations show up in Tetris?
- How and when do they show up as you play? Be specific. For example, in what direction, by how much, and in what context are the objects transformed?
- Which types of transformations do not show up in Tetris?

- (All levels) Have students come up with their own variation on Tetris by changing at least one of the types of transformations used in the game. (Students cannot just add a new transformation to the game—they must “delete” at least one existing one. Other features of the game, such as the way new shapes appear, or the shapes that are included, can be changed.) Students will create prototypes of their game using posterboard and construction paper shapes, and present to the class.
- (All levels) Play a game of “Tetris musical chairs.” Have students play Tetris in class. At random time intervals, alert them that it’s time to pause the game. Based on their paused screen, students will write down the transformations needed to get their piece where they want it to go.

*—Leila Sloman** *

*New Scientist*, February 2, 2024.

To physicist Suman Kulkarni, music is more than just sound, rhythm, and notes. She sees the greatest works of Western classical music as complex networks of information. Kulkarni recently analyzed the works of revered composer Johann Sebastian Bach with tools from the field of information theory, which describes data based on its individual parts and the connections between them. It applies to cryptology just as well as it applies to linguistics. Kulkarni wanted to identify *why* Bach’s music is so revered. Bach “produced an enormous number of pieces with many different structures, including religious hymns called chorales and fast-paced, virtuosic toccatas,” writes Karmela Padavic-Callaghan for *New Scientist*. In this article, Padavic-Callaghan describes Kulkarni’s experiment and analysis.

**Classroom Activities:** *information theory, graph theory*

- (Mid level) The article describes Kulkarni parsing Bach’s compositions into a graph of “nodes” connected by “edges.” Watch this short TED-Ed video about the origins of graph theory for more context and to answer the following questions.
- Based on the article and video, describe what the “nodes” and “edges” are in the Bach graphs. What would it mean when two nodes are connected by an edge?
- Can one node be directly connected to more than two other nodes? If so, what would this mean in the Bach piece?
- How would you expect the graphs of a Bach Toccata to differ from a Bach chorale, based on the information in the article?

- (High level) Learn more about graph theory methods, such as depth-first search (DFS), with this free TeachEngineering lesson. Complete the Making the Connection lesson worksheet, which asks you to define terms and analyze a graph. (An alternative to the “friends in class” criteria could be “students with more than one class together” or “students who share an extracurricular activity.”)

*—Max Levy*

*SFGate*, February 5, 2024.

In 1876, physicist Edward Bouchet became the first African American to earn a doctoral degree at a university in the United States. That long-awaited feat was delayed not by a lack of willing students, but by policies barring those students from equal rights. A new documentary titled “Journeys of Black Mathematicians: Forging Resilience” tells the story of similar hurdles in even more recent memory. “African Americans are not only underrepresented in the field, their achievements have been overlooked because the public perceives the road to excellence for African Americans is limited to sports and the arts,” writes Francine Brevetti for *SFGate*. In this article, Brevetti speaks about the stories he uncovered by speaking with Black mathematicians.

**Classroom Activities:** *equity, math history*

- (All levels) Watch the trailer (or rent the full film on Vimeo) and note the names of interviewees in the documentary.
- Research at least two of these mathematicians, and list five facts about their work or life stories.
- Describe the research or recent work of at least one of these mathematicians, in your words.
- Choose one mathematician and present what you’ve learned about them in class.

- (All levels) Read this profile of Christine Darden from
*Quanta Magazine*: The NASA Engineer Who’s a Mathematician at Heart. Discuss what barriers Darden mentions existing in her career. How have things changed or not changed since Darden worked at NASA?

*—Max Levy*

*Scientific American*, February 17, 2024.

In 1938, mathematicians proved a principle called the Ham Sandwich theorem. The theorem states that you can always “cut” $n$ objects in half simultaneously under certain conditions: If those objects are $n$-dimensional, and if your cut has $n-1$ dimensions. For example, if you have 2 circles in a 2-dimensional plane, there exists a 1-dimensional line that bisects both circles simultaneously. And they don’t have to be circles; they can be *any* shape, including discontinuous shapes like scatters of random points and blobs. “Contemplate the bizarre implications here,” writes Jack Murtagh for *Scientific American*. “You can draw a line across the U.S. so that exactly half of the nation’s skunks and half of its Twix bars lie above the line.” Murtagh’s article explains the Ham Sandwich theorem and how it complicates efforts to prevent political gerrymandering, a geographic trick often used to devalue the votes of minority groups or to favor one party.

**Classroom Activities: ***coordinate systems, geometry*

- (All levels) For each example below, draw a coordinate grid spanning $x=[0,8]$ and $y=[0,10]$ using blank or graph paper. Draw the two shapes/sets and find the line that bisects both. (All squares are upright; that is, their sides are perfectly horizontal and vertical, rather than tilted.)
- One circle with a radius of 8 centered at (1,9); one circle of $r = 2$ centered at (4,5)
- One square with side lengths of 2 centered at (4,9); one oval with a width and height of 2 and 1, respectively, centered at (7,2)
- One set of 4 squares, each with areas of 1, centered at (2,2), (3,3), (4,1), and (6,5); one circle of $r = 2$, centered at (2,7)
- What is easiest and most challenging about these examples?

- (Mid level) Murtagh writes “with sufficiently many voters, any percentage edge that one party has over another (say 50.01 percent purple vs. 49.99 percent yellow) can be exploited to win every district.” Demonstrate how this happens with the following exercise based on the images with purple and yellow dots in the article:
- Describe the tally of purple and yellow votes in each district before and after the straight lines shown in the article. Which group is being advantaged and disadvantaged?
- Starting from the second image, draw four more bisecting lines to create 8 “districts.”
- Is it possible to use the “Ham Sandwich” lines to
*increase*the power of the minority group?

*—Max Levy*

*Science News Explores*, February 29, 2024.

In this article, Lakshmi Chandrasekaran recounts several ways that geometry bleeds into research. One group that she covers, at George Mason University’s Experimental Geometry Lab, modeled four-dimensional shapes by 3D-printed projections of the shapes. Other examples were more practical. Laura Schaposnik at the University of Illinois Chicago is studying the possible geometries of viruses, which can help researchers get ahead of as-yet-undiscovered viruses. “Many people may find it hard to see the appeal or everyday uses of such math,” writes Chandrasekaran. “But modern geometry is full of problems both beautiful and useful.”

**Classroom Activities: ***geometry, tiling*

- (All levels) Read the article. In class, brainstorm other possible applications of geometry. As homework, write a paragraph about one of these possible applications. Be creative—your paragraph can be hypothetical, it can incorporate research about what related geometric work has been done, or it can involve your own experiments, but it must be specific about how geometry shows up.
- (All levels) The article describes projection in terms of shadows. Try to work out what the following shadows would look like, then check your work with physical objects:
- A sphere
- A cube oriented upright
- A cube balancing on one corner, with the opposing corner directly above it
- A rectangular prism balancing on one corner, with the opposing corner directly above it
- (High level, Linear Algebra) Now work backwards by calculating the projections explicitly. Assume light is coming from straight above your objects.

- (All levels) Explore tiling with this lesson plan from PBS. For more on tiling, check out the April digests.

*—Leila Sloman*

- The mathematical muddle created by leap years

*BBC*, February 28, 2024. - ‘Entropy Bagels’ and Other Complex Structures Emerge From Simple Rules

*Quanta Magazine*, February 27, 2024. - How two outsiders tackled the mystery of arithmetic progressions

*Science News*, February 26, 2024. - Mathematicians discover ‘soft cell’ shapes behind the natural world

*New Scientist*, February 26, 2024. - What’s the Best Place to Watch the Solar Eclipse? This Simulator Can Help You Plan

*Wired*, February 24, 2024. - Never-Repeating Tiles Can Safeguard Quantum Information

*Quanta Magazine*, February 23, 2024. - Marvel at Hundreds of Mathematician Max Brückner’s Remarkably Precise Models of Polyhedra

*Colossal*, February 22, 2024. - Can you solve it? The magical maths that keeps your data safe

*The Guardian*, February 19, 2024. - Google’s Chess Experiments Reveal How to Boost the Power of AI

*Wired*, February 18, 2024. - How String Theory Solved Math’s Monstrous Moonshine Problem

*Scientific American*, February 5, 2024. - Can you solve it? Are you smarter than a 12-year-old?

*The Guardian*, February 5, 2024.