- “DeepMind AI outdoes human mathematicians”
- AI takes on Geometry in the Math Olympiads
- $\ell$-adic numbers in
*Scientific American*

We have two items this month on the use of large language models (LLMs) in mathematics, both reported in recent *Nature *articles. In one item, the LLM was able to push forward a research-level problem in combinatorics; in the other, it solved geometry questions from the International Mathematical Olympiad, a math competition for high schoolers. LLMs work by predicting, based on a huge number of samples, the next word in a sentence, or, as in these examples, the next line of code in a program or the next step in a proof. They work amazingly well, but since they have no internal model of reality to work with, the predicted text may have no truth value, the predicted code may not work, and the predicted step may lead nowhere. In each instance, the authors relied on more conventional computer mathematics to keep the LLMs on track.

The first of these applications of LLMs appeared in *Nature* back in December 2023, and was highlighted in a “News” piece by *Nature*‘s math commentator Davide Castelvecchi: “DeepMind AI outdoes human mathematicians on unsolved problem.” The journal article was written by a team of twelve, mostly from Google DeepMind in London, with leaders Bernardino Romera-Paredes, Pushmeet Kohli and Alhussein Fawzi.

The researchers used an LLM to attack a research question that Castelvecchi describes as a generalization of the card game “Set.” The “Set” deck has 81 cards, each bearing one, two or three figures. The figures on a single card are all identical. But between cards, the figures differ: They can be ovals, diamonds, or squiggles; can be colored red, purple, or green; and can be outlined, solid, or shaded. So one card could bear two solid purple diamonds, while another could have three shaded green ovals. There are 3$\times$ 3$\times$ 3$\times$ 3 = 81 possible combinations of number, shape, color, and shading, one on each of the 81 cards in the deck. A SET is three cards which in each of the four aspects are either all the same or all different.

Mathematicians have worked out that if you lay out 21 different Set cards, there must be at least one SET among them. But what happens if we complicate the game? Imagine that each figure has five properties, or six, or $n$. (Keep the number of options for each property fixed at three.) In general, the size of the *cap set*, the largest set of cards possible without allowing a SET, is unknown. In an equivalent geometrical formulation this is called the *cap set problem;* the authors of the *Nature* article tell us that Terence Tao (possibly the world’s leading combinatorialist) once called it “perhaps my favorite open question.”

The authors studied the cap set problem, along with other combinatorics problems, with an AI strategy they call “FunSearch.” (The Fun is for “function.”) Rather than directly look for solutions (like a new cap set), they look for a program that will in turn find solutions. The implementation involves a Large Language Model trained on around one million samples of code, an evaluator that applies the LLM output programs to the problem at hand and grades their performance, and a *genetic* programming component (it uses an analogue of natural selection) that “breeds” the best-rated programs to generate new ones, which are put back into the LLM. At any point, the team can stop the process and pick out the highest-functioning program.

When FunSearch was applied to the cap set problem with 8 “properties” (to continue with Castelvecchi’s Set-game formulation) it generated a program that constructed a 512-element cap set. Previously, 496 was the size of the largest cap set known for $n=8$. The authors remark that FunSearch was able to discover a genuinely new cap set “from scratch,” on its own, and by methods different from those that had previously been used.

The second LLM-related item, “Solving olympiad geometry without human demonstrations,” appeared in *Nature*, January 17, 2024. The team (He He from New York University and Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, and Thang Luong from Google DeepMind in Mountain View, CA) reported on the prowess of their “geometry theorem prover” AlphaGeometry. They tested it on a set of 30 problems taken from the International Mathematical Olympiads since 2000.

The authors make it clear that they have not solved the general problem of how to communicate geometric data to a machine. As they say, this is a “separate and extremely challenging research topic.” Instead, they restrict themselves to problems which can be translated into the input language they use, one that works with “human-like non-degeneracy and topological assumptions,” as they explain it. They report that this language was suitable for 75% of the IMO geometry problems. Problems involving geometric inequalities and combinatorics were also set aside. Of the remaining 30 problems, AlphaGeometry solved 25. Compare to the average Silver medalist at the IMO, who solved 22.9 questions, while the average Gold medalist solved 25.9.

The authors distinguish between a geometry problem that can be solved using only data included in the problem statement, and a problem that requires one or more additional *constructions*: adding to the figure points, lines, etc. that are not specified in the premises. The latter type of problems require creativity on the part of the solver. AlphaGeometry combines a *symbolic deduction engine* (which can be used to derive all the possible consequences of a set of premises) and a *language model* which has been trained on nearly a billion problems and can be used to predict a relevant construction, much as ChatGPT can predict the next word in a sentence. That new material is fed back into the deduction engine, and the process cycles until one of the deduced consequences is the solution to the problem.

The authors give as an example of AlphaGeometry’s work its solution of Problem 3 from the 2015 Olympiad.

Note: The exhibited proof of IMO 2015 P3 uses a nonstandard notation that may confuse readers as it confused me. This is the convention of *directed angles* (thanks to Thang Luong for this link). Part of the convention is (a) the notation $\angle$XOY means the angle involved in turning the segment XO *counterclockwise* to match YO and (b) all angle measurements are taken *modulo* 180$^{\circ}$. In particular the statement $\angle$GMD = $\angle$GO$_2$D must be interpreted this way.

“Simple Math Creates Infinite and Bizarre Automorphic Numbers” by Manon Bischoff ran in *Scientific American*, January 11, 2024. It starts with automorphic numbers (see below) but it’s mostly about $\ell$-adic numbers: taking $\ell = 10$, a 10-adic number is an infinite sum like $5 + 2\cdot 10 + 6\cdot 10^2 + 3\cdot 10^3 + \dots$. Let’s not worry for now about convergence in the usual real-number structure, but accept it as an analogue of an infinite decimal like $a_0.a_1a_2a_3\dots$, which can be written as $a_0 + \frac{a_1}{10} + \frac{a_2}{10^2} + \frac{a_3}{10^3} + \dots$. Continuing the formal analogy, we can add and multiply 10-adic numbers exactly as we do the same operations with decimal expansions. This is the arithmetic of *10-adic numbers*.

10-adic arithmetic can be tricky: consider *automorphic numbers*. An automorphic number is a natural number which reappears at the end of its square. Bischoff explains that when you take the sequence of automorphic numbers $5, 25, 625, 90625, 890625, …,$ and extend it forever, you get a number $n$ which equals its square; it’s infinitely long, but we can calculate that it ends in $…8212890625$. So the 10-adic number $5 + 2\cdot 10 + 6\cdot 10^2 + 0\cdot 10^3 + \dots +$ satisfies $n^2 = n$. But then $(n-1) \times n = n^2 – n = 0$, even though $n-1$ and $n$ are both nonzero, exhibiting non-zero *zero divisors,* which is very bad for arithmetic. But, as Bischoff tells us, the problem goes away if we work with a prime number as our base instead of 10, and consider 2-adic numbers like $1 + 0\cdot 2 + 1\cdot 4 + 0\cdot 8 + \dots$, 3-adic numbers like $1 + 2\cdot 3 + 0\cdot 9 + 1\cdot 27 + \dots$, etc. The digits in a $p$-adic number take the values $0, 1, \dots, p-1$. Any calculation that can be done with real numbers can be duplicated with the $p$-adics—we even have $p$-adic analysis, a flourishing area of research extending calculus to the $p$-adics.

There is however an enormous difference in *topology*, the way the numbers fit together. In the $\ell$-adics two numbers are close if their difference is divisible by a high power of $\ell$. So, for example, the $10$-adic numbers $0$ and $2$ are far apart, while $10^{20}$ and $10^{30}$ are close together; whereas in the usual real-number topology, where two numbers are close if the absolute value of their difference is small, $10^{20}$ and $10^{30}$ are very distant. In particular, in the $\ell$-adics, the higher powers of $\ell$ themselves get closer and closer to 0. This is why a sequence like $5, 25, 625, 90625, \dots$ converges in the $10$-adic topology: the difference between successive terms accumulates more and more zeroes at its end, making it divisible by higher and higher powers of 10; it goes to zero.

*Wired*, January 26, 2024.

According to the American Physical Society, the ancient Greek philosopher Pythagoras suggested Earth was a sphere merely because he liked the idea. But there’s much more compelling evidence out there. In this article for *Wired*, Rhett Allain explains how to use scientific observation and mathematics to deduce that Earth is round, in two different ways.

**Classroom Activities: ***geometry, trigonometry*

- (All levels) Read the section “Water Isn’t Flat” in the article.
- (Mid level) Lake Pontchartrain is more than 40 miles wide. For simplicity, assume that when Allain took his photo, he was 40 miles away from a building on the opposite shore.
- The path between Allain and the building forms an arc on the Earth. Prove that the length of a chord between Allain’s eye and the building is almost exactly 40 miles. Give your intuition as to why this is true.
- Examine Allain’s diagram showing a right triangle with side lengths $R$, $s$, and $R+h$. Draw your own version and add a building across the lake. Find a right triangle that will help you calculate how much of the building is blocked by the horizon.
- Assuming the distance between Allain and the building can be approximated as 40 miles, calculate the height of the part of the building that is blocked by the horizon.
- The Pacific Ocean is about 12,000 miles across at its widest point. Repeat the calculation for someone looking at a building across the Pacific Ocean. Careful: The approximation that the arc length and chord length are the same no longer holds! How will you fix this?

*—Leila Sloman*

*CNN*, January 26, 2024.

Every year, inch-long insects known as cicadas emerge from the ground and take to the skies in a buzzy throng that numbers in the millions. They spend the next several weeks eating, mating, and laying eggs. The next generation will do the same; some descendants emerge the following year, and others, from a species known as **periodical** cicadas, come up only every 13 or 17 years. Biologists believe that periodical cicadas have evolved these long cycles to safeguard their offspring from predators. This spring, two distinct cycles, or broods, will coincide in an eruption of billions of once-dormant cicadas. “It’s a rare emergence of insects some are referring to as cicadapocalypse,” writes Kate Golembiewski. Her *CNN *article describes the once-in-a-lifetime insect event, and how mathematics offers an explanation for the strangely choreographed cycles.

**Classroom Activities:** *prime numbers, data analysis*

- (All levels) Watch this Numberphile video about cicada math.
- Describe in your own words the mathematical reason why 13 and 17 may have benefited cicada evolution. (Note: “The jury’s still out” on whether predators are truly the motivation for the cicada’s life cycle length, writes Golembiewski.)
- What would be the next largest cycle length that offers the same evolutionary benefit?

- (Mid level) The US is home to 12 broods of 17-year cicadas, and three broods of 13-year cicadas. Look up the 15 broods and what years they emerge.
- On a spreadsheet, create a table showing when each brood (columns) emerges in a span of years (rows) from 2024 to 2224. When does the next overlap of broods occur?
- Based on the math, when will these same two broods overlap again?
- How many overlapping cicada emergences occur within 200 years?

*—Max Levy*

*Quanta Magazine*, January 9, 2024.

In the 1940s, computer scientists began encoding data in a way that can reverse whatever data corruption might occur. Without that type of code, known as error correction, the data communication and storage your phone and computer rely on would be useless. As later researchers have improved on this idea, they have imagined new processes like “locally correctable codes,” resilient corrections that only query a few points of the data. “It’s as if you could recover any page torn out of a book by just glancing at a few others,” writes Ben Brubaker. There’s a catch, though: The examples of locally correctable codes that exist are terribly inefficient because they make the encoded message exponentially longer. Researchers have long hoped for a new algorithm that would skirt around this so-called exponential cost. This hypothetical algorithm would be able to correct data errors with only three queries. Yet now, Brubaker writes, computer scientists have proven that such a code is impossible. In this article for *Quanta Magazine,* we learn about work from theoretical computer scientists, and how the mathematics of error correction appear in disparate fields.

**Classroom Activities:** *error correction, exponentials*

- (All levels) Read this explainer about error-correcting code from
*Brilliant*, and familiarize yourself with the NATO phonetic alphabet table.- In groups of two, demonstrate the idea. Each person should write their own sequence of 16 random letters. Out loud, one person should communicate their secret message once, only using standard letters. Next, the other person communicates their own message using the NATO alphabet. Compare the success of communicating in each strategy.
- Explain in your own words why communicating in code with the phonetic alphabet is less prone to misinterpretation.

- (Mid level) As the number of bits $N$ to be relayed in a message grows, the number of bits in the error-correcting code grows exponentially in $N$. This is
**exponential complexity**. Other computer science problems are mathematically more efficient, exhibiting linear complexity, logarithmic complexity, and polynomial complexity.- Using your own online research, describe linear, logarithmic, polynomial, and exponential complexity, and suggest a real-world example for each.
- Write an example mathematical expression for each type of complexity.
- Rank each type of complexity in terms of efficiency.

*—Max Levy*

*Scientific American*, January 30, 2024.

It’s nearly impossible to generate a truly random sequence of numbers. Those who’d like to do so settle for “pseudorandom” sequences, which are not actually random but have many of the properties of randomness. In recent work, mathematicians Christopher Lutsko, Athanasios Sourmelidis, and Niclas Technau found a large family of pseudorandom sequences. Lutsko describes the result in this article for *Scientific American*.

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

- (All levels) Read the first two sections of the article (through “Detecting Randomness”).
- Describe in your own words the randomness test that Lutsko explains in “Detecting Randomness.”
- Suppose you had a sequence of 10 whole numbers between 1 and 10. Come up with a procedure that applies Lutsko’s test to the numbers.
- Read the third section (“Other Tests”). What does the second type of test capture that the first does not? Is there any property of a random sequence that is not captured by the two types of tests described?

- (All levels) Come up with three ways to generate a sequence of 10 pseudorandom numbers between 1 and 5. (Each number should come up with equal probability $1/5$.)
- Generate the sequences and plot them on a number line. Do they look random?
- Apply the procedure you came up with in the first exercise. What did you learn about your sequences?

- (Mid level) In past digests, we’ve analyzed statistics of coin flips. The same ideas apply here. Calculate:
- The probability that if you randomly generate 2 numbers between 1 and 5, you get the same number each time.
- The probability that if you randomly generate 3 numbers between 1 and 5, you get the same number each time.
- If your sequences were truly random, how many times should you see the same number twice in a row? Three times in a row? Compare to your pseudorandom sequences.

*—Leila Sloman*

*CDM*, January 4, 2024.

Have you ever spilled some of your drink out of a mug while walking just a few steps? You might think it’s your fault for not being more careful, but it’s not. It’s just physics. Physicists use the term “frequency” to describe how fast oscillations come and go. A slow-swinging pendulum has a low frequency, and squeaky sound waves oscillate rapidly (at high frequencies). The drink sloshing back and forth in your mug has a certain natural frequency, too. However, math student Sophie Abrahams explains, when the pace of your arms as you walk matches this frequency, a problem arises. The frequencies are said to resonate, which *increases *the swinging. This article in *CDM* covers Abrahams’ explanation and discusses other examples of resonant frequency, such as breaking glass with sound.

**Classroom Activities:** *resonance, frequency*

- (All levels) Watch the first video, “Why do I always spill my coffee?”
- Test Sophie Abrahams’ theory by walking with a full mug of water in three different ways. First, walk at a normal pace holding the mug by the handle. Then, walk at a slow pace holding the mug by the handle. Last, walk at a normal pace holding the mug in whatever way you expect will minimize spillage. Note your observations of the fluid movement and discuss in class.

- (Mid level) Watch the other videos referenced in the article. Define
**oscillation, amplitude, period, frequency,**and**resonance**in your own words as they relate to the following contexts:- Pushing a person on a swing
- Walking with coffee in a mug
- The Rubens tube
- Singing high notes by glassware

- (High level) We can use trigonometric functions, such as sine and cosine, to express model oscillations and resonance. For the following questions, assume that the function $\theta = \cos(t)$ describes the angle ($\theta$) of a swing from vertical at time ($t$).
- What is the frequency of the swinging function?
- Describe in your own words the motion of the swing. What do the variables $\theta$ and $t$ represent? Graph the function, and describe what it says about the swing’s motion.
- Which of the following terms can you add to the above equation to most accurately represent being pushed on a swing with resonance? Explain why and sketch the resulting graph.
- $\frac{1}{2\pi}$
- $\cos(t)$
- $t$
- $\frac{t}{4}\cos(t)$

*—Max Levy*

- Mathematicians Have Just Reversed the Sprinkler

*Gizmodo*, January 31, 2024. - This civil engineer turns to math to make energy more affordable

*Science News Explores*, January 29, 2024. - Lewis Carroll transported to Haryana

*Hindustan Times*, January 29, 2024. - The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

*Quanta Magazine*, January 26, 2024. - The Surprisingly Simple Math Behind Puzzling Matchups

*Quanta Magazine*, January 25, 2024. - A predicted quasicrystal is based on the ‘einstein’ tile known as the hat

*Science News*, January 25, 2024. - This Nomadic Eccentric Was the Most Prolific Mathematician in History

*Scientific American*, January 24, 2024. - A Wild Claim About the Powers of Pi Creates a Transcendental Mystery

*Scientific American*, January 24, 2024. - DeepMind AI solves geometry problems at star-student level

*Nature*, January 17, 2024. - Exploring The Intersection: Math And Machine Learning Forge A Two-Way Street Of Mutual Influence

*IndiaEducationDiary.com*, January 11, 2024. - What the mathematics of knots reveals about the shape of the universe

*New Scientist*, January 5, 2024. - Mathematicians Identify the Best Versions of Iconic Shapes

*Quanta Magazine*, January 5, 2024. - Here’s how much fruit you can take from a display before it collapses

*Science News*, January 4, 2024. - ‘I Got Here For a Reason’: Using Mathematics For Social Justice

*Holy Cross Magazine*.

- Progress in game-of-life studies
- Ominous shakeup for UWV mathematics
- Magnetic skyrmions and hopfions

The “Game of Life” was invented by the late mathematical genius John Conway around 1970, and brought to general attention in Martin Gardner’s “Mathematical Games” column in *Scientific American*, October 1970. Mathematically speaking, it is a *cellular automaton*, but it doesn’t hurt to think of it as a game. Working on a large grid, you mark a few squares in black. These are the live cells; the others are dead. That is Step 1. For Step 2, live cells are created or destroyed according to the following rules. (A cell’s *neighbors* are the eight closest cells).

- A live cell with 0 or 1 live neighbors dies.
- A live cell with 2 or 3 live neighbors persists.
- A live cell with 4 or more live neighbors dies.
- A dead cell with exactly 3 live neighbors comes to life.

Applying the rules to the new configuration leads to Step 3, etc. You can experiment at playgameoflife.com.

The Game of Life proved to be a mathematically fascinating universe, with many remarkable inhabitants, starting with the glider, the Herschel and the Gosper glider gun. Some are *periodic*, returning to their original shape after a certain number of steps. For example, three squares in a row has period 2. The question arose, can any positive integer occur as the period of a game-of-life configuration? That is, for any integer $n$, is there a shape in the Game of Life that returns to itself after precisely $n$ steps? Over the years it was shown that every period was possible except perhaps 19 and 41 and, as Matthew Sparkes reports in *New Scientist* (December 25, 2023), those two last numbers have just been accounted for (“Conway’s Game of Life is Omniperiodic”, posted on ArXiV, December 5, 2023).

The University of West Virginia is closing down its master’s and doctoral programs in mathematics, according to “An ‘Academic Transformation’ Takes on the Math Department,” by Oliver Whong, writing in the November 28, 2023 *New Yorker*. Whong’s story starts in December 2020, with WVU president Elwood Gordon Gee proclaiming that “the perceived value of higher education has diminished” and that WVU should “focus on market-driven majors, create areas of excellence, and be highly relevant to our students and their families.”

“Gee has long argued that land-grant universities, which were created in 1862 by an act of Congress, are meant to ‘prioritize their activities based on the needs of the communities they were designed to serve,’ as he puts it in the book ‘Land-Grant Universities for the Future,‘” writes Whong. The nation’s 105 public and 7 private land-grant institutions include Cornell, MIT, Wisconsin, the University of California … and WVU.

Last September, it was decided that the master’s and PhD programs in mathematics at WVU would be discontinued and a third of the department’s faculty positions eliminated. When Whong asked Gee about the driving force behind the changes, he was told that “it came down to assuring the public, which finances the school, that the university was serving the public’s needs.” A more specific diagnosis came from Maryanne Reed, the WVU provost. The overwhelming majority of math enrollments were in service courses (5000 versus 65 math majors and 23 grad students). Some of those service courses had many low grades: D’s and F’s, along with withdrawals. In an email, Reed told Whong that higher grades “drive first-year student retention and are a primary factor in students’ ability to complete their degrees in a timely fashion. The key point here is that we need to focus on what our students and their future employers want and need.”

Whong reports conversations with several WVU math faculty members, including John Goldwasser, who thinks that students’ mathematical talent can be awakened by a high-quality college program. “I’ve had students in my honors math classes who could be good students anywhere. And many of them didn’t know what their potential was before coming to my class,” Goldwasser told Whong. Goldwasser also mentions the essential belief, shared by many of us in the profession, that “a public university should help to create and shape values, not just reflect the things the majority of people already care about.”

Two kinds of nanoscale physical structures, magnetic skyrmions and hopfions, are attracting attention right now because of their possible application to data storage. Both structures exist inside magnets. We will focus on Hopfion rings in a cubic chiral magnet, contributed to *Nature* (November 22, 2023) by a team at Forschungszentrum Jülich (Fenshan Zheng, Nikolai Kiselev, Filipp Rubakov and collaborators), which reports the first realization of hopfions inside a single magnetic crystal. Getting an idea of what these experiments mean, and how they relate to topology, requires some background. Since skyrmions are simpler to describe, and are necessary ingredients and companions of hopfions, we’ll start there.

Inside a magnet, there is a magnetization vector at each point in space. A skyrmion is a localized 2-dimensional pattern of orientations of the magnetization vector; it is a kink in the magnetization vector field.

In this illustration the magnetization vector field points straight up everywhere, except within a disk. If you trace along the diameter of this disk, the magnetic field vector will gradually rotate clockwise, pointing straight down at the center and pointing up again when you get to the opposite edge. This kink cannot be removed by a continuous deformation –in physical terms, it would cost a lot of energy– and this stability is a topological phenomenon. To understand that, first imagine mapping the disc containing the skyrmion to a sphere about the origin in 3-space: each point of the disc maps to the vector representing the field at that point. Then following the vectors around concentric circles in the disc shows that each circle maps to a circle of latitude on the sphere.

You can imagine the map as stretching the disk over the sphere, with the center of the disk at the South Pole and the periphery at the North Pole. Topology teaches us that, holding the periphery fixed, this map cannot be continuously deformed so as not to cover the sphere. (In topological notation, it represents a non-trivial element of $\pi_2S^2 = Z$, the second homotopy group of the 2-sphere). The skyrmion is trapped in the field, just as a knot cannot be removed from a string if the ends are held fixed.

In a 3-dimensional magnetic solid, skyrmions usually appear aligned together in *skyrmion strings*.

Now for hopfions: a skyrmion string can be bent around to join itself and form a loop. If it is twisted one or more times in the process, the resulting 3-dimensional structure is a hopfion.

For the hopfion illustrated above it is necessary to use a skyrmion string more complicated than the one described by Koshibae and Nagaosa. In particular the magnetization vector field, instead of being constant along the boundary cylinder, rotates as the base point goes around a circumference. (The colors in this image encode the direction of the magnetic vector field at those points; note that the direction-color correspondence is completely different from the one used in the Koshibae-Nagaosa article). The same color-coding is used on the sides of the three skyrmion strings enclosed by a hopfion, and on the surface of the hopfion itself, in the left-hand image below.

In this case the piece of skyrmion string forming the hopfion has been given one complete twist. As a consequence, the colored stripes on the surface corresponding to different directions of the magnetic field are topologically linked, like consecutive links on an anchor chain. That makes the configuration stable under small deformations and guarantees the permanence of the hopfion structure. (The proof is by modern algebraic topology; the magnetization-vector map on the whole solid torus represents a nontrivial element of another homotopy group of $S^2$, this one 3-dimensional.)

]]>*/Film, *December 3, 2023.

An episode of the animated sci-fi sitcom *Futurama* features a machine that allows any two characters to swap bodies. The catch: once two people switch, the machine won’t work for that same pair of bodies again. They can’t switch back. At least, not directly. During the chaos of the episode, nine characters hop around to different bodies. But can they return things to normal? The answer is yes—but it requires some serious math. The show’s writer Ken Keeler, who has a PhD in mathematics, proved that it is possible to return everyone to their own bodies, as long as you add in two new participants who have not used the machine before. The story of Keeler’s result, also known as the “Futurama Theorem,” is described by Witney Seibold in an article for */Film*. In the show, two members of the Globetrotters (a futuristic, math-genius version of the Harlem Globetrotters) discover the theorem, and return everyone to their original bodies to save the day.

**Classroom Activities:** *group theory, symmetric group*

- (All levels) At the beginning of the episode, two people swap bodies. Call them Person 1 and Person 2. Suppose that Person 3 and Person 4 agree to help out by participating in some body swaps. Find a sequence of swaps so that no individual swap is repeated and everyone ends up in their original body. (Keeler’s theorem proves that this is possible.) Hint: This can be achieved with a total of 5 swaps. To keep track of the swaps, make a table whose columns represent the swaps as well as the four bodies (1, 2, 3, and 4), and where each row keeps track of the mind in each body, as below.
- Suppose that only Person 3 agrees to help. Is there still a way to return everyone to their original bodies? Why or why not?
- (High level) The episode shows a sketch of Keeler’s mathematical proof, which is written using symmetric group notation. In this notation, (12) stands for the swap between Person 1 and Person 2, and a sequence of swaps is read right to left. For example, (23)(12) stands for the swap between 1 and 2 followed by the swap between 2 and 3. Try writing out your answer for four people using this notation. Then see if you can write down a solution for five people—that is, once three people have swapped around their bodies, find a way to fix things using two new participants.

- (High level) As a class, work through the Futurama Theorem activity by Cheryl Grood featured on mathcircles.org.

*—Tamar Lichter Blanks*

*The New York Times*, December 6, 2023.

At a United Nations climate conference in December, policymakers debated the future of fossil fuels. Nonrenewable sources contribute 80 percent of the world’s energy supplies, and around 75 percent of greenhouse gas emissions. Fossil fuel emissions must go down to halt climate change, but oil and gas producers contend that technology can capture and sequester the harmful gases. But this scientific solution isn’t as simple as it sounds. “It would be nearly impossible for countries to keep burning fossil fuels at current rates and capture or offset every last bit of carbon dioxide that goes into the air,” Brad Plumer and Nadja Popovich write. “The technology is expensive, and in many cases there are better alternatives.” In this article, Plumer and Popovich analyze the limits of carbon capture technology, and what those limits mean for the future of energy production.

**Classroom Activities:** *rates, data analysis, percentages*

- (All levels) Choose from these classroom activities about climate change and math from NRICH, including one that allows students to compare their carbon footprints, and another that asks students to design an efficient map for delivering energy by drawing a network that maximizes connections but minimizes overall length.
- (Mid level) Approximately 15% of greenhouse gas emissions comes from livestock production, 30% come from automobile emissions, and 40% come from burning coal for energy. Which of the following would reduce total emissions the most?
- Reducing automobile emissions by 20%
- Reducing livestock production by 95%
- Reducing coal use by 40%

- (Mid level) Suppose that the world emitted 50 billion metric tons of greenhouse gases (GHG) in 2023. Create a table based on the following to show how that number changes in the future.
- What will be the amount of annual emissions if the global emissions rate increases by 1.7% every year from 2023 through 2030?
- How many years would it take to get back to 50 billion metric tons per year if, starting in 2030, emissions reduce by a net 1% every year?
- Suppose that burning coal makes up 40% of total emissions and a new carbon capture technology promises to sequester GHG from coal emissions. However, in the first year, the technology can only sequester 0.5% of annual coal emissions. If in every subsequent year, the technology can capture 10% more than it did the previous year, how many years will it take for global emissions to decrease by 10 billion metric tons per year? (Begin your calculations from a total of 50 billion metric tons per year in 2023, and assume that all other emissions don’t change.)

*—Max Levy*

*New Scientist*, December 6, 2023.

In labs around the country, researchers are working on building computers unlike any you have ever used. Traditional computers store and process information as binary digits, or “bits,” 0 and 1. A quantum computer uses principles from quantum physics to carry out calculations at speeds far beyond traditional computers thanks to “qubits” which can be a combination of both 0 and 1 simultaneously. However, when researchers link together many qubits to create a more practical machine, computational errors become more common. In this *New Scientist *article*,* Karmela Padavic-Callaghan writes about a promising breakthrough that benefits from a special type of qubit. “Using thousands of rubidium atoms cooled to near absolute zero, they achieved a record-breaking 48 logical qubits simultaneously, over ten times the previous high,” Padavic-Callaghan writes. “This achievement marks a crucial step toward practical quantum computing.”

**Classroom Activities:** *quantum mechanics, probability*

- (Mid level) Watch this PBS video about the mathematics of quantum computers.
- Explain in your words how a quantum computer works.
- What is special about the mathematics of a qubit compared to a traditional bit?
- In quantum computing, what role does probability play in the measurement of qubits, and how does it affect the outcome of computations?
- If you have a quantum system with 3 qubits, how many different possible states can it represent in superposition?

- (Mid level) Read this article from
*IEEE Spectrum*. If two qubits are entangled, the first qubit can be in a superposition of states $|0\rangle$ and $|1\rangle$, and the second qubit will always have the opposite state of the first one due to entanglement, how many possible states of these entangled qubits exist? What are the possible states?

*—Max Levy*

*BBC*, December 31, 2023.

“Laura is a sprinter,” writes Kit Yates for *BBC.* “Her best time to run 100m (328ft) is 13 seconds, how long will it take her to run 1km (3,280ft)?” If you multiplied 13 seconds by 10 to answer this question, you’re not alone, says Yates. But view this as more than a multiplication problem with set dressing, and you might notice an issue with that strategy: While Laura’s 100-meter sprint time is well behind the times recorded in *World Athletics *list of top 100-meter times for women, a 130-second kilometer is unheard of. “The linear answer would see Laura utterly destroying the world record for running 1km,” writes Yates. That’s just one example of a scenario in which people may be inclined to assume, against the evidence, that things are linear. In this article, Yates argues that this “linearity bias” keeps us from viewing things accurately, with unwanted consequences.

**Classroom Activities: ***nonlinear equations, mathematical modeling, exponentials, statistics*

- (All levels) On the Desmos graphing calculator, let the horizontal axis measure race distance and the vertical axis measure race time. Using data from
*World Athletics*, plot the women’s world records for all races less than 1500 meters.- Using these instructions, find a line of best fit of the form $y = ax + b$. How close can you get $R^2$ to 1? (Hint: Adjust the slider settings so that the step size is 0.001, and $a$ ranges between 0 and 0.5.)
- Now try finding a quadratic curve of best fit, of the form $y = ax^2 + bx + c$. Now how close can you get $R^2$ to 1? Assuming this equation applies to Laura’s running time, how fast would she run the 1000-meter race?
- According to François Labelle, a better formula has the form $y = ax^b$. Using this form, how close can you get $R^2$ to 1? According to this formula, how fast would Laura run the 1000-meter race?
- Do you think these formulas would apply to Laura? Why or why not?
- (High level) Using least squares, calculate the linear and quadratic equations of best fit, and compare the $R^2$ values.

- (Mid level) Yates brings up the example of debt, which grows exponentially over time. Suppose you take out a $100 loan, and every year it accrues 20% interest.
- Without using a calculator or writing anything down, guess how much the debt will have grown after 10 years if you don’t pay anything back.
- How would you calculate how much you will owe after one year if you don’t pay back the loan? Find a general formula for how much you will owe after $n$ years.
- Using your formula, after 10 years, how much will you owe? Was your initial guess based on linearity bias?
- How long will it take for the debt to double?

*—Leila Sloman*

*Scientific American*, December 7, 2023.

In 1979, Adi Shamir came up with a technique which allows a group of people to crack a code, but only if they all work together. In this article, Manon Bischoff walks us through how it works, using a mother who wants to leave an inheritance to her five sons, but who also wants to make sure they split it fairly.

**Classroom Activities:** *finite fields, fitting curves*

- (Mid level) Bischoff explains how in a simpler scenario with two or three sons, the woman could use linear or quadratic curves to make sure the sons have to work together to unlock the combination to her safe. We’ll repeat her example, with a quadratic curve instead of a linear one.
- Have students read the article to understand how this technique works.
- Now students will fit a quadratic curve of the form $y = ax^2 + b + c$. The combination to the safe is the coefficients $(a, b, c)$. Give them two coordinate pairs: $(1, 2)$ and $(-1, 0)$ are both on the curve. What can students figure out about the coefficients based on this information?
- After a few minutes, give the third coordinate: $(0, -2)$. This should be enough to calculate the combination.
- This technique works for any curve with three parameters, as long as you can solve for the parameters. Have students come up with their own combination and parametrized curve, and trade problems with a friend.

- (Mid level) Bischoff notes that if you know the parameters of the curve are integers, that might make the technique less secure. Does this apply to the previous problem? Why or why not?
- (High level) Shamir’s technique uses finite fields. Learn about the modular arithmetic of finite fields using this article from NRICH. Repeat the previous exercise mod 5. How does the arithmetic change?

*—Leila Sloman*

- How magicians control flip of a coin

*Mint*, December 28, 2023. - The Year in Math

*Quanta Magazine*, December 22, 2023. - Asia’s Rising Scientists: Mayuko Yamashita

*Asian Scientist Magazine*, December 21, 2023. - A Close-Up View Reveals the ‘Melting’ Point of an Infinite Graph

*Quanta Magazine*, December 18, 2023. - The Secret Lives of Numbers — a book to make you love maths

*Financial Times*, December 15, 2023. - DeepMind AI outdoes human mathematicians on unsolved problem

*Nature*, December 14, 2023. - Mathematicians Prove the “Omniperiodicity” of Conway’s Game of Life

*Discover*, December 13, 2023. - Tiny balls fit best inside a sausage, physicists confirm

*New Scientist*, December 12, 2023. - A Triplet Tree Forms One of the Most Beautiful Structures in Math

*Quanta Magazine*, December 12, 2023. - How to befriend Maths? Get crafty & play with it

*Times of India*, December 11, 2023. - What Can You Do With an Einstein?

*The New York Times*, December 10, 2023. - How Quantum Math Theory Turned into a Jazz Concert

*Scientific American*, December 6, 2023. - Mathematician Answers Geometry Questions From Twitter

*WIRED*, December 5, 2023.

A press release from UCSD about progress in Ramsey theory was picked up by Rebecca Dyer for ScienceAlert.com with the title: “Mathematicians Crack a Century-Old Problem That’s Perfect For Your Next Party.” The mathematicians in question are Sam Mattheus (Vrije Universiteit Brussel) and Jacques Verstraete (University of California, San Diego). In their preprint, Mattheus and Verstraete write, “Ramsey Theory is an area of mathematics underpinned by the philosophy that in any large enough structure, there exists a relative large uniform substructure.” The simplest example is this question: Suppose you have $n$ points. Join every pair of 2 points by a line colored either red or blue. How large does $n$ have to be so that in any such configuration three of the points are the vertices of a red triangle or three are the vertices of a blue one? Here, the large structure is the randomly-colored $n$-vertex graph, and the uniform substructure is a monochromatic triangle.

More generally, the *Ramsey number* $r(s,t)$ is the smallest $n$ such that if $n$ points are joined two by two by red or blue lines, then there are at least $s$ which are joined two by two by red lines, or $t$ which are joined two by two by blue lines. In other words, $r(s,t)$ is the smallest $n$ such that a red/blue coloring of the *complete graph* on $n$ points must contain a complete red graph on $s$ points or a complete blue graph on $t$ points. The example we just saw was $r(3, 3)=6$. The British mathematician, philosopher and economist Frank Ramsey (some details of his short, intense life are on the University of St. Andrews website) proved in 1930 that for any $s$ and $t$ this number is finite. Finding its exact value, beyond $r(3,3)=6$, turned out to be difficult. The ones currently known are tabulated here, with their sources. They go up to $r(3,9)=36$ and $r(4,5)=25$.

Paul Erdös conjectured an asymptotic formula for a lower bound on $r(4,t)$ as $t$ goes to infinity and offered a \$250 prize for a proof. It remained unclaimed until Mattheus and Verstraete’s calculation this year: they proved that there exists a constant $c$ such that, for $t$ sufficiently large,

$$r(4,t) \geq c\frac {t^3}{\log^4t}.$$

This confirms Erdös’s intuition and nails the exponent in the denominator, which he had left unspecified. To give an idea of the complexity of the problem, Verstraete constructed this image of a red/blue coloring of the complete graph on 24 points which contains neither a complete red graph on 4 points nor a complete blue graph on 5. This image represents for $r(4,5)=25$ what the pentagonal colored graph at the start of this item does for $r(3,3)=6$.

Suppose you are working in a medical lab, and you want to test the efficacy of an antibiotic against some type of single-cell organism. Starting with a solution containing a known concentration of the target cells, you mix in some concentration of the drug, give it time to work, and then measure how many of the cells are still viable. The standard way to perform this measurement is the *CFU assay:* you culture a given volume of the post-treatment solution on some medium and count the number of distinct colonies, assuming that each colony grew from a single surviving viable cell, or *colony-forming unit* (CFU).

The problem with this technique is the counting. If there were too many CFUs in your sample the colonies will run together and be impossible to count; if there were too few, your count will be statistically unreliable. The problem is solved (as explained, for example, in this problem set from the UVM Microbiology and Molecular Genetics department) by diluting the antibiotic to 1/10, 1/100, etc. of the original concentration — a *dilution series *— and then growing samples each time, expecting that one of them will yield a “reasonable number” (in the range 30-300) of colonies.

A high-throughput and low-waste viability assay for microbes (*Nature Microbiology*, November 2, 2023) states that the CFU assay “has remained the gold standard” for measuring viability, but proposes another method that avoids some of its “time-intensive and resource-consuming” aspects. This new method is the Geometric Viability Assay (GVA). The authors, a University of Colorado, Boulder team led by Christian T. Meyer, Anushree Chatterjee and Joel M. Kralj, use the geometry of a cone and the easy availability of cones in the laboratory (polypropylene pipette tips) to replicate the effect of the dilution series at one shot.

The authors argue that the chance of finding a colony in the pipette at $x$ units from the tip is proportional to the volume of the infinitesimal slice between $x$ and $x+dx$ and therefore to $x^2$. This means the assayer can find a “reasonable number” to count by scanning along the pipette and choosing $x$ appropriately. If $N$ is the number of colonies counted in an interval around $x$, then the GVA estimate of the concentration of cells that survived is $N$ divided by the volume of the slice corresponding to that interval, which they calculate by integration as usual. (This is why they make Pre-Meds take Calc I.)

We visited this topic in May, but the mathematical unrest out there continues. The *Wall Street Journal* Editorial Board ran California’s New Old Math (subtitle, “Parents fight to restore eighth-grade algebra in San Francisco’s public schools”) on November 12, 2023. They give the background, going back to the decision in 2014 by the San Francisco Unified School District to stop offering Algebra I in the eighth grade. As they explain it, this was done “in the name of—what else?—equity.” The theory was that separating students into more or less advanced curricular levels would widen the gap between the performance of minority students and their more socio-economically advantaged peers. They cite a March 2023 report by three Stanford researchers (one professor and two graduate students) evaluating the actual results. The *WSJ* editorial does not mention one of the main findings of the report, which was a sharp decrease in AP Calculus enrollment *especially for Asian students*.

The editorial ran in anticipation of a rally supporting a ballot measure that would restore Algebra in grade 8. The editorial concludes by stating that the rally “is an example of parents no longer blindly accepting what school boards, administrators and teachers’ unions tell them,” and that the School Board, meeting again in February, will “face the wrath of the SF Guardians” (who organized the rally) if they don’t vote for the restoration.

A personal, nuanced view of the situation was given by Julie Lynem on the news-site CalMatters: “My son’s decision to retake algebra made me rethink California’s new approach to math” (November 8, 2023). Her son had taken and passed Algebra I in eighth grade, but was dissatisfied with his understanding of the subject and opted to repeat it, so he will likely have not taken calculus when he graduates from high school next year.

That’s the subtext in the discussions about Algebra I in grade 8: *calculus*. For many families, AP Calculus has become an indispensable item in their children’s struggle for admission to a prestigious college. (For Asian-American parents the pressure may be even higher, as documented in this November 26, 2023 article in the *Los Angeles Times*.) But students who take Algebra I in grade 9 have little chance of being able to complete AP Calculus before graduating from high school.

The pressure is not an illusion. The non-profit education news site *The74* posted on November 6, 2023 Even as Caltech Drops Calculus Requirement, Other Competitive Colleges Continue to Expect Hard-to-Find Course. The website *BestColleges* has had a page up since March, 2022: Want to Get Into Harvard? Ace Calculus, while the non-profit *JustEquations *put out a September, 2022 report “Calculating the Odds: Counselor Views on Math Coursetaking and College Admissions.” The executive summary notes that even when colleges explicitly drop the calculus requirement, high school counselors “have concluded, based on their own experiences, that the course is at least strongly expected at highly selective schools.” The report quotes a high school counselor: “It is deeply problematic that college admission offices—many of which are entirely unaware of how actual math content, sequencing, programs work—use calculus as a benchmark for college admission.”

Lynem’s son, now an algebra tutor, doesn’t regret going for mastery instead of what Lynem calls “the race to take calculus”. Lynem’s story reviews the recent history of mathematics education in California, including the new state-wide Mathematics Framework adopted by the California State Board of Education last July. She spoke with the Board president, Linda Darling-Hammond, and quotes her: “Math has been taught as a set of rules and procedures, rather than helping kids use the math in real-world contexts so that students can say, ‘I deeply understand what I’m doing with this, and it’s giving me answers to things I care about.'”

The controversy about Algebra I in grade 8 has nothing to do with solving and graphing equations, etc. and their appropriateness for students at this level. It arises because “Algebra I in grade 8” is seen, *correctly,* by many parents as the first crucial step in the process that will get their children into a prestigious college and thereby preserve or improve their standing in society.

*The New Yorker*, November 10, 2023.

When people say they know how to tie a necktie, they generally think of the one knot—*maybe* two or three—in their repertoire. Boris Mocka thinks of over fifteen hundred. *New Yorker* contributor Matthew Hutson recently introduced Mocka—a doorman in Hutson’s building—to experts in topology, a branch of mathematics. The mathematicians were fascinated by Mocka’s passion for creating new variations of loops and swoops. In this article for *The New Yorker*, Hutson writes about Mocka’s life, his designs, and about how the art of a necktie intersects with an important research area in mathematics called knot theory. “The Gardenia looks like a flower; the Wicker and the Mockatonic look like origami. The Riddler looks like a question mark, and the Exousia requires more than one tie,” writes Hutson. “Many more math techniques would be needed to describe Mocka’s art.”

**Classroom Activities:** *topology, knot theory*

- (Mid level) Watch this Veritasium video about knot theory, and answer the following questions:
- Why are all knots studied by mathematicians “closed loops?”
- What makes two knots mathematically different?
- What is the knot equivalence problem, and why is it so hard to solve?
- What is a prime knot, and how is it different from a composite knot?

- (Mid level) Read the
*New Yorker*article. Based on your reading and the Veritasium video, what is one reason why it’s hard to define Mocka’s ties with knot theory? - (Mid level) Research 3 different knots that you can tie with neckties, or rope. Create them, and then identify which of these knots are cinquefoil, trefoil, or other categories you have learned in the lesson. Think about what the category for each means about their mathematical relationship. For example: the “four-in-hand” and “Windsor” knots are both trefoils, therefore they are mathematically identical.

*—Max Levy*

*The Guardian*, November 5, 2023.

Mathematician Marcus du Sautoy has a message: “In love and economics, business and games, if you know your maths, you’ll end up the winner.” In an article for *The Guardian*, du Sautoy shares mathematical strategies for winning in a wide variety of situations, including some that don’t sound mathematical at first. Euler’s number $e$, for example, shows up in a strategy for finding the right number of people to date before settling on “the one.” Tools from calculus can help an entrepreneur figure out how many units to make of a new product. And statistically, a trivia buff playing “Who Wants to Be a Millionaire?” is better off asking the audience for help than phoning a friend. These and other real-world scenarios have rules and logical patterns that du Sautoy says people can tap into, mathematically, to succeed.

**Classroom Activities:** *economics, game theory, probability*

- (All levels) Read the section of the article titled “How to win at economics.”
- If you were Player A in this scenario, how much money would you offer? If you were Player B, would you accept the split suggested in the article? Do you agree that the golden ratio makes the split feel fairer?
- For another example of Nash equilibrium, watch the Khan Academy video on the prisoners’ dilemma.
- The video illustrates that from Al’s perspective, it’s better to confess, whether or not Bill does so. The same is true for Bill. But when they both confess, they get more jail time than when they both deny. Discuss: How can it be that when Al and Bill make their best individual decisions, it doesn’t lead to the best group outcome? Does this surprise you? Can you think of any other situations where something like this might occur?

- (All levels) Read the section of the article titled “How to win at games.” The article says that if you roll two dice, the most common total you will get is a $7$ because there are six ways to make $7$ out of two dice: $1+6$, $2+5$, $3+4$, $4+3$, $5+2$, and $6+1$.
- How many ways are there to get a total of $2$ when rolling two dice? How many ways are there to get a total of $3$?
- When you roll two dice, how many different outcomes are possible in total? What strategies did you use to try to figure this out?
- (High level) If you roll two dice, what is the probability of rolling a total of $7$?

*—Tamar Lichter Blanks*

*Stat News*, November 30, 2023.

Currently, the Food and Drug Administration allows labs that produce and then use their own diagnostic tests to proceed free from the usual regulations. This fall, the FDA proposed changing this policy, citing the tests’ frequent false positives. But the FDA, and the public, may find that they keep getting false positives even when diagnostic testing is subject to the strictest quality standards. In this article for *Stat*, mathematician Manil Suri and epidemiologist Daniel Morgan explain that false positives are par for the course when it comes to medical testing, especially when the condition being tested for is rare. “Practically *all* tests, not just [lab-developed tests], carry the risk of false positives, which can render the results effectively useless when the condition is rare enough,” they write.

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

- (Mid level) Learn about sensitivity and specificity of diagnostic tests with this short interactive lesson from
*SchoolYourself.* - (Mid level) Navigate to Morgan’s online calculator, which shows the probability that a patient has a disease in a variety of situations.
- Under “What disease are you testing for?” select “Flu (Influenza).” Under “What is the pre-test probability?” select “Non-Influenza Season.” Under “Symptoms,” select “No clinical symptoms.”
- Before selecting any other options, scroll down to “What Test will you order?”, and read the numbers for sensitivity and specificity. Based on the “Chance of Flu” number given on the left-hand side, try to figure out how likely it is that a patient who has
**no symptoms**actually has the flu if they test positive on each test- during flu season
- not during flu season.

- Now, work through all three tests. For each test, select “Result > If positive.” Look at the “Chance of Flu > After test” number. Does the result surprise you?
- Repeat the exercise during flu season. How much do the numbers change? Why do you think they changed?

- (Mid level) Now, create your own 2 x 2 grid as in the
*SchoolYourself*lesson, with the columns “Actually has the disease,” “Does not have the disease,” and the rows “Tests positive,” and “Tests negative.”- According to Morgan’s data, during non-flu season, 0.5% of people have the flu, and during flu season, about 4% of people have the flu. Suppose you test 1,000 random people. Using the sensitivities and specificities listed on Morgan’s calculator, fill in the squares of your grid for each test, during flu season and not during flu season.

*—Leila Sloman*

*NPR Short Wave*, November 27, 2023.

When John Urschel started his Ph.D. in mathematics at the Massachusetts Institute of Technology, he already had a high-profile job playing football for the Baltimore Ravens. In this episode of *Short Wave*, Regina Barber interviews Urschel about how he balanced those two demanding careers. and about his current research in linear algebra.

**Classroom Activities: ***linear algebra*

- (Mid level, Algebra II) Barber describes linear algebra by example: As the type of math used to solve systems of equations such as $x + y = 3$, $x – y = –1$.
- Read Section 5.2 from this online textbook from
*LibreTexts*to learn how to solve a system like this using a method called “substitution.” Do exercises 5.2.1, 5.2.2, 5.2.3, and 5.2.19. - Now read Section 5.3 to learn a method called “elimination”. Do exercises 5.3.4, 5.3.7, 5.3.10, and 5.3.22.
- Solve Barber’s example using both substitution and elimination. Show your work.

- Read Section 5.2 from this online textbook from
- (High level) In the episode, Urschel talks about “higher-dimensional versions of lines.” Before viewing the next activity, discuss the following questions in small groups, then answer individually.
- What does he mean by “higher-dimensional”?
- How would you define a “higher-dimensional version of a line”? Reminder: You can think about a line as a type of curve, or you can think about it in terms of the equation that defines it. You may find one of these perspectives easier than the other!

- (Mid level, Algebra II) Generally, a “higher-dimensional line” refers to adding more variables into a linear equation. For example, $x + y + z = 1$ has three variables, instead of just two, and defines a plane in three-dimensional space.
- Plot this plane in the GeoGebra 3D calculator. Do you think this deserves to be called a higher-dimensional line? Why or why not?
- Play with the coefficients of the plane, and observe how the image in GeoGebra changes. How does the idea of a line’s “slope” generalize to higher dimensions?
- Solve the following system of equations using both substitution and elimination.

$$x + y + z = 1, \quad x + y – z = 1, \quad x + 2y + 3z = 3.$$

*—Leila Sloman*

*The Conversation*, November 1, 2023.

Cancer is not one disease. Cancer differs in the body parts it resides in, the genetic mutations that propel it, how it interacts with the body, and who it infects. This means that oncologists must think about cancer in many different ways. One of those ways is math. Of the five unconventional ways of thinking about cancer described by Vivian Lam in *The Conversation*, some are biological, like evolution and inflammation. But in the mathematical lens, one oncologist tells Lam that the mathematical randomness occurring in cancer biology, called epigenetic entropy, may be key to learning how to prevent cancer. “Epigenetic entropy shows that you can’t fully understand cancer without mathematics,” he says.

**Classroom Activities:** *stochasticity*

- (All levels) The mathematics of cancer mentioned in the article deals with stochasticity, a word for randomness. Create a table with 100 rows and two columns. Each row represents the outcome of a coin flip—either “heads” or “tails.” In the first column, write what you expect 100 successive coin flip results to look like. Imagine that your goal is to convince someone that your list is really a list of 100 coin flips. In the next column, do 100 coin flips and record the outcome on each row.
- Compare the two columns. How are they similar? How are they different? Did anything in the real coin flip surprise you?
- Now, watch this video about stochasticity, which discusses the results of an experiment very much like the one you just performed on yourself. Discuss what you’ve learned.
- Revisit the October digest on coin flipping for alternative activities and videos on coin flip outcomes and manual attempts to mimic them.

- (All levels) Play this educational game about epigenetics, which will teach you the basic science of what is being randomized in this area of cancer.
- (Mid level) The article describes how stochasticity appears in other processes like the stock market and epidemiology. Take a few minutes to research how stochasticity plays a role in both fields. What parameter is random in each field? How can we measure the randomness? Based on your reading, describe how the mathematics of epigenetics is related to the math of epidemiology and finance.
- In what ways do the examples seem different?
- Could researchers of one field (e.g. stochasticity in epidemiology) learn from the work and discoveries of researchers in another (e.g. stochasticity in cancer epigenetics)? Discuss why or why not as a group.

*—Max Levy*

- What is the largest known prime number?

*Live Science*, November 23, 2023. - Pierre de Fermat’s Link to a High School Student’s Prime Math Proof

*Quanta Magazine*, November 22, 2023. - Is the Lottery Ever a Good Bet?

*Scientific American*, November 17, 2023. - Simple Formula Makes Prime Numbers Easy, but a Million-Dollar Mystery Remains

*Scientific American*, November 16, 2023. - After 90 Years, Mathematicians Finally Solved the Most Notorious Ramsey Problem

*Popular Mechanics*, November 7, 2023. - Finding my culture in ‘Hidden Figures’

*The Slate*, November 7, 2023. - Mathematical Genius Katherine Johnson to be Inducted into National Aviation Hall of Fame Class of 2024

*WV News*, November 7, 2023.

- “Unusual Geometries” at Oklahoma State
- Numbers in the human brain
- Polymer helicity and knot chirality

A posting at OKState News on October 17, 2023 announced the opening of “Unusual Geometries,” an exhibition at the Oklahoma State University Museum of Art (runs until December 16). The show promises its guests “the hidden beauty and playfulness found within the world of geometry.” Among the art on display is an image with the title “Cohomology fractal for the SnapPy manifold s227,” by David Bachman (Pitzer College, Claremont), Saul Schleimer (Warwick) and Henry Segerman (OKState).

What does this picture show? “SnapPy manifold s227” is the name of a certain mathematical object: it’s a finite-volume three-dimensional space with a hyperbolic metric. The authors have put together a YouTube video explaining in elementary terms what these words mean, and how the image corresponds to a cohomology class. They work their way up from two dimensions, where things are much easier to visualize. There we talk about area instead of volume. An example of a 2-dimensional space with finite area is the sphere, or more interestingly, the torus (a donut shape).

One way to get a (1-dimensional) *cohomology class* on the torus is to start with a closed curve on the surface, like the black one in the figure below. Give it an orientation: decide which direction along the curve you will count as “positive.” The arrow on the black circle shows which orientation is chosen in this example. Then pick any other closed, oriented curve on the torus. You can assign to this second curve a whole number by summing the *intersection numbers* of the two curves. Each intersection contributes either $+1$ or $-1$ depending on what direction the curves are headed when they cross each other, as illustrated in the figure. The intersection sum associates a whole number to each closed oriented curve on the torus. This association is the cohomology class.

Bachman, Schleimer, and Segerman’s image is of the inside of a three-dimensional space. There the analogue of a closed curve is a closed, oriented surface, and like our black curve this surface defines a 1-dimensional cohomology class. Their space is *hyperbolic*, which means that it has constant negative curvature. In two dimensions, such a space would be saddle-shaped everywhere. The area of a circle around such a point increases more rapidly with the radius $r$ than the $\pi r^2$ we’re used to in the plane. If every point is a saddle point, this area blow-up makes a finite-area surface of constant negative curvature impossible to realistically portray on the page, even though there are many of them.

In their video, Bachman, Schleimer, and Segerman ask you to imagine that you are living in this space, and that when you look through that surface in the positive direction, everything behind it looks darker; whereas looking in the negative direction makes everything appear lighter. The picture on exhibit is what you would see.

Additionally, the authors have an expository paper about the construction, written for the participants in the interdisciplinary Bridges Conference on math and art, 2020.

A news item in *Nature* titled “Your brain finds it easy to size up four objects but not five — here’s why” highlights an October 2 paper from *Nature Human Behaviour*. *Nature *staff writer Mariana Lenharo reaches back in the *Nature* archives for early research on the topic. In Volume 3 (1871) W. Stanley Jevons published “The Power of Numerical Discrimination”, including the table replicated here.

As Jevons’s experiment shows, our ability to grasp immediately how many objects we are faced with (called *subitizing*) works up to four items and becomes more and more unreliable afterwards. Lenharo tells us how the authors of the *Nature Neuroscience* article, a group of German epileptologists and physiologists led by Florian Mormann (Bonn) and Andreas Nieder (Tübingen), were able to improve on Jevons by using a larger number of subjects (17) and technology that he likely could never have imagined. For instance, Mormann and Nieder’s group had access to patients with microelectrodes implanted in their brains. This allowed the team to monitor the activity of individual neurons. It has been known for some time that single neurons in the primate brain can encode numbers of items (a review article published in 2009 by Nieder and Stanislas Dehaene cites work dating back to 2002). In the new experiment, subjects were asked to guess whether the number of dots shown on a screen was odd or even while researchers tracked neural activity.

The new study shows that the neurons associated with the numbers 1-4 are different from the others. As Lenharo explains it, “neurons specializing in numbers of four or less responded very specifically and selectively to their preferred number. Neurons that specialize in five to nine, however, responded strongly to their preferred number but also to numbers immediately adjacent to theirs.” She quotes Nieder: “The higher the preferred number, the less selective these neurons were,” giving a nice cellular-level perspective on Jevons’s experimental results. The article’s abstract ends with the suggestion that this difference may be connected to “capacity limitations” in our attention and working memory. These limitations are probably familiar to everyone: trying to drive and eat an ice-cream cone while discussing relativity is not a good idea.

First the definitions.

A spiral curve in 3-space has right or left *helicity*. Right helicity is the same kind as a standard corkscrew: when you turn it to the right (clockwise) it goes forward. For a point on the curve, moving forward (in either direction) means rotating clockwise. The mirror image of a curve with right helicity is a curve with left helicity. In plumbing, screws with this structure are called *left-threaded* or *reverse-threaded*.

When you take the mirror image of certain knots, their over- and under-crossings get interchanged: Two threads that cross one another will switch positions. The thread that was on top will now be on the bottom, and vice versa. This is a topologically different object; these knots can’t be deformed into their mirror images. These knots are said to have *chirality* (literally, handedness): they exist in a left-handed and a right-handed form.

How is handedness calculated? The string bearing the knot is first given an orientation (it doesn’t matter which). Using the orientation, an index $\pm 1$ is assigned to each crossing in a 2D drawing of the knot: $+1$ if the rotation from the positive direction in the “over” strand to that in the “under” is counter-clockwise, and $-1$ if it is clockwise. If the sum of these indices is non-zero, the knot is *chiral* (has a handedness), left if it is negative, right if it is positive.

These related concepts appear in “Can Polymer Helicity Affect Topological Chirality of Polymer Knots?”, published in *ACS Macro Letters* (American Chemical Society) on January 27, 2023 and picked up in *Nanowerk News* on October 4. The authors, a group of physicists and chemists in Mainz led by Peter Virnau and Kostas Daoulas, used computer simulations to investigate the interplay when long polymer chains of one helicity or another (or none) spontaneously form knots. The results are summarized in this graph printed in their abstract. The strength of the helicity strongly predicts the handedness should a particular knot occur, except that there is a striking difference of behavior between the two five-crossing knots $5_1$ and $5_2$.

The authors explain the difference in behavior between $5_2$ and the torus knots $3_1$ and $5_1$ in terms of “braids” naturally occurring when the polymer crosses itself.

]]>The Game Theory of the Auto Strikes

*Wired*, October 1, 2023.

On September 15, the United Auto Workers — a labor union that represents workers from the automotive and aerospace industries — began a strike against Ford, General Motors, and Stellantis. The union was holding out for raises, pensions, and automatic cost-of-living adjustments to their wages. The companies, meanwhile, claimed those demands were too costly. In this article for *Wired*, Aarian Marshall walks through what the mathematical field of game theory — which studies scenarios in which rational players have competing interests — can contribute to the analysis of the strikes.

**Classroom Activities: ***game theory, modeling*

- (All levels) Try this online lesson plan from Population Education simulating two simple game theory scenarios.
- (Mid level) After playing each game, discuss in small groups what the two games in the lesson plan have in common with a strike scenario, and how they differ.

- (Mid level) Read the
*Wired*article.- Brainstorm in small groups everything that should go into “the pie.”
- (Mid level) Come up with a game using poker chips that captures the key features of the strike scenario. First, brainstorm as a class the features that the game should include. If the discussion becomes too complicated, remind students that mathematical models balance realism with simplicity, and ask for their suggestions of how to do so.
- How many teams will the game have? What roles will the teams play, and what roles will individuals within the teams play?
- Who is in charge of distributing the poker chips? Is it one of the teams, or someone else?
- What constitutes cooperation or defection? What are the effects of cooperation and defection?
- What should happen in each round? When should the game end?
Break into small groups to work out the game’s details. Have each group outline the best strategy for each side.

- (Mid level) On October 30, the UAW entered a tentative agreement with the last of the three companies. (The agreement still needs to be approved by union members.) Read the linked articles about the agreements and compare to the prediction made by Marc Robinson in the
*Wired*article. What did he get right? What did he get wrong? Do you think there was any information he missed?

*—Leila Sloman*

The mathematical theory that connects swimming sperm, zebra stripes, and sunflower seeds

*Popular Science*, September 27, 2023.

One peculiar pattern arises over and over in nature, appearing in plants, animal coloration, and even sand on the beach. Unique stripes and spots emerge mysteriously. The phenomenon occurs when two chemicals or forces compete, interact, and move around one another. This process is called “reaction-diffusion” for the way chemicals react with one another and diffuse, or spread, throughout the environment. What results are “Turing patterns,” named for mathematician Alan Turing, who discovered the principle at play. In recent work, scientists found a surprising new realm where reaction-diffusion appears—the movement of sperm. “Nature replicates similar solutions,” a researcher told *Popular Science* reporter Laura Baisas. The team noticed the same motion in bull sperm and green algae. “We show that this mathematical ‘recipe’ is followed by two very distant species.” In this article, Baisas describes how new mathematical models helped unpack the mechanics of sperm’s flagellar movement. They learn that the waviness emerges spontaneously, and the findings could have applications from robotics to fertility.

**Classroom Activities:** *reaction-diffusion, simulations*

- (Mid level) Watch The Mathematical Code Hidden In Nature and describe, in your own words, how you think a “mathematical code” can explain zebra stripes.
- (Mid level) Read this tutorial on reaction-diffusion models, then play with this reaction-diffusion simulator.
- Recreate the “mitosis” and “coral growth” simulations described in the tutorial.
- Describe in your words what the “kill rate” and “feed rate” would correspond to in leopard spots. Would the ratio of kill:feed be higher for leopard spots or zebra stripes?

*—Max Levy*

Coin flips don’t truly have a 50/50 chance of being heads or tails

*New Scientist*, October 17, 2023.

Flipping a coin seems like the perfect example of a fair random game, with each flip having a 50/50 chance of landing on heads or tails. But according to a recent experiment, there is a bias. In a study posted online this October, researchers found that a flipped coin lands on the same side it started on about 50.8% of the time. In practice, that means that if a coin starts with its heads side up, then you are slightly more likely to win if you bet on heads. To get their data, the researchers recorded the outcomes of 350,757 coin flips. This experimental result confirms an earlier theoretical model, as Matthew Sparkes describes in this article for *New Scientist*.

**Classroom Activities:** *probability, coin flipping*

- (All levels) Have each student flip a coin 20 times. Each time, have them record the starting position of the coin and what side it lands on. They should end up with two sets of numbers: How many times the coin landed on heads vs. how many tails, and how many times it landed on the same side it started on vs. the opposite side.
- Have each student calculate the percentage of their flips that landed on heads, as well as the percentage of their flips that landed on the same side.
- Next, add up a tally for the whole class: How many heads (H) and how many tails (T)? How many flips landed on the same side (S) and how many on the opposite side (O)?
- Have each student calculate the total percentage of flips that landed on heads, as well as the total percentage of flips that landed on the same side.
- Discuss: How close to 50/50 were each of the results? What do you think would happen if everyone flipped a coin 1000 times instead of 10 times?

- (All levels) Without actually flipping a coin, try to write down a random sequence of 20 imaginary coin flips (for example, HTHHT…). Then, watch this video about random sequences of coin flips from Numberphile.
- Looking back at your sequence, can you detect any predictable patterns or trends?
- Compare your imagined sequence to the 20 coin flips you recorded in the first activity. Are there any noticeable differences?
- Discuss: What do you think are some challenges that make it difficult to generate randomness?

*—Tamar Lichter Blanks*

Don’t Worry About Global Population Collapse

*Bloomberg Opinion*, September 30, 2023.

Last year, the United Nations announced that the world’s population passed 8 billion people, and current projections predict a peak at 11 billion before next century. However, it’s hard to trust whether these estimates will prove correct. Innovations that help the planet cope with population growth are difficult to predict, and it’s essential to keep the margins of error in mind. It’s even harder to tell whether population reduction is good or bad. It could be good news for the future, since Earth’s resources like land, food, and water are finite. “And yet alarms are sounding,” writes F.D. Flam for *Bloomberg*. “While environmentalists have long warned of a planet with too many people, now some economists are warning of a future with too few.” In this article, Flam speaks with demographers and a mathematician about the debate around population growth.

**Classroom Activities:** *error margins, growth rate*

- (All levels) Complete population growth activities from PopEd, such as “Stork and Grim Reaper”, or a free trial of this population growth game by Labster.
- (Mid level) Watch this Khan Academy lesson on exponential growth. What other examples of exponential growth in nature or society can you think of?
- (High level) Simulate population growth under the conditions below using spreadsheet software, then answer the questions that follow. Assume that we can describe global population as $P = P_0 e^{rt}$, where $P$ is total population (in millions); $P_0$ is initial population (in millions); $r$ = growth rate; $t$ = time in years.
- If the initial population is 8 billion ($P_0 = 8000$), and the rate, $r$, is 0.01, in how many years will the population pass 11 billion?
- Now suppose that the rate, $r$, decreases by a factor of 1% every year. In how many years will the population pass 11 billion? Discuss why.
- Based on the reading, what would explain the rate decreasing with time?
- The model we assumed here is greatly simplified. How would the math have to change if we wanted the population to peak and then fall?

*—Max Levy*

Joseph J. Kohn, Who Broke New Ground in Calculus, Dies at 91

*The New York Times*, October 24, 2023.

Scientists Say: Imaginary Number

*Science News Explores*, October 2, 2023.

In this obituary for the *New York Times*, Kenneth Chang covers the work of Princeton University mathematician Joseph J. Kohn, who passed away last month. Kohn studied functions that were defined over complex numbers — a set that includes familiar numbers like 2, $\pi$, or $\sqrt{5}$ as well as the imaginary square root of $-1$. “They’re called ‘imaginary’ because they don’t count or measure things, the way real numbers do,” Katie Grace Carpenter wrote earlier in the month in an explainer of imaginary numbers for *Science News Explores*. Despite their counterintuitive nature, imaginary numbers are crucial for studying electric circuits and quantum mechanics, she adds.

**Classroom Activities: ***complex numbers, calculus*

- (Mid level) Read the
*Science News Explores*article. Calculate the following numbers, leaving them in the form $a + bi$ where $a$ and $b$ are real numbers.- $(3i)^2$
- $(1 + i)^2$
- $i^4$
- $(1 + i)^3$

- (Mid level) The complex numbers include the real numbers, the imaginary numbers, and any sum of a real number plus an imaginary number. They can be plotted on a graph as pairs of coordinates. To see how, read the first two sections of this lesson from
*Maths is Fun*.- On the complex plane, plot the numbers you calculated in the previous exercise.
- Plot the numbers $1$, $i$, $-1$, and $-i$. Now multiply them all by $i$, and plot the results. What is the effect of multiplication by $i$?
- Try the same exercise, but this time multiply the results by ${1 + i \over \sqrt{2}}$. What happens?

- (High level) In this article for
*Scientific American*, three researchers describe how essential complex numbers are in quantum mechanics, the physics of particles that are extremely small. Read the article and reflect: What did the researchers prove in the work they describe? What is the role of complex numbers in physics? What questions do you have?

*—Leila Sloman*

**Some more of this month’s math headlines:**

- A New Generation of Mathematicians Pushes Prime Number Barriers

*Quanta Magazine*, October 26, 2023. - All Natural Numbers Are Either Happy or Sad. Some Are Narcissistic, Too

*Scientific American*, October 24, 2023. - Mathematicians Found 12,000 Solutions to the Notoriously Hard Three-Body Problem

*Popular Mechanics*, October 18, 2023. - A random walk through mountains, surprising objects and keeping an open mind

*The Irish Times*, October 14, 2023. - This mathematician is making sense of nature’s complexity

*MIT Technology Review*, October 13, 2023. - AI is helping mathematicians build a periodic table of shapes

*New Scientist*, October 13, 2023. - Echoes of Electromagnetism Found in Number Theory

*Quanta Magazine*, October 12, 2023. - Non-Western art and design can reveal alternate ways of thinking about math

*Science News*, October 6, 2023. - For the love of maths: France opens its first museum dedicated to mathematics

*Euronews Culture*, October 3, 2023. - A Big Data Approach to Life Science

*The Scientist*, October 2, 2023.

* *

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In this article headed “Move Over, Euclid” for *New* *Scientist*, Kate Kitagawa explains that the history of mathematics tends to concentrate on Europe at the expense of mathematical knowledge from other places. For example, the Pythagorean Theorem was known not just by the ancient Greeks, but also in ancient Babylonia, Egypt, India and China. Liu Hui (3rd century CE) wrote commentary in the *Nine Chapters on the Mathematical Art* that “includes the earliest written statement of the theorem that we know of,” writes Kitagawa. Unless we happen to know about Euclid (died c. 270 BCE) and his *Elements*: Book I, Proposition 47 states that “in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle” (from David E. Joyce’s online edition).

Lui Hui’s proof is a non-obvious cut-and-paste argument quite alien to Euclid’s way of working; it illustrates how the same mathematical truth can be discovered in completely separate cultures, and so provides a valuable lesson to students of the history of mathematics. The chronology is really irrelevant but it should be kept straight.

I missed a related item last April. Two high-school students discovered a new trigonometric proof of that same Pythagorean Theorem. The news was covered in *The Guardian* and by Leila Sloman for *Scientific American*. (Note: Leila Sloman edits this column.) What made this noteworthy was that the identity $\sin^2\theta + \cos^2\theta=1$ for any angle $\theta$, a basic element of trigonometry, is an application of Pythagoras’s Theorem, so at one time, it seemed that any such proof would have to be circular; the mathematician Elisha Loomis stated in his 1927 book *The Pythagorean Proposition* that no trigonometric proof would be valid. (In fact, since then, a couple of valid trigonometric proofs had been discovered, but this was a new one.) What proved Loomis wrong in this case was the Law of Sines (any triangle with sides of length $a, b, c$ and opposite angles $\alpha, \beta, \gamma$ must obey the identity $\frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}$) which is independent of the Pythagorean Theorem. The two students, Calcea Johnson and Ne’Kiya Jackson, used the Law of Sines, an intricate geometric construction (they call it the “Waffle Cone”) and a calculation involving infinite series to nail their result.

Their proof has not been published at the date of this writing, but an ingenious YouTuber was able to hazard a reconstruction based on a diagram of theirs that appears briefly in a video clip embedded in the *Guardian* article.

Maybe not as natural as we think. “Diverse mathematical knowledge among indigenous Amazonians”, published August 21, 2023 in *PNAS*, investigates the origins of mathematics in humans. The authors, a team of seven led by David O’Shaughnessy and Steven Piantadosi of UC Berkeley, worked with communities of the Tsimane’, a Bolivian indigenous people “for whom formal schooling is comparatively recent in history and variable in both extent and consistency.”

The article reports on several studies. The first is a meta-analysis over several studies in which Tsimane’ children are asked to move a certain number of counters ($N$). This test is used to group children into *knower levels* according to how high an $N$ they respond to correctly. Those who make it to 8 are termed *Full Counters.*

This study shows how for children accurate comprehension of the meaning of number-names depends critically on the number of years of education. For example, almost none of the 6-8-year-olds with less than one year of schooling were comfortable with cardinalities above 4. (For adults, the picture was different. Almost everyone with one or more years of education was a “full counter.”)

Another study involving addition shows a more complicated picture. As the authors tell us, many developmental theorists explain number learning as essentially modeled on the axiomatic system mathematicians have derived for the integers, starting from 0 and using the *successor function* $S(x)$ to define $1$ as $S(0)$, $2$ as $S(S(0))$ and so forth, so that the operation of “adding one” is a basic and natural thing. The authors tested this idea in a single particularly remote Tsimane’ village, presenting addition questions to adult participants who had little or no formal education. Each test item was posed in two ways: formally (using mathematical language, as in school) and as a more practical word problem, for example, one involving prices. They used a set of 12 *augends* (the $a$ in $a+b$) together with the *addends* (the $b$) 1 and 5.

The authors point out that $1+1$ was not obvious for everyone (although $+1$ sums overall got more accurate answers). They also point out how much better performance was when the augend was divisible by 5. The reason they give for this anomaly is commercial: in this village the base unit of sales and purchasing is most often 5 Bolivanos. At the end of the article, they refer to their work and to other investigations with other cultures as evidence supporting “the theory that number’s emergence was tied to more concrete, practical uses in specific cultural circumstances, rather than being predetermined by innate logic.”

Aitor Morales-Gregorio, Alexander van Meegen and Sacha J. van Albada (Jülich Research Centre, Jülich, Germany) investigated how neuron density varies among different mammals’ cerebral cortices. Their findings are reported in the August 15, 2023 issue of *Cerebral Cortex*. Sampling a natural parameter like neuron density in the cortex (or, for example, individual height in a human population) typically gives a range of values; the question is, how are those values distributed? This means, given an interval of possible values, determining the probability that the value for a given sample will lie in that interval. For these natural parameters the probability distribution is usually described by a *probability density function* $f(x)$, a positive function defined over the range of possible values so that the probability of a measurement landing in the interval $[a,b]$ is equal to the area under the graph of the function and over that interval. (So the total area under the graph has to be 1.)

Many measurements of interest in the natural world are *normally* distributed (their probability density function has the same shape as the blue graph –the “bell-shaped curve”– in the image below). In particular, the Central Limit Theorem implies that, in general, if a large number of random samples are taken from a population, then the distribution of values will tend to be normally distributed. But this article’s authors found that this was not the case for the neuron density in samples from mammalian cerebral cortices from seven different species (humans, mice and five types of monkeys). The distribution turned out to be *lognormal*, with density function graph like the red curve below. The authors describe these findings as “a new organizational principle” of brain structure (“cortical cytoarchitecture”).

The authors analyzed data from studies published over the past 16 years covering one or more of the species in their survey. Experimental protocols varied; the easiest to describe was for the baboon: “the cortex was subdivided into small regions of approximately equal size and shape … ” with 142 samples taken.

]]>

Here’s why mathematicians are so interested in cake cutting

*Science News,* September 8, 2023.

What’s the best way to cut a cake so that everyone is sure they got a fair share? In an article for *Science News*, Stephen Ornes takes a deep dive into the theory of cutting cake and the many ways that mathematicians think about the topic. The problem is simplest when just two friends share a cake. In that case, they can use a strategy called “I-cut-you-choose.” The first friend splits the cake into two pieces that she feels are equal, so that she would be happy with either piece; the second friend then chooses whichever piece he prefers. This strategy works great for two people, but for three or more friends, the problem is more complex. Ornes describes the development of the cake problem over time, from a 1948 paper to modern-day research.

Beyond dessert, the insights about cake cutting can be applied to real-world problems about dividing things fairly. Researchers have used the cake model to find more fair ways to design voting systems, admit students to schools, assign chores, and split rent.

**Classroom Activities:** *ratios, fractions, optimization*

- (All levels) Draw a picture of a cake whose left half is chocolate frosted and whose right half is vanilla frosted. Try the “I-cut-you-choose” strategy with a partner. Take turns deciding how to cut the cake in each of the following scenarios. (There may be multiple right answers—just make sure you would be equally happy with each of the two pieces you cut.)
- How would you cut the cake if you both have no preference between the chocolate frosting and the vanilla frosting? Give a few different options for how you could cut the cake.
- How would you cut the cake if you both love chocolate, and you think that each bite with chocolate frosting is worth twice as much as a bite with vanilla frosting?
- (Advanced) As in the last scenario, imagine that you both love chocolate, and you think that each bite with chocolate frosting is worth twice as much as a bite with vanilla frosting. How can you cut the cake so that one of the pieces is only chocolate, but you are still equally happy with either piece?

- (All levels) The article discusses a strategy for sharing cake among three people called the “last-diminisher” method.
- Read the description in the slideshow within the article titled “How the ‘last-diminisher’ method works.”
- How could Carla believe that she has a piece that’s worth 1/3 of the value of the cake, but still be jealous of someone else’s piece? What would that mean about her perception of the other pieces?
- Discuss with another student: what do you think are the benefits of this method? What did you find confusing? If you and your friends prefer different parts of the cake (for example, if one person loves fruit and another person does not), how might that affect the way you approach the cutting problem?

*—Tamar Lichter Blanks*

Were these stone balls made by ancient human relatives trying to perfect the sphere?

*Science*, September 5, 2023.

1.4 million years ago, a human ancestor known as *Homo erectus* accumulated 600 plum-sized stone balls in one corner of northern Israel. Researchers discovered these oddly symmetrical rocks beside hand axes at an archaeological site in the 1960s, and similar artifacts have appeared elsewhere around the world. Scientists have proposed that these could have occurred naturally, or that they could be debris left behind while making other tools. But now, one research team claims that these “spheroid” orbs were crafted intentionally—and the evidence is in their math. In this *Science* article, Phie Jacobs explains how archaeologists used 3D analysis software to measure angles of the stone balls, calculate curvatures, and find the center of mass. Jacobs writes that the practical purpose remains “enigmatic,” while suggesting another possible purpose: “the sheer joy of creating symmetry.”

**Classroom Activities:** *archaeology, center of mass*

- (All levels) Do this Science Buddies activity about finding an object’s center of mass.
- (High level) This research comes from a lab that studies “computational archaeology.” Based on the reading, describe how this field might differ from what people normally imagine archaeology entails.
- Explain how the stone balls’ angles, center of mass, roughness, and curvature can suggest that they were intentionally crafted.
- Watch this TED-ED video (What can you learn from ancient skeletons?), and list all the ways that mathematical analyses appear.

*—Max Levy*

Mathematician Solves 50-Year-Old Möbius Strip Puzzle

*Scientific American*, September 12, 2023.

The Möbius strip — the one-sided loop — has been around since the nineteenth century, and mathematicians love to trot out its counterintuitive features. But they still haven’t fully plumbed its mathematical depths. “Until recently, researchers were stumped by one seemingly easy question about Möbius bands: What is the shortest strip of paper needed to make one?” writes Rachel Crowell in this article for *Scientific American. *In August, mathematician Richard Schwartz of Brown University finally solved this question, announcing that the strip needs to be $\sqrt{3}$ times as long as it is wide.

**Classroom Activities: ***geometry*

- (All levels) The writers of this article from
*The Conversation*point out two counterintuitive properties of Möbius strips, both of which can be explored in class using construction paper, tape, pencil, and scissors. Create your own Möbius strips with these materials.- The authors write: “If you take a pencil and draw a line along the center of the strip, you’ll see that the line apparently runs along both sides of the loop.” Confirm this for yourself.
- Now try to guess what will happen if you cut the Möbius strip along the center line. Then cut to find out if you were right. Are you surprised by the result? What experiments do you want to try next?
- (High level) For more about Möbius strips, read the full article. For another example of a one-sided surface, read this article about the Klein bottle from
*Plus*.

- (All levels) In this video, mathematician Eugenia Cheng cuts a bagel along a Möbius strip. Before watching, try to figure out what will happen. Then, either watch the video or try it in class. What comes to mind after you see the result?
- In the video, Cheng says that the bagel “can’t separate into two, because the cutting surface only had one side.” Does this make sense to you? Why or why not?

- (All levels) Crowell is reporting on a new result about the aspect ratio of the Möbius strip — the ratio of its length to its width. Have students make Möbius strips in class using tape, scissors, and a flexible material like fabric. Compete to see who can make a strip with the shortest ratio of length to width. How close to $\sqrt{3}$ can students get?

*—Leila Sloman*

3 reasons we use graphic novels to teach math and physics

*The Conversation*, August 17, 2023.

Traditional math and physics textbooks can be boring and bombard you with information all at once. Mathematicians Sarah Klanderman and Josha Ho believe this information overload actually hinders learning. So, they propose an alternative: graphic novels that teach student’s concepts before calculations. “This approach helps build their intuition before diving into the algebra. They get a feeling for the fundamentals before they have to worry about equations,” they write. In this article for *The Conversation*, Klanderman and Ho describe the benefits of graphic novels as educational tools that make math accessible for students, parents, and people studying to become teachers. “It boosts their confidence and shows them how math can be fun—a lesson they can then impart to the next generation of students.”

**Classroom Activities:** *storytelling with math*

- (All levels) Draw a short comic (stick figures are fine!) that tells a story while illustrating a concept you recently learned in math or science class.
- (All levels) Suppose that you had more time to create a bigger comic or graphic novel. What scientific concept would you choose to explain?
- Check out these story synopses. What is intriguing or not about each?
*Quantum physics: “**Suspended in Language”**Women in STEM: “**Astronauts: Women on the Final Frontier**”**Thermodynamics:**“Max The Demon”*__Calculus:__*“The Manga Guide to Calculus”*

- How would you use what you like (and don’t like) to make your story better?

- Check out these story synopses. What is intriguing or not about each?

*—Max Levy*

The story of zero: How ‘nothing’ changed the world

*CBC Ideas*, September 26, 2023.

In the beginning, there was no such thing as zero. Zero wouldn’t come about until the ancient Sumerians, says this episode of *Ideas*. Once humanity invented the concept of the number zero, massive progress in technology, medicine, engineering, and mathematics itself became possible.

**Classroom Activities: ***calculus, asymptotes*

- (Mid level) The episode describes the mathematical issues that arise when you try to divide by zero. Without doing any calculations, what do you think the answer to $ 1 \div 0$ should be? What about $3 \div 0$? What about $ -1 \div 0$? Justify your answers.
- Calculate $ 1 \div 0.5$, $1 \div 0.2$, and $1 \div 0.1$, and plot your results on a graph. What are the results hinting at? Do they match your prediction for the value of $1 \div 0$?
- Repeat the exercise, this time calculating $1 \div (-0.5)$, $1 \div (-0.2)$, and $1 \div (-0.1)$. Do the results change your response?
- Write down what you’ve learned from your calculations, and whether you’ve changed how you think about dividing by zero. What else do you want to know?
- In the above exercise, you started to plot the
**vertical asymptotes**of the graph $y = 1/x$. Learn about asymptotes with this online lesson by Richard Wright of Andrews University.

- (High level, Calculus) In the episode, one mathematician describes calculus as “the art of properly dividing by zero.” What does that mean to you? Do you agree or disagree? What makes calculus different from the division-by-zero examples above?
- In calculus, the derivative, or the instantaneous slope of a function $f(x)$ at the point $a$, is defined as

$$\lim_{h \to 0} \frac{f(a + h) – f(a)}{f(h)}$$

By plugging in $h = 0.1$, $0.01$, and $0.001$ to the following functions, estimate their derivatives at $a = 0$:- $f(x) = x$
- $g(x) = x^3$
- $h(x) = \sin(x)$

- How well do your results match the derivative formulas $f’(x) = 1$, $g’(x) = 3x^2$, and $h’(x) = \cos (x)$? Can you prove that those formulas are accurate?
- Can you think of a function whose derivative at zero is undefined? Why does this happen?

- In calculus, the derivative, or the instantaneous slope of a function $f(x)$ at the point $a$, is defined as

*—Leila Sloman*

**Some more of this month’s math headlines**

- The mathematical theory that connects swimming sperm, zebra stripes, and sunflower seeds

*Popular Science*, September 27, 2023. - How Mathematical Objects Are like People and Other Mysteries of Intersection Theory

*Scientific American*, September 26, 2023. - Emmy Noether: the woman who developed one of the most beautiful theorems in physics

*ZME Science*, September 21, 2023. - Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math

*Quanta Magazine*, September 21, 2023. - Can you solve it? The man who made India’s trains run on time

*The Guardian*, September 18, 2023. - A catalog of all human cells reveals a mathematical pattern

*Science News*, September 18, 2023. - In Ukraine, Mathematics Offers Strength in Numbers

*The New York Times*, September 12, 2023. - A Tower of Conjectures That Rests Upon a Needle

*Quanta Magazine*, September 12, 2023. - How scientists test whether humans are causing our extreme weather

*The Washington Post*, September 11, 2023. - Mathematicians find 12,000 solutions for fiendish three-body problem

*New Scientist*, September 8, 2023. - Duolingo confirms its app will soon include both math and music lessons

*TechCrunch*, September 6, 2023. - Math is hard — even for teachers. What if they conquered their math anxiety?

*The Associated Press*, September 5, 2023. - The Math Revolution You Haven’t Heard About

*EdSurge*, September 5, 2023. - Alan Turing and the Power of Negative Thinking

*Quanta Magazine*, September 5, 2023.