Frank Wilczek’s column in the the *Wall Street Journal* (April 14, 2022) had the title “A Quantum Leap, With Strings Attached; The Inca system of quipu—tying a series of knots to record information—is providing a surprising model to modern physics and quantum computing.”

This textile document is a typical quipu in that its data is numerical and recorded in a decimal system. (For another example and more details, see Nicole Rode’s YouTube video from the British Museum). While they were used by earlier pre-Columbian Andean cultures, most of the surviving specimens date from the period of Inca domination, c. 1400–1532 CE. (We can only guess what the numbers recorded on quipus were actually counting —these civilizations left no written records).

Wilczek compares quipus with the information storage and transmission systems we encounter today: “written human language, the binary code of computers and the DNA and RNA sequences of genetics,” and remarks that the Andean system involves “something unique: topology, the science of stable shapes and structures.” In fact the difference between one knot and another, which is *contrastive* in quipus, to borrow a term from linguistics, is one of the most basic examples of a purely topological concept.

The connection between quipus and “modern physics and computing theory” comes precisely through topology. The equivalents of Andean cords are the *world-lines* of particles. Wilczek asks us to suppose our particles are only moving in two dimensions, and that we add a third dimension to represent time. Then as time progresses the successive positions of a particle trace out a curve: this is its world-line. And if several particles are observed at once, their world-lines can tangle (“these are not our ancestor’s strings”) and form what mathematicians call *braids*.

“There are certain particles, called anyons, whose quantum behavior keeps track of the braid that their world-lines form [and therefore could be used to store information]. The anyon world-lines form a quantum quipu.” Wilczek’s terminology has to be taken with a grain of salt. Knots and braids are very different mathematical objects, although fundamentally related (see the drawing above and Alexander’s Theorem on Wikipedia); quipus use one and not the other. Nevertheless it is striking that the topology of curves in space turns up both in an antique recording system and in the latest quantum science.

The science really is very new. Anyons were only experimentally detected two years ago (Wilczek himself had conjectured their existence, and named them, some 40 years back). He tells us that “the simple quantum quipus that were produced in those pioneering experiments can’t store much information” but that only last month “Microsoft researchers announced that they have engineered much more capable anyons.” This is presumably the research described in the Microsoft Research Blog on March 14.

Siobhan Roberts contributed Is Geometry a Language That Only Humans Know? to the March 22, 2022 *New York Times*. The subtitle is more specific: “Neuroscientists are exploring whether shapes like squares and rectangles — and our ability to recognize them — are part of what makes our species special.” The neuroscientists in question are Stanislas Dehaene (Université Paris-Saclay and Collège de France) and his collaborators.

The first part of Roberts’s article concerns the research that Dehaene and his team published last year in *PNAS*: “Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity.” In a typical experiment they report, subjects were presented with a display of polygons. Five of the six were similar, differing only in size and orientation; the sixth was like the others except that its shape had been changed by moving one vertex. Subjects were asked to pick out the oddball.

The “normal” polygons were chosen from a family of eleven quadrilaterals that can be ranked, starting with a square, by how unsymmetrical they are.

The first experiment, involving 605 French adults, showed that the number of errors they made “varied massively” with the lack of symmetry/orthogonality/parallelism of the “normal” exemplar.

The team repeated the experiment with French kindergarteners and with Himba adults (“a pastoral people of northern Namibia whose language contains no words for geometric shapes, who receive little or no formal education, and who, unlike French subjects, do not live in a carpentered world.”) The results correlated strongly with those of French adults. “Both findings converge with previous work to suggest that the geometric regularity effect reflects a universal intuition of geometry that is present in all humans and is largely independent of formal knowledge, language, schooling, and environment.”

The experimenters had access to a colony of baboons (*Papio papio*) in the south of France; they managed to train the baboons to where they had a “clear understanding of the task”—they could recognize the oddball apple in a group of watermelon slices, and even a regular hexagon in a group of non-convex polygons, but “although error rates differed across the 11 shapes, with a consistent ordering across baboons, […] they correlated weakly and nonsignificantly with the geometric regularity effect found in human populations.”

After speaking with Moira Dillon (a psychologist at New York University) Roberts puts this research in a historical context: “Plato believed that humans were uniquely attuned to geometry; the linguist Noam Chomsky has argued that language is a biologically rooted human capacity. Dr. Dehaene aims to do for geometry what Dr. Chomsky did for language.” But Frans de Waal (a primatologist at Emory University) cautioned her: “Whether this difference in perception amounts to human ‘singularity’ would have to await research on our closest primate relatives, the apes.”

Roberts reviews connections between this research and work in artificial intelligence, and then moves on to Dehaene *et al.*‘s latest project, essentially figuring out what in the human mind makes geometric regularity so significant. Here’s a clue, quoting from Dehaene: “We postulate that when you look at a geometric shape, you immediately have a mental program for it. You understand it, inasmuch as you have a program to reproduce it.” The team explored an algorithm, DreamCoder (the authors overlap with Dehaene’s collaborators) that “finds, or learns, the shortest possible program for [drawing] any given shape or pattern.” Then they tested human subjects on the same shapes. “The researchers found that the more complex a shape and the longer the program, the more difficulty a subject had remembering it or discriminating it from others.”

*Discover Magazine*, April 7, 2022

If you’re a sports fan, you’ve likely witnessed the “home-field advantage”: Teams tend to win more often when they’re playing at their home venue. Statistical studies have shown that this phenomenon is real. As Cari Shane explains, the primary cause of the home-field advantage is that hometown fans exert a subconscious sway over referees, leading to a few more calls in favor of the home team. Travel-related factors contribute, too. But in recent years, the magnitude of the advantage has been decreasing, statistical analyses show. Rule changes such as the addition of instant replays and coaches’ challenges are leveling the playing field. Plus, travel isn’t as hard on athlete’s bodies as it used to be, thanks to increased budgets and advances in sports medicine.

**Classroom activities: ***statistics, sports*

- (Introductory Statistics) Suppose that the Middletown Wanderers won 20 out of their 30 home games last season, and won 15 out of their 30 road games that same season.
- Create a list of possible reasons that these figures might not represent a true home-field advantage. (For example, think about where where the team faced their most highly ranked opponents, or how the team’s performance might have varied throughout the season.)
- Assuming that none of the factors you listed are at play, use a two-proportion Z test at at the 5% significance level to determine if there is statistical evidence that the Wanderers had a home-field advantage last season.

- (Introductory Statistics) Gather win-loss data from a recent season for all the teams in a local sports league. Calculate the percentage of games in which the home team won.
- Are there any confounding factors that might skew the data?
- If not, use a one-proportion Z test at the 5% significance level to determine if the league as a whole had a home-field advantage that season.

*Related Mathematical Moments:* Holding the Lead.

*—Scott Hershberger*

*Wired*, April 1, 2022

How fast should you drive to work? You can find a precise answer to that question, at least if your goal is to save money. In an article for *Wired*, physicist and science popularizer Rhett Allain explains how to calculate the most cost-effective driving speed given your car’s fuel efficiency, your hourly pay rate, and the price of gas. Allain says that the cheapest speed is not too slow (if you’re idling, you’re getting 0 mpg) and not too fast (because of factors like air resistance and friction), but at a sweet spot in the middle. As a bonus, Allain shares a snippet of code that does the math, which you can edit to calculate your own optimal speed.

**Classroom activities**:* fractions, ratios, physics, optimization, coding, programming*

- (All levels) Let’s develop an intuitive understanding of the equation for average velocity described in the article. Try to answer the following questions based on your experience. Then compare those with the answers you get from the equation $v = \Delta x / \Delta t$, where $v$ stands for average velocity, $\Delta x$ stands for the distance a car traveled, and $\Delta t$ stands for the amount of time the trip took. (Note: Speed is the absolute value of velocity, so in the following two examples you can treat “velocity” and “speed” interchangeably.)
- If Car A and Car B both drove for 1 hour, but Car A traveled 15 miles and Car B traveled 45 miles, which car had the higher average speed?
- If Car A and Car B both drove 40 miles, but Car A’s trip took 1 hour and Car B’s took 2 hours, which car had a higher average speed?
- (High level) In general, when you increase the absolute value of $\Delta x$, does average speed increase or decrease? What about when you increase the value of $\Delta t$? Does this match with your intuition about what it means for something to travel at a high or low speed?
- (High level) The article mostly discusses “speed” rather than “velocity.” Why?

- (High level) Experiment with the interactive coding tool in the article by clicking the pencil icon and changing the values of the variables
`basempg`

,`G`

,`R`

, and`dx`

, then clicking the play/run icon. Try some of the following:- Compare the cheapest speeds for a car that has an EPA listed mileage of 25 mpg versus a car that has a mileage of 50 mpg. For which car is the cheapest speed faster?
- Make a prediction about whether increasing the pay rate
`R`

will increase or decrease the cheapest speed. Then test your prediction using the code. - Does the commuting distance
`dx`

have any effect on the optimal speed? (The answer to this question is in the article.)

*—Tamar Lichter Blanks*

*Popular Mechanics*, April 25, 2022

In an April 20 tweet that attracted over 75,000 likes, attorney Paul Sherman claimed that “It is physically impossible to exceed the 70-pound domestic weight limit for a small flat rate box” shipped by the US Postal Service. A small flat rate box filled with pure osmium—the densest naturally occurring element—would weigh in at around 61.5 pounds, he wrote. In this article, writer Caroline Delbert compares osmium to other dense materials. While osmium may top the scales on Earth, trying to mail the material that neutron stars are made of would leave you on the hook for a hefty overweight charge.

**Classroom activities:*** density, mass, volume, media literacy*

- (Pre-algebra, Algebra I) Let’s check Sherman’s calculations and do some of our own.
- Find the inside dimensions of a USPS small flat rate box. What is the volume of the box in cubic inches? What is the volume in cubic centimeters?
- The density of osmium is 22.59 g/cm
^{3}. What mass of osmium could you fit in the box? How does this compare to the USPS limit of 70 pounds? - (Mid level) The density of a neutron star is on the order of 10
^{14}g/cm^{3}. What mass of neutron star material could fit in the box (in kilograms and in pounds)? What volume of neutron star material would weigh 70 pounds? - (Upper level) Ask each student to pick a substance (water, rock, cotton candy, etc.) and look up its density. Calculate the weight of a small flat rate box filled with that substance. Then calculate the volume of 70 pounds of that substance.

- (High school) Ask students to find examples of claims on the Internet that cite simple calculations. Have them try to confirm or refute the claims, either by doing the calculations themselves or searching for other reputable sources online.

*—Scott Hershberger*

*Ars Technica*, April 6, 2022

To Ben Orlin, a teacher and author, playing is essential to learning. That’s especially true in math. Orlin recently wrote a book called *Math Games with Bad Drawings*. It’s an illustrated collection of multiplayer games in which intriguing puzzles emerge from simple mathematical rules. You just need household items like pencil and paper, your hands, the internet, and Goldfish crackers, writes Jennifer Ouellette in *Ars Technica*. Ouellette interviewed Orlin about his new book, which includes more than 50 “math-y” games. According to Orlin, the book doesn’t try to create educational puzzles for math-lovers. He instead sees each game as a thought experiment enjoyable for anyone. “Games are constantly generating new puzzles,” he says. “With a puzzle, you solve it, you’re done. A game is like a fountain of puzzles that’s constantly pouring out new puzzles for you.”

**Classroom Activities: ***math games*

- (All levels) In partners, play Ultimate Tic-Tac-Toe. Discuss what is “math-y” about it and how your strategy is different from your strategy in standard Tic-Tac-Toe. (You can find more info on Wikipedia.)
- (All levels) In partners, play Sequencium, one of Orlin’s own games.
- Show your final board to another pair of students and compare the strategies that you used.
- If both players are able to move until the board fills up, what is the maximum possible score? What is the minimum possible score?

- (All levels) Read about the game of Set via examples from pop culture on Orlin’s webpage, and watch a PBS video about the geometry of Set.
- (Advanced) Try to solve the challenge problem at the end of the PBS video (then see the solution).

*—Max Levy*

*CNET*, April 1, 2022

Math is a language, and humans aren’t the only creatures who understand it. Nature is full of animals that seem to perform simple mathematical operations. Animals as big as lions tally their competitors; animals as small as bees distinguish small numbers from large ones. The list is always growing as scientists put more animals to the test. In an article for *CNET*, Monisha Ravisetti writes about new research involving fish. Researchers from Germany taught cichlids and stingrays to add and subtract 1 from numbers up to 5. Their experiment worked like this: Fish can’t read our numerals, so the team showed the fish some number of blue or yellow shapes. Blue meant “add 1,” and yellow meant “subtract 1.” The fish earned treats for choosing correct answers and eventually, they learned. “The fish passed with flying colors,” writes Ravisetti.

**Classroom activities:** *counting, symbolic math, numerals, box plots, p-values*

- (All levels) Let’s count like fish. Read the article to learn how the scientists tested the fish. (More info in the full study here.) In groups of two (or a class) challenge each other to create a prompt and pick the correct answer. For example, what would be the correct response to a symbol with a blue square, a blue circle, and a blue triangle?
- (Mid level) Each student can create their own language of symbolic arithmetic different from the colors and shapes here. This math language should communicate addition, subtraction, multiplication, and division. They can use shapes and colors or something else entirely, like emojis as digits. Get creative! Then, students can form small groups, teach each other, and quiz each other. Discuss the advantages and disadvantages of each system as compared to our standard system.
- (Upper level, Statistics) Discuss the significance of Figures 5 and 7 in the scientific paper and whether they provide compelling evidence for their claim that these fish can “add and subtract 1.”

*—Max Levy *

**Making math fun outside the classroom**

*Philadelphia Tribune,*April 26, 2022**Elegant Six-Page Proof Reveals the Emergence of Random Structure**

*Quanta Magazine,*April 25, 2022**When Exponential Growth Isn’t Enough**

*The Wall Street Journal,*April 21, 2022**New Proof Illuminates the Hidden Structure of Common Equations**

*Quanta Magazine*, April 21, 2022**What tangled headphones can teach us about DNA**

*Popular Science,*April 18, 2022**Bad Math Is Steering Us Toward Climate Catastrophe**

*OpenMind,*April 14, 2022**Infinity has long baffled mathematicians – have we now figured it out?**

*New Scientist*, April 13, 2022**After a Texas teacher saw his students struggling with math, he turned to rap music**

*NPR*, April 11, 2022**Untangling Why Knots Are Important**

*Quanta Magazine,*April 6, 2022**Could computer models be the key to better COVID vaccines?**

*Nature*, April 5, 2022**Math in 3-D: Q&A with Abel Prize Winner Dennis Sullivan**

*Scientific American*, April 5, 2022**Father-Son Team Solves Geometry Problem With Infinite Folds**

*Quanta Magazine,*April 4, 2022

- Steven Strogatz on $\pi$ in
*The**New York Times* - On math and money
- “We’re all still living in Euclid’s world”
- Symmetry in biology, why?

On March 14, 2022, the *Times* ran “Pi Day: How One Irrational Number Made Us Modern” by Steven Strogatz. Strogatz focuses on the nature of $\pi$ as a number that can only be reached by an *infinite* process, so that while we can calculate more and more digits of the decimal expansion of $\pi$, the knowledge of exactly where $\pi$ sits on the number line will always be out of reach.

As Strogatz tells us, the mathematician who first made the process explicit was Archimedes (c. 287–c. 211 BCE). The Greeks had already been using the approximation of curves by polygonal lines—for example, this is how Euclid proved that the area of a circle is proportional to the square of its diameter. But determining the exact constant of proportionality does not seem to have been on his agenda. (Euclid did not explicitly consider the ratio of circumference to diameter.) It was left to Archimedes, who was an engineer as well as a mathematician, to try to nail it down.

Strogatz leads us through the process, imagining measuring the length of a circular track by walking around it and counting your steps, and multiplying that number by the length of your stride. As he explains it, each of your steps is a shortcut “in place of what really is a curved arc,” so the product you obtain will underestimate the actual length of the track. But by taking smaller and smaller steps you can get better and better estimates.

Archimedes started with an inscribed regular hexagon as a first approximation to a circle (say, of radius $r$).

Since a regular hexagon is made up of six equilateral triangles, the perimeter of a regular hexagon inscribed in a circle of radius $r$ is equal to $6r$. This gives a lower bound of $3$ for the ratio of circumference to diameter. As Strogath tells us, Archimedes went on to calculate the perimeters of inscribed regular polygons with 12, 24, 48 and 96 sides.

What allowed Archimedes to do these calculations is the (relatively) simple relation between the side of an inscribed regular $n$-gon and the side of the $(2n)$-gon obtained by putting a new vertex halfway between each pair of original adjacent vertices.

The red chord cuts the radius bisecting it into two segments of lengths $x$ and $y$, with $y=r-x$ . The calculation involves two applications of the Pythagorean Theorem. In triangle $*$, the theorem gives $x^2 = r^2 – (\frac{a}{2})^2$. Then in triangle $**$, it gives $b^2 = (\frac{a}{2})^2 + (r-x)^2$. The two steps, each involving extraction of a square root, give $b$ in terms of $a$ and $r$.

Archimedes repeated this calculation three more times, starting with the hexagon and $a=r$ and ending with the side of a $96$-gon. This allowed him to set $3 + \frac{10}{71}$ as a lower bound for $\pi$, while a similar calculation with circumscribed polygons yielded $3 + \frac{10}{70}$ as an upper bound. As Strogatz describes it, “The unknown value of pi is being trapped in a numerical vise, squeezed between two numbers that look almost identical, except the first has a denominator of 71 and the last has a denominator of 70.”

For Strogatz the accuracy of Archimedes’ estimates is less important than the model he gave of linear approximations to curves and the importance of iterative calculations giving sharper and sharper bounds on a number of interest. As he puts it, “Archimedes paved the way for the invention of calculus 2000 years later.”

The multimedia web portal *Big Think* ran a piece by Michael Brooks on February 2, 2022, with a nonstandard perspective on mathematical research: “More math, more money: How profit-seeking has sparked innovations in mathematics.” Brooks, the author of *The Art of More: How Mathematics Created Civilization* (Penguin Random House, 2022), tells us here that while we often think of math as somehow above everyday, sordid life, in fact “math and money are like Bonnie and Clyde.”

The article starts off with a reference to the “The Anatomy of a Large-Scale Hypertextual Web Search Engine” on the Stanford website, where Sergey Brin and Larry Page give among other things the mathematical definition of *PageRank*, an idea we know turned out to be worth a big chunk of change. But it didn’t start there. Brooks links to an article on Mesopotamian Bronze Age Mathematics by the eminent historian of science Jens Høyrup to show us how back in Ur, about 4000 years ago, King Shulgi used his knowledge of arithmetic to “implement a kingdom-wide, tamper-proof accounting system,” prevent fraud in the collection of taxes, and fill his coffers.

Moving rapidly through the centuries, Brooks takes us to When pirates studied Euclid, where Margaret Schotte (York University, Toronto) tracks in wonderful detail the merchant and privateers in 17th-century Europe as they attended special academies to learn how to use astronomy and spherical trigonometry “to deliver goods faster or, in the case of the pirates, perform better interceptions.”

Additional illuminating examples cover algebra, calculus, and statistics. The moral of the story: “No one should be aiming to become a singer or a sports star. Math is a much more reliable road to riches.”

“The geometry of ancient Greece has stood for more than two millennia, even after relativity and quantum mechanics” is the rest of the title of a short article by Frank Wilczek. The Nobel prize-winning physicist is a regular contributor to *The Wall Street Journal*, where the piece ran on February 4, 2022.

Euclid’s *Elements* was written around 300 BCE. It built geometry up from some more or less obvious definitions and from a few “axioms,” supposed to be self-evident assumptions about how our geometrical world works. As Wilczek reminds us, the *Elements*, beyond training generations of students “not only in the science of space and measurement but in the art of clear thinking and logical deduction,” provided the framework for Newton’s physics and later for Maxwell’s theory of electromagnetism.

But then something happened. Euclid’s Axiom 5, the “parallel postulate” (equivalent to the statement that the angles in a triangle add up to $180^{\circ}$), felt different from the others; in the 19th century, mathematicians realized that it could be modified in two different ways and still give a coherent, uniform geometry. In one direction the sum of the angles in a triangle is always *more than* $180^{\circ}$. This corresponds to the situation on the surface of the Earth, where navigators build triangles out of segments of great circles, the spherical equivalent of straight lines. In the other direction, the sum is always *less than* $180^{\circ}$. This second geometry was discovered by Gauss (who didn’t publish it) and independently by Lobachevsky and Bolyai. As opposed to the curvature of a spherical surface, which we call *positive*, these “hyperbolic” surfaces have negative curvature, like the surface of a saddle, curling up in one direction and down in the other.

Not many years later Bernhard Riemann came up with the concept of what we now call a *Riemannian manifold,* where the curvature can vary from point to point. The equivalent of a straight line on such a surface (a path giving the shortest distance between two of its points) could be quite far from straight. Wilczek gives an example from sports: “an Alpine skier racing down a bumpy mountain will keep doing her best to go straight down, but over the course she will trace a curve.” Riemann himself was interested in the geometry of the three-dimensional space we live in, and his manifolds can have 2, 3, 4, … any number of dimensions.

Wilczek leads us through the next development. It started with Einstein’s special theory of relativity, developed to explain the non-intuitive fact that your measurement of the speed of light does not depend on how fast you are moving. The mixture of space and time coordinates that Einstein required was given a geometric formulation by Minkowski who stated in 1908, as Wilczek quotes him, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Einstein used this formalism for his next step, the general theory of relativity, where space-time has *curvature*, analogous to the Riemannian concept we just saw, and intimately related to gravity. (The measurement, made during the 1919 total solar eclipse, of how much light rays—by definition, straight lines in space-time—would curve when passing close to the sun was experimental proof of the theory.)

This sounds pretty far removed from similar triangles and the Pythagorean theorem, but Wilczek assures us that “Einstein’s framework is still recognizably Euclidean, extended and adapted to bring in time and large-scale curvature.” In fact, he considers this “the most striking example of what Eugene Wigner called ‘the unreasonable effectiveness of mathematics in the natural sciences.'”

Kate Golembiewski wrote “Life’s Preference for Symmetry Is Like ‘A New Law of Nature,'” including the above image, for *The New York Times.* It appeared in the Trilobites section online on March 24, 2022, and a shorter version appeared in print on the 29th. The article reports on “Symmetry and simplicity spontaneously emerge from the algorithmic nature of evolution” by Iain Johnston, Kamaludin Dingle, and collaborators, published in *PNAS* on March 15.

As we know, symmetry is everywhere in nature: Starfish have 5-fold symmetry, flowers have bilateral or rotational symmetry (of various orders), virus capsids have icosahedral symmetry, our left hands are almost exact mirror-images of our right. The question is *why*. “Biologists aren’t sure,” Golembiewski tells us, “—there’s no reason based in natural selection for symmetry’s prevalence in such varied forms of life and their building blocks.”

It turns out that the study of *algorithms* can point to an answer. As Golembiewski relates, the authors of the *PNAS* paper concluded that symmetry is evolutionarily favored because it makes the storage and replication of genetic data easier and more reliable. “Dr. Johnston […] likens it to telling someone how to tile a floor: It’s easier to give instructions to lay down repeating rows of identical square tiles than explain how to make a complex mosaic.” This is an explanation distinct from natural selection! Chico Camargo, another of the authors, told her: “It’s like we found a new law of nature. […] This is beautiful, because it changes how you see the world.”

*Dot Esports*, February 28, 2022

It was only a matter of time until someone came up with an even nerdier version of Wordle, the viral word game. Enter: *Nerdle*. Data scientist Richard Mann created Nerdle for people who enjoy math games, according to writer Sage Datuin. In Nerdle, players guess *equations *instead of words. Each equation contains digits, operations ($+$, $-$, $\times$, and/or $\div$), and one equals sign. Players must guess equations until they have the right pieces to find the mystery equation. Much like Wordle, you only get 6 guesses, and each guess has to be mathematically correct—or else the game will tell you, “This guess does not compute!” In this article, Datuin highlights the differences between Nerdle and Wordle and shares some tips on how to win.

**Classroom activities:** *Nerdle, combinatorics*

- (All levels) Play Nerdle individually or as a class. For an easier game, go to the menu and select “mini-Nerdle” to activate a smaller board. Discuss what makes the game hard and what strategies you can use to solve the puzzle in as few guesses as possible.
- (Algebra, Statistics) Wordle relies on a fixed dictionary containing 2,315 possible words, but how many different equations are possible in Nerdle? For simplicity, let’s assume that the format will always be a sum or product of two two-digit numbers, and that a three-digit solution is ok, such as 12+34=46 or 12$\times$34=408.
- Consider the spaces on the left side of the equation that contain digits—how many are there, and how many options do we have to fill them?
- Next, how many options do we have for the operation? Now, combine these measures according to the Fundamental Counting Principle.
- Discuss why we don’t need to count the permutations of the digits on the right side of the equals sign.
- (Advanced) How does the calculation change if we allow subtraction? What about division?

*—Max Levy*

*Quanta Magazine*, March 24, 2022

Is it possible for every person at a party to shake hands with an odd number of people? Patrick Honner suggests you try it out at your next social gathering. In this article, he explains the solution using the tools of graph theory. The puzzle connects to the concept of subgraphs, which is an open area of research. The latest advance in describing odd subgraphs, Honner writes, came just last year.

**Classroom activities: ***graph theory, even and odd numbers*

- (All levels) Before students read the article, split them into groups of 5 or 6 and ask them to try to shake hands with an odd number of people. (You can use elbow bumps or verbal greetings instead.)
- Draw graphs representing each group’s handshakes.
- Discuss which groups were able to succeed and why, using the idea of a graph’s “degree sum.”

- (Mid level) Watch this 3Blue1Brown video about the three-utilities problem, another popular puzzle in graph theory.
- The three-utilities problem is impossible on a flat surface, but a solution
*does*exist on a coffee mug. Why doesn’t a similar trick work for the shaking hands puzzle?

- The three-utilities problem is impossible on a flat surface, but a solution
- (Upper level) Complete the exercises at the end of the article (answers are provided).

*—Scott Hershberger*

*Scienceline*, March 4, 2022

Every computer you own is in a sense the same under the hood: Desktops, laptops, and phones all run on the same kind of math. Encryption is designed with this in mind. But quantum computers are different, exploiting properties of quantum physics to do their calculations. This allows them to solve certain problems that have been the basis of cryptography until now, such as factoring large numbers efficiently. Quantum computers are hard to build—they require extremely cold metals and other specialized engineering—but if they get big enough, they’ll be able to break much of the encryption that keeps digital communications private. The US National Institute of Standards and Technology is running a competition to find new methods of encryption that are safe against both regular and quantum computers. Most of the finalists rely on mathematical objects known as lattices, as Daniel Leonard writes in *Scienceline*.

**Classroom activities:** *cryptography, quantum computing, number theory*

- (All levels) Work in pairs through the lattice-based encryption algorithm described in the graphic in Leonard’s article.
- One person, You, should choose a secret number $S$ and perform Steps 1–3 and the first part of Step 4.
- The other person, Friend, should choose a secret message, $0$ or $1$, and under Step 4 carry out Friend’s Steps 1 and 2 (labeled in blue).
- You should then use Steps 5–6 to uncover the secret message.
- You and Friend exchange roles and repeat the above steps so that each can try both sides of the algorithm.
- You and Friend share the public steps—the first part of Step 4 and Friend’s (blue) Step 2—with another pair of students. Can they determine what secret message Friend sent? If not, what missing piece of information would allow them to find out the message?

- (Advanced) Watch a video by minutephysics about how quantum computers break encryption.
- Check the “repeating property” used in Shor’s Algorithm for $g=5$, $N=31$, and $p=3$ by showing that $5$, $5^4$, and $5^7$ all have the same remainder after dividing by $31$.
- Show that for any integers $g$, $N$, and $p$ satisfying the equation $g^p = m \cdot N + 1$ for some integer $m$, the numbers $g^{x+p}$ and $g^x$ have the same remainder after dividing by $N$ (where $x$ is any integer).
*(Hint: use the fact that $g^{x+p} = g^x \cdot g^p$.)*

*Related Mathematical Moments: ***Securing Data in the Quantum Era****.**

*—Tamar Lichter Blanks*

*Scientific American*, March 14, 2022

March 14, or $\pi$ (Pi) Day, has turned $\pi$ into the most famous number. Writing in *Scientific American*, mathematician Alissa S. Crans takes issue with pi’s fame. “I’m tickled that honoring something mathematical has become a widespread phenomenon,” she writes. “But, at the same time, I’m disappointed that this numerical celebrity seems to be somewhat of an accident.” Crans wants people to understand that math is more than just weird numbers. Math is full of fascinating mysteries and clever solutions to deceptively hard problems. In that spirit, the International Mathematical Union has turned March 14 into the annual International Day of Mathematics. In this article, Crans shares examples of other fascinating facets of math worth celebrating every March.

**Classroom activities:** *pi*, *cake-cutting, irrational numbers, infinity*

- (All levels) Explore all the ways that NASA uses $\pi$ to solve its space travel problems.
- (Geometry) Solve this puzzle about a space probe photographing a dwarf planet in our solar system.
- (All levels) Check out NASA’s many other $\pi$-related puzzles.

- (Mid level) Crans writes that math can teach you the scientifically perfect way of cutting a cake so that every person gets the same amount, with the fewest cuts. Watch this Numberphile video about the cake-cutting problem to see how this works.
- (High level) The decimal representation of $\pi$ contains a non-repeating sequence of digits after its decimal point because it is an irrational number. But Crans notes that $\pi$ is not special in this regard. She writes: “If you asked a genie to choose a number truly at random, the likelihood it would pick an irrational number is 100 percent!” Discuss why this is true.
*(Hint: it has to do with different sizes of infinity and the fact that the rational numbers have measure zero.)*

*—Max Levy*

**In Music and Math, Lillian Pierce Builds Landscapes**

*Quanta Magazine*, March 30, 2022**Katherine Johnson’s great-granddaughter inherited the late mathematician’s talents**

*Today,*March 29, 2022**Dennis Sullivan, Uniter of Topology and Chaos, Wins the Abel Prize**

*Quanta Magazine,*March 23, 2022**How the way we’re taught to round numbers in school falls short**

*Science News*, March 22, 2022**The Evolving Quest for a Grand Unified Theory of Mathematics**

*Scientific American*, March 21, 2022**U.S. News Ranked Columbia No. 2, but a Math Professor Has His Doubts**

*The New York Times,*March 17, 2022**To Keep Students in STEM fields, Let’s Weed Out the Weed-Out Math Classes**

*Scientific American,*March 15, 2022**How Much Pi Do You Really Need?**

*WIRED,*March 14, 2022**This professor studies each swimmer as a math problem. It’s helped them to be faster**

*NPR*, March 12, 2022**Math’s ‘Oldest Problem Ever’ Gets a New Answer**

*Quanta Magazine,*March 9, 2022**8 real-life applications for math equations you learned in high school**

*Watauga Democrat,*March 7, 2022**In New Math Proofs, Artificial Intelligence Plays to Win**

*Quanta Magazine,*March 7, 2022

- The United Nations and the geometric mean
- 99 ways to prove your theorem
- “We solved the problem.”
- 2D tori in rat neurospace

The Google “geometry” alert for February 7, 2022 pulled up a webpage from the United Nations Development Programme with the title “Why is the geometric mean used for the HDI [Human Development Index] rather than the arithmetic mean?” (Read more about the Human Development Index here, including technical notes.) The *arithmetic mean* of $n$ numbers $x_1, \dots, x_n$ is just the usual average $\mu = \frac{1}{n}(x_1+\cdots +x_n)$, whereas their *geometric mean* is $\gamma = \sqrt[n]{x_1 \cdots x_n}$, the $n$th root of their product. The concept of geometric mean goes back at least to Euclid, who used the “mean proportional” (without explicitly defining it) between two lengths $a$ and $b$ to mean the length $c$ such that $a$ is to $c$ as $c$ is to $b$. In modern notation, $a/c=c/b$ so $c^2=ab$ and $c=\sqrt{ab}$, the special case when $n=2$. The geometric mean could also be called the logarithmic mean, since $\log \gamma = \frac{1}{n}(\log x_1 + \cdots + \log x_n)$, which suggests how the geometric mean reduces the impact of larger terms. For example if $x_1=1$ and $x_2 = 100$, then $\mu = 50.5$ whereas $\gamma$ is only 10.

As the UN explains it, they use $\gamma$ in computing the HDI because “Poor performance in any dimension is directly reflected in the geometric mean. In other words, a low achievement in one dimension is not linearly compensated for by a higher achievement in another dimension.” In mathematical terms, for example dividing $x_i$ by $2$ will divide $\gamma$ by $\sqrt[n]{2}$ regardless of the size of $x_i$ compared to the other factors. In human development terms, “a 1 percent decline in the index of, say, life expectancy has the same impact on the HDI as a 1 percent decline in the education or income index.”

There are at least 99 styles in which to prove a mathematical statement; they are all illustrated in Philip Ording’s book *99 Variations on a Proof* (Princeton, 2019), reviewed by Dan Rockmore (paywall protected) in the *New York Review of Books*, January 13, 2022. The statement in question is the following:

- If $x^3-6x^2+11x-6=2x-2$, then $x=1$ or $x=4$.

This is not an especially striking specimen of mathematics, but that is the point. Ording is taking us on a mathematical trip following the path of the French literary gymnast Raymond Queneau, whose *Exercises de Style*, first published in 1947, tells the same unremarkable story (young man with peculiar hat seen arguing on a bus, later spotted getting advice about moving a button on his overcoat) in 99 different styles (telegraphic, exclamatory, botanical, retrograde, …). Rockwell, a mathematics professor at Dartmouth, obviously likes Ording’s book and does his best to bring the *NYRB* readership around to where they might enjoy it, too. He starts with the assertion “Mathematics is writing,” dragging the subject into their purview (although most other mathematicians would probably not agree with the characterization; he would have been safer with “mathematics almost always involves communication”). And when you have writing you have *style:* Different mathematicians communicate in very different ways. Ording’s styles, though, like Queneau’s, are not just personal or aesthetic. Rockwell gives us examples, among which:

- #10
*Wordless.*(See below.) - #16
*Ancient.*A proof in Babylonian style, written entirely in Babylonic cuneiform. - #34
*Medieval.*Formatted as a medieval manuscript, starts with “Suppose that the intensity of a quality is as the cube of its extension and 9 times that less 6 times its square. It will be demonstrated that when this quality achieves an intensity of 4, its extension is 1 or 4.” (Here the equation is rewritten as $x^3-6x^2+9x=4$.) - #43
*Screenplay.*In standard movie-screenplay style, a competition set in 1548 Milan between the algebraic giants of the day, Girolamo Cardano (represented by his protégé Ludovico Ferrari) and Niccolò Tartaglia.

Rockwell quotes extensively from the commentaries that Ording has attached to each proof “outlining the inspiration of the variation and reflecting on the culture of mathematics through the ages.” He adds, “The less mathematically inclined reader might benefit from reading these commentaries first.”

The *New York Times *on February 1, 2022 included an article by William J. Broad with the headline “A Patron’s Vital Spark Revived a Math Quest.” The quest was the proof of Fermat’s Theorem (that $x^n+y^n=z^n$ has no non-zero integer solutions for $n\geq 3$). As Broad tells us, “according to the usual narrative” the problem was solved by Andrew Wiles toiling in secret for seven years. But in fact the credit should be shared with Mr. James M. Vaughn, Jr., a rich Texan whose money “drew top mathematicians to the puzzle after great minds had given up, succeeded in bringing the moribund field back to life, and may have helped make Dr. Wiles’s breakthrough possible.” James Vaughn himself is less circumspect, and told Broad, “We solved the problem. If we hadn’t put the program together as we did it would still be unsolved.”

Vaughn’s statement is not as far-fetched as it might seem. As Broad explains, Vaughn (who took graduate-level math at the University of Texas) was inspired by Eric Temple Bell’s *The Last Problem* (1961) and decided to devote part of his considerable inherited wealth to a foundation that would fund research on the problem, starting in 1972. At that time “no mathematician would take his financial aid or even admit to being interested in working on the conundrum.” Vaughn persevered and in 1981 funded a major conference on the topic at MIT, with 76 participants, including the prominent algebraists Kenkichi Iwasawa, Barry Mazur, and Alte Selberg. Andrew Wiles was one of the organizers. Broad calls our attention to the salience of elliptic curve theory among the subjects treated and implies that Vaughn was responsible for this emphasis, which was unpopular among some of the attendees but turned out to be crucial for the final resolution of the problem.

As Broad tells us, one reason that James Vaughn’s role went untold at the time Wiles’s proof was big news is that since 1992 his foundation had switched its support to the arts. Blame it on the IRS and an auditor who was convinced that mathematical research was a boondoggle; Vaughn and his wife were apparently threatened with huge fines and incarceration. As Vaughn put it: “We figured it was too dangerous. You don’t have that kind of trouble if you give to a ballet company.”

“Toroidal topology of population activity in grid cells” ran in *Nature *on January 12, 2022. The authors, an international team led by Richard Gardner and Erik Hermansen (Norwegian University of Science and Technology, Trondheim), are investigating topological models for the firing pattern of *grid cells *and utilizing topological data analysis techniques. Grid cells reside in the medial entorhinal cortex (MEC), a key component of the neural system that maps an animal’s position within its physical environment. It was known from previous work involving some of the same authors that a grid cell fires, in an individually characteristic hexagonal pattern, depending on the location of the animal. Fig. 1 shows the firing (red dots) of a single grid cell in the MEC of a rat exploring (grey path) a square enclosure. Those authors commented in that earlier paper, “Implausible as the idea might have seemed, cells with regular, periodic place fields are found in the medial entorhinal cortex.”

The grid cells are organized in modules. In the new paper, Gardner *et al.* demonstrate that the periodic place fields of the cells of a single module fit together to map the floor of the enclosure onto a torus. To do this they followed the progress of a rat trained to forage for food in a square enclosure (1.5m on a side); instead of one grid cell as above, they used a set of 149 belonging to the same module. Fig. 2. shows the firing patterns of 36 of those cells. Each of the squares should be compared with Fig. 1. above; it shows how that particular cell is registering the rat’s position in the enclosure. This image and the next two are from the open-source article *Nature* **602** 123-128 (2022). They are used under the Creative Commons license, and have been cropped to fit the format of this column.

At each instant in time (about four times a second) the outputs from the cells determine a point in 149-dimensional space. The rat’s exploration path thereby generates a cloud of some 30,000 points in what we can call “neurospace.” The team used the latest mathematically motivated, computer-intensive methods to analyze the shape of this cloud.

They first performed principal component analysis, treating the time slots as observations and the cells as variables, to reduce the number of dimensions from 149 to six.

Then they applied UMAP (Uniform Manifold Approximation and Projection) to reduce the number of dimensions to three and allow the cloud to be visualized. UMAP is described by its developers as “a novel manifold learning technique for dimension reduction … constructed from a theoretical framework based in Riemannian geometry and algebraic topology.”

For an independent check on the topology of that cloud of dots in neurospace, they applied persistent homology theory to the six-dimensional result of principal component analysis. This theory is designed for detecting the topological “shape” of clouds of points, in homological terms: the number of connected components (dimension 0), the number of independent non-bounding closed curves (dimension 1), the number of independent non-bounding closed surfaces (dimension 2), etc. The results are often expressed as a “barcodes” for each dimension, illustrating how structures persist during the course of the analysis. As shown in Fig. 4, persistent homology analysis picks up 1-dimensional zero-th homology, 2-dimensional first homology and 1-dimensional second homology, identifying the cloud as being a 6-dimensional thickening of a 2-dimensional torus.

]]>*USA Today, *February 8, 2022

What’s the best strategy for Wordle, the viral word-guessing game? If you’re a frequent player, maybe you start by figuring out many vowels as possible, or by ruling out some of English’s most common consonants, or maybe you have a favorite go-to word as an opening guess. Many people have used math and statistics to analyze the game. This *USA Today* article highlights math YouTuber Grant Sanderson, who developed a strategy based on information theory. For his computer algorithm, SALET was the starting word that led to the right answer in the smallest average number of guesses—but Sanderson emphasizes that the best strategy for a computer and the best strategy for a human are not the same thing.

**Classroom activities:** *information theory, logarithms*

- (Pre-calculus, Calculus, Computer Science) Watch the 3Blue1Brown video on solving Wordle with information theory (and the follow-up video with a correction—both videos are long but give an excellent introductory lesson on information theory). Let’s explore some other basic examples.
- How many bits of information are contained in one roll of a standard six-sided die? What about in two consecutive rolls of a die? Why does the answer not depend on the numbers that you roll?
- How many bits of information are contained in the choice of a random integer between 2 and 12 (inclusive)?
- If you roll two dice and take the sum, what is the expected information (i.e. entropy)? (Hint: count the number of ways you can roll each number between 2 and 12, then use the equation onscreen at 11:38 in the first video.) Why is this number smaller than the previous answer?
- Discuss the Wordle strategies that you like to use and how they connect to information theory.

- (All levels) Read more about the history and applications of information theory in this
*Quanta Magazine*article.

*—Scott Hershberger*

*Ars Technica, *February 3, 2022

As an ocellated lizard grows up, some of its scales change from green to black or black to green. By adulthood, it has a two-toned pattern of scales that looks like a miniature maze. This complex design resembles something unexpected: a tool from physics. Researchers recently showed that this lizard’s pattern aligns with a simple mathematical model, called the Ising model, that is also used to represent magnets, melting sea ice, and more. In an article for *Ars Technica*, Jennifer Ouellette describes how the Ising model works and how it describes the lizards’ scales, including the way the colors flip over time.

**Classroom activities:** *physics, combinatorics, lattices*

- (High school) Learn about the Ising model by reading the first part of the section “Definition of Ising Model” in these course notes by Stanford graduate student Jeffrey Chang.
- How many possible arrangements of up and down arrows are possible in the $5\times 5$ grid pictured?
- How many possible arrangements are there for an $n \times n$ grid, where $n$ is a positive integer?

- (High school, Geometry, Algebra II) Given any two points $p=(x_1, y_1)$ and $q=(x_2, y_2)$ in the Cartesian plane, the set of points of the form $ap+bq=(ax_1+bx_2, ay_1+by_2)$, where $a$ and $b$ are integers, is an example of a lattice, similar to the lattices used in the Ising model.
- By drawing dots on graph paper, sketch the lattice for $p = (1, 0)$ and $q = (0, 1)$. Make another sketch for the lattice corresponding to $p = (2, -2)$ and $q = (1, 3)$. Which types of lattice are they? Which type most closely resembles the scales on the lizards?
- What is the area of the parallelogram whose vertices are $0, p, q$, and $p+q$ when $p = (2, -2)$ and $q = (1, 3)$?
- Give an example of points $p$ and $q$ for which all of the lattice points lie on a single line in the plane.

*—Tamar Lichter Blanks*

*NPR Short Wave*, February 10, 2022

Mathematics and drag: It’s a unlikely combination, but TikTok star Kyne has captivated viewers with her stunning makeup and outfits, flair for performance, and easy-to-follow explanations of math concepts. Kyne Santos studied mathematical finance at the University of Waterloo and now creates math drag videos full-time. For an episode of the podcast *Short Wave*, Emily Kwong interviewed Kyne about how she both brings math to a wide audience and represents STEM in the drag scene. “My whole message is just that math can be really interesting,” Kyne said. “Math can be beautiful. Math can be fun. And math can be extremely relevant to our world.” (Also see an article about Kyne in the January 2022 issue of *Notices of the AMS*).

**Classroom activities: ***exponential growth, exponential decay, history of mathematics, diversity and inclusion*

- (Algebra II, Pre-calculus) In one of her most popular videos, Kyne explained that if you folded a piece of paper 42 times, it would be as tall as the distance from the Earth to the Moon. Let’s explore this.
- Find a stack of paper. How many times can you fold a single sheet in half? Can you figure out a way to use a ruler to determine the thickness of a single sheet?
- How thick would the sheet of paper get if you folded it 20 times? What about 42 times?
- Each time you fold a sheet of paper in half, its horizontal area decreases. If you started with a sheet of paper the size of a football field (360 ft by 160 ft), what would its area be after 20 folds? What about after 42 folds?
- No matter how you fold a sheet of paper, its volume (length times width times height) should stay the same. Do your calculations agree with this statement?

- (All levels) Kyne mentions that Euclid is one of her math heroes. Discuss the stories of inspiring mathematicians past and present.
- Here are some famous Black mathematicians, women mathematicians, and Arab mathematicians.
- Lathisms, Mathematically Gifted & Black, Spectra (the Association for LGBTQ+ Mathematicians), and Indigenous Mathematicians feature many of today’s experts across pure and applied mathematics.

*—Scott Hershberger*

*Quanta Magazine*, February 8, 2022

Can you turn a circle into a square? Don’t fall for the simplicity of the question—it has baffled mathematicians for almost *2,500 years*. The problem, known as “squaring the circle,” challenges you to cut a circle into any (finite) number of pieces and reassemble them into a square of equal area. It quickly turns philosophical. You’ll wonder how to make a corner out of something round, and *why* those two things are so fundamentally different. Researchers have made some headway in the last century by imagining cutting unorthodox shapes impossible to create with real scissors. Around 1990, one researcher showed how to “take a circular space and make it straight,” a result described as “jaw-dropping.” And this year, mathematicians found an even better way to solve the problem. In this article, Nadis describes the long-running journey and fascinating result.

**Classroom activities ***squaring the circle, transcendental numbers*

- (All levels) To learn more about circle squaring, watch this Numberphile video.
- (All levels) Try to square the circle yourself. Use a compass to draw a circle. Use scissors to cut it out. Now, using only that circle—and every part of that circle—try to cut and reassemble pieces into a square. How close can you get?
- (Pre-calculus) The ancient problem originally asked if you could construct a square with the same area as a given circle using just a compass and straightedge (rather than cutting the circle into pieces). This is impossible because $\pi$ is what’s known as a transcendental number. Watch this Numberphile video for an introduction to transcendental numbers.

*—Max Levy*

*FiveThirtyEight*, February 8, 2022

People sometimes joke that American football is poorly named. The sport is mostly a game of *hands*—throwing, catching, carrying, and tackling. But it is often a *foot,* in fact, that decides football games. Kickers will boom the ball from over 50 yards away in the most critical moments. And this year, according to data analyzed by Alex Kirshner, kickers have done so better than almost ever before*. *“There’s been no recent postseason in which NFL teams have asked so much of kickers in such fraught situations,” writes Kirshner. “They’ve responded almost perfectly.” In this article, Kirshner uses metrics like field goal attempts, makes, and distance to compare the past 22 years of NFL playoff data, revealing what he deems “wild success.”

**Classroom activities:** *sports statistics, data analysis*

- (All levels) Let’s analyze the data from the table in the article titled “Kickers have been nearly automatic this postseason.” First, sort the table by season, and then copy the values from the top-left corner (2000) to the bottom-right corner (77.8). Paste them into a spreadsheet (e.g. Microsoft Excel or Google Sheets).
- What is the
**average**number of field goal attempts in a postseason in the years between 2000 and 2021? (See here for help with Excel formulas) - (Mid level) What is the
**average**distance of field goal attempts over that entire span? (Hint: you’ll need the numbers in columns 2 and 4.) What is the**range**in the postseason averages over that same span? Does this value suggest that distance varies a lot or only a little from year to year? - (Mid level) Plot the average distance data on your spreadsheet as a scatter plot (x-axis=season; y-axis=avg. distance). Discuss whether it looks like postseason kicking is improving. Add a trendline and the $R^2$ value to the plot—how strong is the evidence?

- What is the
- (High level) Have students create and analyze their own data in groups with an online typing test. (Alternative options: Pacman or a simple mouse-based field goal kicking game). If they want to participate, each student can test their typing speed and share their words-per-minute score. Collect all the data. Calculate the average and standard deviation, first by hand, and then via Excel or a graphing calculator. (Find basic statistics formulas here
__.__)

*—Max Levy*

**Let’s learn about pi**

*Science News for Students*, March 1, 2022**All Ocean Life Follows This Massive Pattern—Except Where Humans Have Interfered**

*Scientific American*, March 2022 issue**MU Professor, High Schooler and Undergraduate Mathematician Lead the Research on Variants**

*Flatland,*February 25, 2022**Mathematician Answers Math Questions from Twitter**

*WIRED,*February 18, 2022**Does Calculus Count Too Much in Admissions?**

*Inside Higher Ed*, February 14, 2022**[Op-Ed] Virginia students and businesses need modern mathematics**

*Richmond Times-Dispatch*, February 12, 2022**Scientists Say: Geometry**

*Science News for Students*, February 7, 2022**Promoting mathematics to girls in Ghana**

*Nature*, February 7, 2022**Op-Ed: Math anxiety is real — and we’re passing it on to our kids**

*The Los Angeles Times*, February 2, 2022**More math, more money: How profit-seeking has sparked innovations in mathematics**

*Big Think*, February 2, 2022

- Gödel’s incompleteness theorem in
*The Guardian* - “Fun with Math” in
*The New Yorker* - Rock-paper-scissors and evolutionary game theory
- “What physics owes to math”

Alex Bellos’s Monday puzzle in *The Guardian* for January 10, 2022 was derived from explanations of Gödel’s incompleteness theorem due to the logician Raymond Smullyan. (Smullyan published *Forever Undecided: A Puzzle Guide to Gödel *in 1987). The setting for the puzzle, as Bellos presents it, is a hypothetical island he calls “If.” Natives of If are either “Alethians” or “Pseudians;” they are indistinguishable except that Alethians always tell the truth, while Pseudians always lie. This sounds like the traditional Liar Problem, but there is a wrinkle: on the island is a ledger where every native is listed, along with his or her tribe. Anyone can consult this ledger. You arive on If and a person, Kurt, comes up to you stating: “You will never have concrete evidence that confirms I am an Alethian.” The puzzle: is Kurt an Alethian, a Pseudian or neither? Think about it before checking Bellos’s solution and before reading on.

Now comes the connection with Gödel’s incompleteness theorem, which states, as Bellos puts it, “that there are mathematical statements that are true but not formally provable.” Suppose you are the first non-native ever to visit If, so you know that everyone you meet is Alethian or Pseudian. Kurt pops up and says, just as before, “You will never have concrete evidence that confirms I am an Alethian.” But now just as he speaks the Ledger burns to ashes.

Where are we? Kurt cannot be a Pseudian, because with no Ledger that statement has to be true. So you know Kurt is an Alethian. But you can never have concrete evidence of that fact because if you did, his statement would be false, and it can’t be false since he is Alethian. Think about it. [Thanks to Jonathan Farley for bringing this item to my attention. -TP]

Dan Rockmore’s contribution to “Talk of the Town” in the January 17, 2022 issue of the *The New Yorker* was an item titled “Fun with Math.” He recounts attending “a recent evening of math dinner theater” organized by Cindy Lawrence, director of New York’s National Museum of Mathematics. The event featured Peter Winkler, a mathematics professor at Dartmouth and expert on math puzzles. Among the puzzles and phenomena that Winkler served up for discussion during dinner:

- “On average, how many cards does it take to get to a jack in a shuffled deck of fifty-two cards?”
- “What’s the best way to use two coin tosses to determine which of two coins, one fair and one ‘biased,’ is fair?”
*Simpson’s paradox*in statistics, best described by an example: for Berkeley’s graduate programs in 1973, overall “men were admitted at a higher rate than women, but, program by program, women were admitted at a higher rate.” (See MinutePhysics for a more detailed explanation.)

Apropos of Simpson’s paradox Marilyn Simons, a guest with a PhD in economics, remarked, “I think that, to a lot of us who even *think* we know statistics, the way we process statistics is not deeply informed.” Elsewhere Rockmore quotes her as saying that her husband Jim (identified as “a financier and a former mathematician”) doesn’t like puzzles: “He says that if he works that hard he wants to get a theorem out of it.”

“Non-Hermitian topology in rock-paper-scissors games” by the three Tsukuba physicists Tsuneya Yoshida, Tomonari Mizoguchi and Yasuhiro Hatsugai was published January 12, 2022 in *Scientific Reports*. This is a physics article, but it applies a nice piece of mathematics, *evolutionary game theory,* to the familiar rock-paper-scissors game.

The game consists of two players; at a signal each shows a clenched fist (“rock”), a flat hand (“paper”) or a vertical hand with the first two fingers displayed (“scissors”). The winner (rock smashes scissors, scissors cut paper, paper covers rock) gets one point, the loser loses one. If both players show the same symbol, each gets zero.

The article contains this image:

Here R, P and S have to stand for rock, paper and scissors, but how is this diagram related to the game? We need to make a detour into e*volutionary game theory*. This is a method for simulating the process of evolution in populations. Here the population is split among three subspecies; let’s call them Ravens, a fraction $s_1$ of the population, Penguins with $s_2$, and Swifts with $s_3$, where the fractions $s_1, s_2, s_3$ add up to 1. These correspond to the three “pure strategies” in the game: at every encounter, a Raven will play “rock,” a Penguin will play “paper” and a Swift, “scissors.” The *state vector* ${\bf s}=(s_1, s_2, s_3)$ encapsulates the current mix in the population.

Evolution occurs in time. Suppose the population is in state ${\bf s}$ at some moment. Where will it be just a litle later? For example, suppose ${\bf s}=(\frac{1}{2},\frac{1}{2},0)$. That means half the population are Ravens and half are Penguins. So a Raven will meet a Penguin with probability $\frac{1}{2}$, and can expect to lose half a point. Likewise a Penguin will meet a Raven with probability $\frac{1}{2}$ and so can expect to gain half a point. The Penguins have an advantage. If the object is to model evolution, then the Penguin’s advantage in that state should translate into their population increasing at the expense of the Ravens. That gives a clue to the meaning of the red arrow at the point $(\frac{1}{2},\frac{1}{2},0)$: the population mix at that point is shifting to the right. More Penguins, fewer Ravens.

To make this more precise, keeping the language of evolution, we measure the *fitness* of one of the groups at some state ${\bf s}=(s_1,s_2,s_3)$ of the population by the expected gain or loss in points at the next encounter. So the fitness of the Ravens at state ${\bf s}$ will be the probability of meeting a Penguin times $-1$ plus the probability of meeting a Swift times 1. We write this as

$F(\mbox{Ravens}|{\bf s})= -s_2 + s_3.$ Similarly $F(\mbox{Penguins}|{\bf s})= s_1 – s_3$ and $F(\mbox{Swifts}|{\bf s})= -s_1 + s_2.$

Finally we set up a dynamical system by stating that the proportion of the population in any group will increase or decrease exponentially with growth coefficient equal to the fitness of that group (which can be positive or negative) at that instant in time. Writing that statement as a differential equation gives the *replicator equation* for rock-paper-scissors:

$$\frac{ds_1}{dt}= s_1(-s_2 + s_3), ~~\frac{ds_2}{dt}= s_2(s_1 – s_3), ~~\frac{ds_3}{dt}= s_3(-s_1 + s_2).$$

In vector form, the equivalent equation is

$$\frac{d{\bf s}}{dt}= (s_1(-s_2 + s_3), s_2(s_1 – s_3), s_3(-s_1 + s_2)).$$

Now we can interpret the first image in this item, which shows the state space for rock-paper-scissors as an evolutionary game. The arrows represent the direction of evolution, with the magnitude encoded by color saturation. The central cross marks the equilibrium $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$; the blue loop is the solution curve obtained by numerically integrating the replicator equation starting at the point ${\bf s} = \frac{1}{3}(1-\delta, 1+\delta/2, 1+\delta/2)$, with $\delta=0.1$.

For this game, all the solution curves are closed loops. In fact, they are level curves of the function $f(s_1,s_2,s_3)=s_1s_2s_3$. This follows from the equality $\displaystyle{\frac{d{\bf s}}{dt}\cdot \nabla f= 0}$, which is actually fun to check. Try it.

The evolutionary game described here is just the starting point for the article by Yoshida *et al.* They consider perturbations of this system that break its symmetry—surprisingly, the resulting phenomena have parallels in condensed-matter physics.

On January 12, 2022 *Le Monde* ran a guest article with the title “What physics owes to math” (text in French) written by two physicists, Jean Farago and Wiebke Drenckhan, from the Institut Charles-Sadron in Strasbourg. The authors begin with the famous quote from Galileo about the Book of Nature being written in the language of mathematics, and go on to observe: “The millenia elapsed between the birth of mathematics and its use in physics demonstrate that this contiguity between natural phenomena and the mathematical laws of our human rationality was far from being obvious.”

Farago and Drenckhan mention that one of the most antonishing examples of the “intimate” relationship between mathematics and physics comes from complex numbers. Starting in the 16th century, mathematicians found that calculating solutions to polynomial equations with whole-number coefficients required the use of an ‘imaginary’ number $i$ with $i^2=-1.$ “How could anything be more abstract than this fictitious number, given that ordinary numbers always have a positive square ($2^2=(-2)^2=4$)!” But fast-forward to 1929 and Schrödinger’s equation $i\hbar\partial_t\psi=H\psi$, which doesn’t work without it. We read that no one was more surprised by “this irruption of $i$ in the corpus of physical laws” than Schrödinger himself, and that he described his reaction, in a footnote, by quoting an unnamed Viennese physicist, “known for his ability to always find the *mot juste*, the cruder the *juste*r,” and who compared the appearance of $i$ in that equation to one’s involuntary (but welcome) emission of a burp. Our authors add: “This shows us that a contiguity can sometimes also exist between humor and physics.” [My translations. -TP]

“What physics owes to math” could have mentioned an article from *Nature* last month: “Quantum theory based on real numbers can be experimentally falsified,” written by an international team with corresponding author Miguel Navascués (IQOQI, Vienna). Physical experiments are expressed in terms of probabilities, which are real numbers. So why can’t there be a “real” quantum theory? The authors show that complex numbers are actually needed, by devising “a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.”

*The Guardian, *January 24, 2022

Time and time again, mathematical ideas developed decades or even centuries ago find unexpected—and profitable—uses in industry. In an opinion piece for *The Guardian*, mathematician David Sumpter gives a quick tour of how mathematics has transformed the modern world and speculates about which branch of mathematics will lead to money-making applications next. Fractal geometry, chaos theory, and random walks are all possibilities, he writes. “You don’t need to be a mathematical genius yourself in order to put the subject to good use. You just need to have a feeling for what equations are, and what they can and can’t do.”

**Classroom activities: ***fractals, probability, matrices*

- (All levels) Introduce students to fractals with K-12 activities from the Fractal Foundation. Topics include fractal triangles, coastlines, exponents, and more.
- Ask students to find examples of fractals in their own home or neighborhood and share them with the class.

- (Middle school) Introduce students to random walks using coin flips with this lesson plan from the National Museum of Mathematics.
- (Linear Algebra) When teaching eigenvalues and eigenvectors, use Google’s PageRank algorithm as an example (see these Cornell notes). Discuss: what are some other situations where a similar approach might be useful? What are the limitations of PageRank?
- (Advanced) Have students complete the exercises at the bottom of the Cornell notes.

*—Scott Hershberger*

*Slate, *January 14, 2022

During this winter’s Omicron wave, you may have taken rapid COVID-19 tests. But interpreting the results is not always straightforward. In an article for *Slate*, mathematician Gary Cornell explains two statistical terms and how they relate to COVID tests. A high *specificity* means that a test gives very few false positives. A high *sensitivity* means that a test gives very few false negatives. Rapid COVID tests have a high specificity, but a not-so-high sensitivity, Cornell writes—so a positive test result means you are almost certainly infected, but a negative result cannot give total confidence that you are in the clear.

**Classroom activities: ***statistics, specificity and sensitivity*

- (High school) Introduce students to sensitivity and specificity with this lesson plan from Penn State. The lesson also discusses
*positive predictive value*and*negative predictive value*, which answer an important question: What is the chance that a person who tests positive is infected, or that a person who tests negative is not infected? - (High school) A test’s predictive value depends in large part on how common the condition is in the population. (For example, a recent New York Times investigation found that prenatal tests for rare disorders give far more false positives than true positives.) For each of the following hypothetical diseases, suppose you take a test with 99.9% specificity and 80% sensitivity. If you test positive, what is the probability that you actually have the disease? If you test negative, what is the probability that you do not have the disease?
*(Hint: create charts like those in the lesson plan.)*- Disease W: prevalence is 10 out of every 100,000 people
- Disease X: prevalence is 10 out of every 1,000 people
- Disease Y: prevalence is 10 out of every 100 people
- Disease Z: prevalence is 40 out of every 100 people

Discuss how these results could be relevant during the different stages of a pandemic.

*—Scott Hershberger*

*Quanta Magazine*, January 13, 2022

Despite all we know about math, the field is still full of mysteries. Some may seem hopelessly abstract, but others have to do with fundamental concepts we all recognize, like primes. “Prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics,” writes Kevin Hartnett*.* A prime number (like 3, 5, 23, or 419) is only divisible by 1 and itself. The mystery of primes is that they seem to follow no discernible pattern. Yet a 160-year-old idea called the Riemann hypothesis suggests that there *is* a pattern to be found, and mathematicians are hard at work trying to crack it. There is even a million-dollar prize on the line. In this article, Hartnett describes a groundbreaking new step toward solving this stubborn mystery.

**Classroom activities: ***prime numbers, finding patterns, sequences*

- (All levels) Learn more about the Riemann hypothesis with these videos from
*Quanta Magazine*and Numberphile. - (Mid level) Write out the prime numbers under 100, one below the other. There should be 25 of them. One column to the right, in the space between each consecutive prime, write the result of subtracting the smaller number from the larger number. Do you see a pattern? Discuss why or why not. Compare this to the pattern you get when subtracting consecutive Fibonacci numbers.
- (High level) Compute the values of the function $f(n) = n^2 + n + 41$ for $n = 1, 2, 3, \text{and } 4$. Referencing your earlier work, what appears to be happening with this function? Can you find a counterexample to your conjecture?
- (High level) Have each student come up with their own simple rule to create a sequence of numbers. The rule should involve just addition, subtraction, multiplication, or division and should involve either one, two, or three consecutive terms of the sequence. Ask students to swap sequences in partners and see if they can figure out each other’s patterns.

*—Max Levy*

*The Conversation*, January 4, 2022

With a few folds, a piece of paper can become a piece of art—and maybe more. In an article for *The Conversation*, mathematician Julia Collins writes about how origami can inspire mathematical discovery. Collins starts with a small, parallelogram-shaped bit of origami called the sonobe unit. With six of those units, you can build a cube. With more of them, you can create other mathematical shapes such as Platonic solids, Archimedean solids, and Johnson solids. You can also explore principles in the mathematical field of graph theory, like the Four-Color Theorem, by building a shape out of sonobe units of different colors. Origami may even be useful for technology like unfolding solar panels in space.

**Classroom activities:** *origami, geometry, technology*

- (Middle school / high school) Follow Collins’ instructions or the video linked in her article to build sonobe units out of a square pieces of paper.
- Students can work alone or in groups to build cubes or more complicated geometric shapes out of the sonobe units.
- Discuss: what does it mean for a shape to have symmetry? Why does the sonobe unit lend itself to building objects with symmetry?

- (Middle school / high school) Watch the video “See a NASA Physicist’s Incredible Origami,” which is linked to in the article. What are some examples of technology that might be inspired or improved by origami?

*—Tamar Lichter Blanks*

*CTV News, *January 13, 2022

Throughout the COVID-19 pandemic, mathematical models have helped policymakers estimate infection risk based on factors like vaccination status, indoor versus outdoor setting, and crowd density. But one of the most important factors for determining transmission risk is also potentially misleading: positivity rate, or the percentage of tests that come back positive. The hyper-transmissible Omicron variant is straining test supplies, so a larger slice of positive cases is going unreported—skewing positivity rates. Mathematicians modeling the spread of COVID-19 are struggling to keep up. “We’re still adapting to flying blind in terms of reported cases,” one mathematician told *CTV News* reporter Sarah Smellie. In this article, Smellie explains how mathematicians need to adapt their models to keep up with the constantly changing pandemic.

**Classroom activities: ***exponential growth, logarithms, data analysis*

- (Algebra II) The doubling times (how long it takes for the number of infections to double) for Omicron are “some of the fastest we’ve seen in the pandemic”—between 1.5 and 3 days in some regions. Imagine a city of 10 million people where two people are sick. If nobody is vaccinated or takes any precautions to prevent the spread, how many days would it take for 10% of all inhabitants to catch the disease if:
- The cases doubled every 2 days
- The cases doubled every 3 days
- Discuss the implications for public health interventions.

- (High level) Collecting enough data points is an important part of having a reliable model. To see why, gather two different colors of marbles (or pieces of paper or other item)—one will represent negative cases and the other positive. Place 10 marbles in each of four identical bags or boxes according to the following ratios of
*positive:negative*: 1:9, 2:8, 3:7, 5:5. Now, scramble the bags and remove one marble from each bag. Write down a guess of which bag corresponds to which ratio. Repeat this until no marbles remain. How many rounds did it take until you were correct about all of the bags? Discuss how this relates to the challenge of determining COVID infection rates by sampling from different areas of the country.*Remote-friendly version: Use**Wheel of Names**with 4 different ratios of names “positive” or “negative” instead of marbles and bags.*

**Related Mathematical Moments: ****Resisting the Spread of Disease.**

*—Max Levy*

**After twice being denied tenure, this Naval Academy professor says she is seeking justice**

The Washington Post, January 31, 2022**The Texas Oil Heir Who Took On Math’s Impossible Dare**

The New York Times, January 31, 2022**Why mathematicians sometimes get Covid projections wrong**

The Guardian, January 26, 2022**How Infinite Series Reveal the Unity of Mathematics**

Quanta Magazine, January 24, 2022**Take an online journey through the history of math**

Science News, January 18, 2022**Love Wordle? Here’s How to Use Math to Dominate Your Friends at the Viral Word Game**

Popular Mechanics, January 14, 2022**Fairer Elections in Pa. Could Depend on 12 Mathematicians**

NBC Philadelphia, January 17, 2022**ArXiv.org Reaches a Milestone and a Reckoning**

Scientific American, January 10, 2022**Secret maps revelation, testimony of mathematicians deal blows to GOP defense as NC redistricting trial concludes**

NC Policy Watch, January 7, 2022**Minnesota mathematicians, data scientists use new technology to shape political districts**

*Minneapolis Star-Tribune,*January 1, 2022

*The Conversation, *December 1, 2021

Pop superstar Adele has a habit of titling her albums in a peculiar way. Each album title is an integer that represents how old the singer was when she began writing the songs. David Patrick of the Art of Problem Solving searched the On-Line Encyclopedia of Integer Sequences to see if Adele’s albums match any known integer sequence. Nine sequences turned up—some of them easy to describe and others less so. For *The Conversation, *Anthony Bonato explains the rules underpinning one of the possible “Adele sequences.”

**Classroom Activities: ***sequences, prime numbers*

- (Middle school / high school) Explore the OEIS by having students come up with the first four terms of an integer sequence. They can choose four integers based on something in their own life, or they can choose four integers less than 40 that could represent the ages at which a musician releases new albums.
- Have students search the OEIS to see if their sequence matches any known sequences in mathematics, then share their findings with the class.
- Discuss: what do the results suggest about the mathematical significance of the “Adele Sequence”?

- (Number Theory, Problem-solving) The sequence A072666 that Bonato analyzes depends on the sequence of prime numbers. Ask students to prove that there are infinitely many primes using the following hints (full solution here):
- Assume that there are finitely many primes which can be denoted $p_1,p_2, \dots, p_n$.
- Consider the number $(p_1 \times p_2 \times \dots \times p_n) + 1$.

—*Leila Sloman*

*Science News for Students*, December 16, 2021

Turbulence in water affects where fish eggs end up. Massive icebergs drift on ocean currents. Wildfire smoke from the U.S. West can make its way to Europe. In all of these cases, mathematics helps researchers predict what will happen, offering potential solutions to environmental threats. As Rachell Crowell explains, the computer models used to simulate how objects drift depend on the math of *differential equations*. These equations relate quantities that change as time passes and vary at different locations in the environment. Crowell interviews several researchers about what they have learned about the science of drifting and why it is important.

**Classroom activities: ***mathematical modeling, differential equations, heat equation*

- (Algebra I, Algebra II) The article mentions that satellites helped reveal drifting wildfire smoke. Satellites are also useful in studying the fire itself. Explore one real-life example with this assignment from NASA’s Jet Propulsion Laboratory.
- (Differential Equations) The article also mentions that researchers use the heat equation to predict icebergs’ melt rate. When introducing partial differential equations, show students this 3Blue1Brown video that visually explains the heat equation.

*—Scott Hershberger*

*The New York Times*, December 10, 2021

Math shapes the world around us—and it shapes the *shapes* around us too. A trio of researchers from France and the United Kingdom are known for investigating the math and physics of how seashells form. Their newest advance deals with ammonites, an extinct group of mollusks. A type of ammonites called Nipponites had weird shells that twisted, turned, coiled, and bulged in unexpected directions. “The first time you look at it, it’s just this tangled mess,” mathematician Derek Moulton told reporter Sabrina Imbler. “And then you start to look closely and say, oh, actually there is a regularity there.” In this article, Imbler shares how Moulton’s team came up with a mathematical model that explains the origins of these ammonites’ weird shapes.

**Classroom activities: ***growth rates, symmetry, golden ratio*

- (All levels) Nipponites appear so strange because they are very asymmetrical, whereas nature is normally full of symmetry. Ask students why symmetry tends to be beneficial in the living world. (As an example, think of symmetry in birds and fish as they move.)
- (All levels) Nipponites provide one example of an initial symmetry giving way to an asymmetric result. Have students explore another example using a thin string. Hold the string above a table so it hangs freely. Now lower it gently so that one end touches the table, and continue to lower it until it has folded over itself a few times. Inspect the bundle and take a picture. Repeat this a few times and share your observations. Does the bundle look the same every time or does it appear unique? Discuss how this compares to what happens inside an ammonite shell.
- (Algebra II, Pre-calculus) To learn more about how a simple mathematical rule can give shells and flowers fascinating shapes, watch this Numberphile video on the Golden Ratio.

*Related Mathematical Moments:* Going Into a Shell.

*—Max Levy*

*Newsweek,* November 30, 2021

When Lotfi Zadeh first came up with the idea for fuzzy logic, a way of enshrining in mathematics the uncertainty and imprecision of life, it was not received well by everyone. But almost sixty years later, Zadeh’s impact on the world is indisputable. On November 30, a Google Doodle celebrated this impact, marking the 57th anniversary of Zadeh’s seminal paper, “Fuzzy Sets.” For *Newsweek*, Soo Kim articulates what was so important about that work: “Considered an early approach to artificial intelligence, Zadeh’s fuzzy logic structure formed the basis of various modern everyday technologies including facial recognition, air conditioning, washing machines, car transmissions, weather forecasting, stock trading and rice cookers.”

**Classroom Activities: ***set theory, logic, fuzzy logic*

- (High school) Teach an introduction to set theory with this online lesson, assigning the “Try It Now” boxes as problems. For extra practice, assign the following problems:
- Let $S = \{ 1, 10, 19 \}$. Write down all the subsets of $S$.
- Suppose the universal set is all the integers. Let $A = \{ \text{even integers} \}$ and $B = \{ \text{multiples of 3} \}$.
- Find 3 numbers in $A^c \cap B$.
- Find 3 numbers in $A \cap B^c$.

- (Advanced) Prove that for general sets $A$ and $B$, $(A \cap B)^c = A^c \cup B^c$.

- (High school) In classical logic, an item either belongs to a set or it doesn’t—there is no in-between. By contrast, fuzzy logic allows items to be partly in sets. Have students read this
*Britannica Kids*explainer on fuzzy logic. Ask them to think of three situations in which fuzzy logic might be more useful than classical logic and justify their claims.- (Advanced) Have students read the first four pages of Zadeh’s original paper and then come up with an example of two membership functions $f_A(x)$ and $f_B(x)$ in the real numbers such that $A$ is a fuzzy subset of $B$.

*—Leila Sloman*

*Nature, *December 1, 2021

Mathematics might sometimes seem very dry. You follow fixed rules to find right answers and check your work carefully to avoid wrong answers. That process doesn’t seem creative, but when it comes to discovering new proofs and theorems, creativity is arguably the most important trait. In an article for *Nature*, Davide Castelvecchi explains how math researchers have recruited a new ally to track down creative solutions to math’s mysteries—artificial intelligence. Researchers used machine learning, a type of AI, to comb through huge datasets and find patterns related to the study of knots and symmetry. It takes creativity and intuition to find these hidden patterns. “As mathematical researchers, we live in a world that is rich with intuition and imaginations,” one researcher said. “Computers so far have served the dry side. The reason I love this work so much is that they are helping with the other side.”

**Classroom activities** *machine learning, knots*

- (All levels) The machine-learning algorithm helped researchers solve a question about knots that eluded researchers for decades. Read more about this study’s “knot” result and/or watch this Numberphile video about knot theory.
- (Middle level) A central question that researchers ask is whether a set of knots that appear different are actually equivalent. With a shoelace, string, or yarn, try making some of the distinct knots found here. Discuss what properties make them different. (For example, how many times the thread crosses over itself.)
- (High level) Discuss
*why*machine learning can speed up innovation, based on this article and others. (Here’s a primer on machine learning.) One researcher told the writer: “Without this tool, the mathematician might waste weeks or months trying to prove a formula or theorem that would ultimately turn out to be false.” If artificial intelligence can help find patterns quickly, what other areas of science or society could benefit from these quick solutions? In what situations would machine learning be more dangerous or unethical?

*Related Mathematical Moments:* Being Knotty.

*—Max Levy*

**Our Favorite Things: Math and Community in the Classroom**

Short Wave, NPR, December 28, 2021**The Year in Math and Computer Science**

Quanta Magazine, December 23, 2021**How Mathematicians Cracked the Zodiac Killer’s Cipher**

Discover Magazine, December 18, 2021**Google Doodle honors French mathematician Émilie du Châtelet**

CNET, December 16, 2021**Mathematician Hurls Structure and Disorder Into Century-Old Problem**

Quanta Magazine, December 15, 2021**Quantum physics requires imaginary numbers to explain reality**

Science News, December 15, 2021**Shirley McBay, Pioneering Mathematician, Is Dead at 86**

The New York Times, December 14, 2021**Abstractions Are Good for Goodness’ Sake**

The Wall Street Journal, December 10, 2021**Using Math to Rethink Gender**

Short Wave, NPR, December 1, 2021

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- The exponential function in
*Slate* - New mathematical ideas from artificial intelligence
- Math history online

The online magazine *Slate* invited Gary Cornell to explain to all of us why the Omicron variant of Covid-19 was spreading so fast. (This was back on December 17, 2021). He contributed “The Math That Explains Why Omicron Is Suddenly Everywhere” with an illustration containing a graph like this one:

Cornell explains that the current pattern of Omicron cases doubling every two days is an example of *exponential growth,* and that we humans are not wired to process that phenomenon reliably: “when it comes to exponential growth, your gut feelings *are* going to be wrong and you need to stop and do some (elementary) math.” The characteristic of exponential growth is that the increase during the next time period is proportional to the value right now. To illustrate this phenomenon Cornell retells the story of the king and the chessboard, which ends up with the king owing his opponent one grain of rice on the first square, two on the second, four on the third, and so on, totaling “more than 18 quintillion grains of rice, which would roughly cover the planet and would be the world’s output of rice for about 1,000 years.” He reminds us that cases of the Omicron variant, which were doubling every two days, have the same growth potential. And even if the hospitalization rate is as low as 1%, after less than two weeks the number of hospitalizations would be greater, day by day, than the number of infections from two weeks before.

“For the first time, machine learning has spotted mathematical connections that humans had missed.” This is the beginning of Davide Castelvecchi’s news piece in the December 9, 2021 issue of *Nature*. The article Castelvecchi refers to, “Advancing mathematics by guiding human intuition with AI,” was published in the same journal a week before. The authors were 11 members of Alphabet Inc.’s Deep Mind laboratory in London working with three academic mathematicians from Oxford and Sydney. The team, led by Deep Mind’s Alex Davies and Pushmeet Kohli, states: “We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures.” They present two examples, one involving knots and one representation theory, of their process in action.

The authors set up a general framework for approaching the process, as follows. They take $z$ to be a variable element of some class of mathematical entities (knots, polyhedra, groups, …) and consider two “mathematical objects” $X(z)$ and $Y(z)$ associated with $z$. Are these two objects related, and how? They think of such a relationship as a function, and ask if there is an $f$ such that $f(X(z))\approx Y(z)$ for all $z$.

They give an example of what they have in mind. Let $z$ range over the set of 2-dimensional polyhedra; they set $X(z)$ to be the vector $(V(z), E(z), Vol(z), Surf(z))$, where $V$ and $E$ are the numbers of vertices and edges, $Vol$ and $Surf$ the volume and the area of the polyhedron; and $Y(z)$ to be the number $F(z)$ of faces of $z$. In this case Euler’s formula $V-E+F=2$ means that the function $f(X) = (V, E, Vol, Surf)\cdot (-1, 1, 0, 0) +2$, where $\cdot$ is the dot-product of vectors, does the trick, since in fact $-V+E+2 = F$. Notice that in this example the volume and surface area are not part of the calculation. Eliminating spurious information is part of the “attribution techniques” mentioned above, used in an iterative process of refining the form of the relation to be found.

The training process relies on a large set of specimen values of $z$, a large bank of possible functions as candidates for $f$, and an initial guess of which objects $X(z)$ and $Y(z)$ have the potential of being part of a true mathematical statement of the form $f(X(z))= Y(z)$. If the machine finds an $f$ that works more often than expected by chance, the human partners examine the form of $f$ to see how it can be improved. Besides discarding spurious variables as above, a technique the authors use is *gradient saliency:* imitating optimization in calculus by examining the derivative of outputs of $f$ with respect to the inputs. “This allows a mathematician to identify and prioritize aspects of the problem that are most likely to be relevant for the relationship,” they write. They emphasize the interactive aspect of the iterative process, where “the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.”

Applied to knot theory, the process yielded a theorem the authors describe as “one of the first results that connect the algebraic and geometric invariants of knots.” They comment: “It is surprising that a simple yet profound connection such as this has been overlooked in an area that has been extensively studied,” echoing Euler’s remark about his $V-E+F=2$ discovery: “It seems extremely amazing that, while Stereometry along with Geometry has been studied for so many centuries, nevertheless some of its most basic elements have been unknown until now” (Demonstratio … , p. 141).

Castelvecchi tells us that Alex Davies, one of the project leaders from Deep Mind, “told reporters that the project has given him a ‘real appreciation’ for the nature of mathematical research. Learning maths at school is akin to playing scales on a piano, he added, whereas real mathematicians’ work is more like jazz improvisations.”

“Online exhibit adds up the history of mathematics” by Erin Blackmore ran in the *Washington Post* on November 28, 2021. Blackmore reports on a collaboration among the National Museum of Mathematics, Wolfram Research, and the Overdeck Family Foundation. The History of Mathematics Project has nine interactive exhibits (Counting, Arithmetic, Algebra, Geometry, etc.), each with around six items. Each item has extra graphics, interpretations, and an interactive component. For example, the Rhind Papyrus appears in the Arithmetic exhibit along with a Wolfram-powered app that visualizes how each of the fractions 2/(odd number between 3 and 101) can be expressed, Egyptian-style, as a sum of fractions with numerator 1. If you want, you can see the new denominators in the Egyptian hieratic script used on the Papyrus. “Whether you come to try your hand at some ancient math homework or to enjoy imagery from artifacts from around the world,” Blackmore writes, “you’ll come away with a greater appreciation of how math developed — and how much modern math owes to our brainy ancestors.”