I love math, and I love music. I’ve played piano since I was 6 and composed music since middle school. So when I was planning my week at the JMM, I knew I had to attend the MAA Contributed Paper Session on Mathematics and Music.

Music inherently contains many levels of remarkable mathematical structure. And throughout history, some composers have devised clever ways to sneak even more math into their music (once, I represented exponential growth musically). Béla Bartók (1881-1945), a great Hungarian composer, might have been one of them—but historians are divided on the issue. In his talk “Bartók, Fibonacci, and the Golden Ratio: Fact or Fiction?,” Gareth Roberts of the College of the Holy Cross presented the evidence for and against the presence of the Fibonacci sequence in Bartók’s 1936 work Music for Strings, Percussion and Celesta. 

The Fibonacci sequence begins 1, 1, 2, 3, 5, … , with each number being the sum of the previous two. Bartók excelled in math and physics and collected plants like sunflowers and fir cones whose structures feature Fibonacci numbers. But he was notoriously tight-lipped about his compositional methods, leaving no written records that could confirm the use of Fibonacci numbers in his music. 

The first page of the score of Bartok’s Music for Strings, Percussion and Celesta.

According to Roberts, the controversy was sparked by music theorist Ernő Lendvai, who in the 1950s published an analysis that seemed to demonstrate extensive use of Fibonacci numbers in the first movement of Music for Strings, Percussion and Celesta. The piece lasts 89 measures, with its climax (loudest moment) at the end of bar 55. The exposition in the opening lasts 21 bars, and the string mutes are removed in measure 34. Of course, 21, 34, 55, and 89 are all Fibonacci numbers.

But the case is not so clear-cut, Roberts explained. A later analysis by musicologist Roy Howat pointed out various flaws in Lendvai’s reasoning. The written score of the piece contains 88 measures (Lendvai argued that an extra bar of rest should be tacked on). Although the dynamic climax does occur in bar 55, the tonal climax is in bar 44. Some of the string mutes are removed in bar 33, some in 34, and the rest in 35. The exposition, Howat claims, ends in bar 20. And Lendvai neglected to mention the titular celesta, which enters after bar 77—nowhere near a Fibonacci number. 

Top: Lendvai’s analysis of the structure of the piece. Bottom: Howat’s analysis.

The third movement of Music for Strings, Percussion and Celesta begins with a more unambiguous instance of the Fibonacci sequence, a xylophone playing a rhythmic pattern of 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1. (Apparently, this movement was used in the soundtrack of The Shining.)  But the bottom line of Roberts’ talk was that it’s a stretch to say that Bartók intentionally incorporated Fibonacci numbers into the structure of the first movement. 

Roberts teaches this example in his undergraduate course on mathematics and music. The controversy draws students in, and the resolution offers lessons applicable beyond finding patterns in music. Lendvai’s analysis is a perfect example of cherry-picking data and confirmation bias.

It reminds me of the statistically insignificant “Bible code” that people have claimed predicts the future. Pick your favorite composer. Examine their oeuvre, looking for your favorite sequence of numbers. If you massage the data enough, considering all the possible ways your sequence could be hidden in the music, you’ll surely find it somewhere. But does that mean that the composer put it there intentionally? Unlikely. That being said, it seems plausible that Bartók played these sorts of mathematical games in other pieces.

Attending this talk gave me potential inspiration for my future composing. I think it would be fun to write a piece built around the Fibonacci sequence. And if I do so, I’ll make sure I explain it in writing to save future musicologists the trouble. 

The COVID-19 pandemic has made painfully clear the importance of studying zoonotic diseases, infectious diseases that are passed from animals to humans. Today, Linda Allen, a mathematical biologist at Texas Tech University, gave an AMS-MAA Invited Address on how mathematical modeling helps us understand emerging zoonoses, particularly viruses. 

Before a virus passes from a natural animal host to humans, it generally passes through an intermediate animal host. For example, SARS originated in horseshoe bats, jumped from them to civets, and jumped from civets to humans. Seasonality in ecological interactions can affect a virus’s spread within the intermediate host population as well as its transmission to humans. In her talk, Allen focused on how seasonality impacts the probability and timing of a spillover—the moment when an intermediate animal host first infects a human with the disease. 

Mathematicians harness differential equations and Markov chain methods to model the spread of infectious diseases. In the context of COVID-19, most of us have heard about susceptible-infected-recovered (SIR) models, which are characterized by a transmission rate $\beta$ and a recovery rate $\gamma$. Allen applied an SIR model to a spillover event by including both an animal-animal transmission rate $\beta_{aa}$ and an animal-human transmission rate $\beta_{ah}$. She modeled seasonality by considering both transmission rates (as well as the animal recovery rate) to be continuous periodic functions. 

Allen explained how an SIR model can be used to study the spillover of an infectious disease from animals to humans.

If seasonal effects are strong, the time of year when the virus infects the intermediate animal host has a major influence on the probability of a spillover into humans. Allen presented the results of computations and simulations with different relationships between the transmission rates and animal recovery rate. 

Unsurprisingly, if the seasonal peaks in the two transmission rates align, the probability of a spillover is highest near those peaks. If the peaks don’t align, the resulting trends in spillover probability are not as intuitive. Plus, the timing of the maximum spillover probability can depend on the number of initially infected intermediate hosts. 

Allen gave an example of the model applied to H5N1 (a type of avian influenza) in domestic poultry and humans. Data from the World Organization for Animal Health show sharp annual spikes in outbreaks among domestic poultry. The results of Allen’s approximate calculations as well as Markov chain simulations clearly showed the resulting seasonality in spillover probability.

The probability of a spillover shows seasonal variation. In the left column, the animal-animal transmission rate (dashed red) and animal-human transmission rate (black) are plotted as functions of time. In the other two columns, the time of the first animal infection is on the x-axis. The middle column shows the probability of a spillover if one animal host becomes infected initially. The right column shows the probability of a spillover if five animal hosts become infected initially.

Researchers estimate that over 60% of human infectious diseases are zoonotic. Of those, around 75% are emerging or reemerging, so understanding the dynamics of spillovers is crucial if we hope to prevent future pandemics. Worryingly, the frequency of zoonoses is rising due to deforestation, climate change, globalization, and other factors. 

Allen closed her talk with some thoughts on how we can promote public health. She mentioned three interconnected tasks: collaboration between mathematicians and experts in diverse fields like ecology, geography, and epidemiology; cooperation within and between agencies at the local, national, and international levels; and education of the public about the sources and prevention of zoonoses. Given the ongoing pandemic, now is the time to build momentum in all three of these areas. Maybe next time a dangerous zoonotic disease emerges, humanity will be better-prepared. 

In all her time studying math as an undergraduate, Jacqueline Dewar never heard about any woman who had made important contributions to math. The first time she encountered a woman mathematician in the curriculum was when Emmy Noether was mentioned in a graduate course on rings and fields. When Dewar became a professor at Loyola Marymount University, she decided to design an interdisciplinary course on women in math in order to counteract negative stereotypes and promote interest and persistence among future women mathematicians and math educators.

Dewar first taught her course in 1979, drawing inspiration from Toni Perl’s book Math Equals: Biographies of Women Mathematicians + Related Activities. The book highlights nine mathematicians: Hypatia, Émilie du Châtelet, Maria Agnesi, Mary Somerville, Sophie Germain, Ada Byron Lovelace, Sonya Kovalevskaya, Grace Chisholm Young, and Emmy Noether. Dewar’s course includes material about all of them, plus statistician Florence Nightingale. Originally designed as a math course for liberal arts students, the course eventually became an upper-division elective for math majors. Yesterday at the MAA Contributed Paper Session on Promoting Women in Mathematics, Dewar gave a talk, “Encouraging Women in Mathematics Through an Interdisciplinary Course,” describing the contents of the class and the impact it has had through the years.

The mathematical content of the course doesn’t focus on particular concepts, but rather three broad recurring themes: inductive and deductive reasoning, representing a single concept in multiple ways, and math as a study of patterns (not just numbers). Similarly, Dewar asks her students to find commonalities in the biographies of the 10 women, connect their experiences to the present day, and engage critically with scholarship on gender issues in math education.

The math and biographies are woven together throughout the course. Key assignments include a short paper synthesizing three readings on gender equity, small group math work, an individual research project, and an in-class report on a modern woman mathematician. 

After teaching her class for decades, Dewar knows that it has made a positive impact, one that continues to spread. In follow-up surveys of her students, she found that the class gave them an opportunity to do math in a supportive environment and prepared them to discuss the state of women in math in the 21st century. And its influence on future teachers carried over into their classroom practice. By 2012, three former students teaching in Los Angeles schools had used Dewar’s course materials with 165 secondary students and 20 elementary students. 

I know I certainly would have benefitted from taking Dewar’s course. Although I have learned about the contributions of some of the mathematicians she discusses, I unfortunately don’t know anything about others. Luckily for me (and anyone else who wants to read up on the topic), Dewar has made her course materials freely available online. Or if you want just a quick summary, check out the slides from her talk and her course description and selected bibliography. 

In lieu of the traditional Noether lecture, the Association for Women in Mathematics held a panel on equity, ethics, and bias in mathematics research. The event featured Loretta Cheeks, president and CEO of Strong Ties and DS Innovation, Kristian Lum of the University of Pennsylvania, Maria De-Arteaga of UT Austin, and Suresh Venkatasubramanian of the University of Utah.

Yesterday’s events at the Capitol Building brought misinformation and polarization to the top of my mind. So when an audience member asked about how mathematicians can effectively persuade policy-makers that an algorithm is unethical, Dr. Cheeks’ words about speaking to people with different perspectives felt timely and important. Her research explores the space between the computational sphere and the “informal space” of unstructured information–hints, subjective narratives, and the like. She noted that people are getting more and more of their information online, from indirect sources. This introduces an additional layer of bias beyond hearing, say, a friend’s personal story. Cheeks pointed out that once someone has very deeply held beliefs, their commitment to them goes beyond the familiar phenomena of confirmation bias and groupthink. It’s only by finding someone who can relate to their perspective that a connection can be made.

These communication problems can even arise from simple bids for algorithmic transparency. One participant asked how translating policy goals into something that can be used in an algorithm complicates things. Dr. De-Arteaga noted that this can cause extra friction between groups when their goals don’t align perfectly. Algorithms require highly specific and quantifiable inputs and objectives, while policy often benefits from ambiguity that glosses over disagreements between policymakers. Moreover, explaining algorithms can have unintended consequences: recent research suggests explanations may make users overconfident by giving them an unearned sense of understanding.

This highlights the importance of other fields in dealing with the points where mathematics meets society. A social scientist is likely to know what kinds of explanations cause overconfidence, and can analyze the content of data more effectively than a mathematician. What’s more, their scientific training involves thinking critically about assumptions. Dr. Venkatasubramanian believes, however, that mathematicians are still uniquely valuable in discussions of algorithmic ethics. The precision of mathematical thinking lends itself well to reasoning about systems’ capabilities and limitations.

Moreover, mathematics and data still holds special standing in the eyes of many. Dr. Venkatasubramanian and Dr. Lum both recalled the extra credibility they command in settings with lawyers and politicians. And when citizens face people in positions of power in meetings or in court, they can greatly increase their leverage by having data and analyses at their disposal, says Dr. Lum.

All the panelists agreed that it is far too early for mathematicians to be able to draw up a rigid, universal set of ethical guidelines like the AMA Code of Medical Ethics. But there are plenty of ethical and equity questions that modellers can ask themselves before they forge ahead with their work–about the bias in the data, about what assumptions are going into the model and the questions being answered, and about what, specifically, is being output and optimized for. As algorithms become more and more vital to the functioning of our society, mathematicians will have a special role to play that goes beyond the fallacy of math as truly objective.

This afternoon, I dropped by the MAA Contributed Paper Session on Recreational Mathematics: Puzzles, Card Tricks, Games, and Gambling. The first two talks focused on the board game Catan and chess, taking inspiration from social distancing in a way only mathematicians can.

Catan lends itself to fun combinatorics: The board consists of 19 hexagonal resource tiles, 18 number tokens placed on the resource tiles, and 9 ports. The arrangement of these components is randomized for each game. As my family members often comment when we play Catan, part of the fun comes from the fact that the board is different every time. For mathematicians, it’s natural to ask how many arrangements of the board are possible. 

Since each game of Catan involves randomizing the locations of the resource tiles, number tokens, and ports, it lends itself naturally to fun combinatorics.

In a 2019 paper, Jathan Austin, Brian Kronenthal, and Susanna Miller computed the number of nonequivalent boards (boards that cannot be transformed into each other via rotations and reflections), arriving at a staggering $1.5404939\times10^{28}$ possibilities. But this computation ignored a subtle rule of the game of Catan: four of the number tokens are red, and no two red tokens can be placed on adjacent resource tiles. In the language of our post-2020 world, the red tokens must be “socially distanced.”

In their talk today, “Counting Socially-Distanced Catan Configurations,” Austin, Kronenthal, and Miller sketched out their approach to the refined problem of counting all legal (socially distanced) arrangements of the Catan board. They viewed the board as a trio of concentric rings—the center tile, the middle ring of six tiles, and the outer ring of 12 tiles—which broke the possible configurations of the red tokens into five cases. From there, the counting arguments needed for each case were fairly straightforward, involving simple combinations and the use of the principle of inclusion-exclusion.

The number of allowable configurations of the red tokens can be counted by considering five cases. In each ordered triple in the leftmost column, the first entry is the number of red tokens on the center tile, the second is the number of red tokens on the middle ring, and the third is the number of red tokens on the outer ring.

It turns out that there are 532 possible configurations of the red tokens. Combining this result with techniques from their earlier work, the speakers arrived at a total of $2.1144034\times 10^{27}$ legal configurations of the Catan board. If you played one game of Catan every second, you would need 4.9 trillion lifetimes of the universe to get through every socially distanced configuration. 

As a side note, what’s the probability that a Catan board arrangement chosen completely at random will satisfy the red token social-distancing rule? Dividing the number of legal configurations by the total number of configurations, you get about 0.137. 

Satisfied that my relatives are right when they say that every game of Catan is different, I moved on to the next talk, “Social Distancing on the Chessboard” by Doug Chatham of Morehead State University. On an otherwise empty $n\times n$ chessboard, a rook or a queen can move between any two squares in at most two moves. But if you add some pawns to the board, you can cause certain squares to be separated by a larger number of moves. 

Chatham translated this situation into the language of graph theory. A given arrangement of pawns on an $n\times n$ chessboard determines a graph whose vertices are empty squares and whose edges are the pairs of squares $(A, B)$ for which a queen can move directly from $A$ to $B$. Then the distance between squares $A$ and $B$ is the minimum length of a queen’s path from $A$ to $B$, and the diameter of a given queen graph is the maximum distance between two squares, running through all pairs of squares on the chessboard.

A chessboard with pawns on some squares can be thought of as an undirected graph.

With this set-up, Chantham asked the following question: How many pawns $p$ are needed so that some placement of those pawns on an empty $n\times n$ chessboard produces a board with a queens graph of diameter $d$? (He also studied the same question for rooks.)

As those readers familiar with graph theory will know, the structure of an undirected graph can be encoded in an adjacency matrix where each entry corresponds to a pair of vertices. A 1 represents an edge connecting the vertices, and a 0 represents no edge connecting them. From a graph’s adjacency matrix $A$, it’s easy to calculate the diameter of the graph: It’s the minimum $k$ for which $I+A+A^2+\dots + A^k$ has no zero entries.

Chantham used a simple algorithm to have a computer find the value of $p$ (which he called the “diameter separation number”) for given values of $n$ and $d$. The results are below.

The (queen) diameter separation number for some values of n and d.

Chantham also stated (without proof) two propositions. The first was that for any $n\geq 4$, only one pawn (when properly placed) is needed to yield a diameter separation number of 3. The second was that for any $n\geq 4$, three pawns (when properly placed) yield a diameter separation number of 4.

Left: a board with one pawn and a diameter separation number of 3. Right: a board with three pawns and a diameter separation number of 4.

The graph theory of queens and pawns on a chessboard is rich with variations to explore. What happens to the diameter separation number for fixed $d$ as $n$ increases? What happens if the board is rectangular but not square, or if we use bishops instead of queens or rooks?

In neither talk did the speakers address why anyone should care about their results. To an audience of fellow math fans, it’s obvious: The problem was there and it could be solved using math, so the process of arriving at the solution is inherently interesting. But I would add another reason why recreational problems with no apparent applications to “serious” math are worthwhile. 

A lot of people enjoy Catan or chess—including many people who don’t like math (or at least think they don’t like math). Fun puzzles like the social-distancing problems discussed today could be great educational and outreach tools that connect math with something that people already understand and enjoy. The Catan puzzle is an unconventional combinatorics problem accessible to students who are learning about counting and combinations. The chessboard puzzle offers a hands-on exploration of graph theory, with plenty of room for people to form their own conjectures and try to prove them or find counterexamples. 

After the JMM session recordings are posted, I hope to go back and watch some of the talks on recreational math that I missed and see what other education and outreach activities they inspire.

When physicians attempt to use targeted therapy against bacterial infections and cancer, one of the greatest challenges they face is the rapid evolution of the disease to resist treatment. I attended a fascinating talk yesterday afternoon about a novel approach to this problem that draws on an idea from physics: counterdiabatic driving.

Oncologist Jacob Scott of the Cleveland Clinic explained counterdiabatic driving using the analogy of a waiter carrying a full glass of water on a tray. If the waiter starts walking while holding the tray horizontally, the water might spill. But if the waiter tilts the tray forward as they start walking, they can keep the water at equilibrium and avoid spilling it.

A waiter carrying a glass of water provides an analogy for counterdiabatic driving. As the waiter walks to the right, the water leaves equilibrium in situation B but stays in equilibrium in situation C.

In physics, counterdiabatic driving — this sort of preemptive tweaking of parameters to keep a system in instantaneous equilibrium — shows up in Brownian motion of a bead in an optical trap and in adiabatic quantum computing. I encountered the concept during my undergraduate research on parity-time symmetry in quantum mechanics, so the title of Scott’s talk, “Controlling the speed and trajectory of evolution with counterdiabatic driving,” immediately caught my attention.

As Scott explained, the various genetic mutations that allow a disease to evolve resistance to a drug can be thought of as forming a fitness landscape. The shape of this landscape will be different for each drug, so evolving resistance to one drug affects the sensitivity to other drugs. This leads to an important but often-overlooked fact in treating diseases: Evolution is not commutative. That is, the sequence in which drugs are administered has a big impact on the final outcome. For example, it’s possible that using Drug A followed by Drug B would render Drug C ineffective, but using Drug B followed by Drug A would cause Drug C to be highly effective. It all depends on the path that evolution takes through the fitness landscapes.

A conceptual representation of fitness landscapes. Which peak (purple pentagon or green star in large figure) the disease reaches during the first treatment will determine which second-line drug (smaller figures) is most effective.

Here’s the core question of Scott’s research: Is it possible to steer evolution to maximize the effectiveness of second- and third-line drugs? It turns out that the answer is yes. Simply changing the order in which drugs are administered is a rudimentary way to accomplish this, and counterdiabatic driving provides a more sophisticated strategy. When administering the first drug, altering the dosage over time in a specific way can guide the disease genotype toward a desired point in the fitness landscape where the second-line drug will prove particularly effective. 

The details get more complicated, of course. The individual cells in a bacterial infection or a cancer tumor don’t evolve as a homogenous unit, but rather as a distribution of genotypes. As a result, the math of diffusion comes into play in the evolutionary dynamics. Plus, in the theoretical models, it takes infinite time for the disease to reach the desired point in the fitness landscape. But patients can’t wait forever before switching from the first-line drug to the second-line one, so researchers have to determine how close is close enough. 

We’re still in the early days of applying these ideas to actual diseases, but results so far have been promising. For example, Scott mentioned a study of 15 empirical fitness landscapes of E. coli that showed that it is possible to steer the bacterial population to avoid the emergence of antibiotic resistance. 

Counterdiabatic driving, an idea that originated in physics, can be used to steer the evolution of a disease and make treatment more effective.

I thought it was fascinating to see how an idea that I first encountered in the context of physics has direct applications to medicine, and I can’t wait to hear about future developments. If you want to learn more, Scott’s slides and references are available online. The talk was part of the AMS Special Session on the Mathematics of RNA and DNA.

This afternoon, I stopped in at the AMS Special Session on Mathematics Courses Designed to Develop Mathematical Knowledge for Teaching High School. I managed to catch Andrew Ross, Stephanie Casey, and Melody Wilson’s talk “Developing Statistical Knowledge for Teaching with an emphasis on Equity Literacy”. They talked about designing a course for the MODULES(S2) project, which aims to improve the training of secondary-school teachers.

A revision of statistics curricula for teachers-in-training is due for a few reasons, Casey explained. First of all, statistics is a key topic–its importance in research and industry is growing fast even as teachers report feeling less prepared to teach statistics than other content areas. And more generally, teacher education is moving away from the traditional separation between content and pedagogy and toward more holistic practices.

One way to address these issues efficiently is by using real-world statistics problems that address equity in class. Ross presented an example scenario from class: the state of Pennsylvania came up with a formula for funding schools equitably, enabling data scientists to compare actual school funds to the formula’s recommendation. By analyzing things like the median income of students or the proportion of white students against their funding situation, pre-service teachers can experience for themselves the entwining of data with equity and justice.

In survey results at the end of the MODULES(S2) course, teachers-in-training reported feeling more equipped to practice statistics. They also indicated that, compared to the start of the course, they thought more about equity issues such as race and felt teachers had more of a role in student success. As one teacher-in-training reported: “Including topics about equity and social justice can engage and encourage students … they do have something to contribute to these ideas.”

Although these days I work mostly on my computer, there is something special about the chalkboard. The expansiveness of the surface has a way of absorbing me into my work, whereas the outside world and all its distractions are in full view when typing on a laptop or writing on scratch paper.

So it was gratifying to see that experience celebrated in Jessica Wynne’s new book, Do Not Erase, when I logged into my first JMM event this afternoon. Wynne, a photographer and professor at the Fashion Institute of Technology, photographed over 150 mathematicians’ chalkboards for this project. Each photo in the book is paired with an essay by the mathematician whose board is pictured. The book will be available this June, though you can preorder it now with a 30% discount using code JMM21.

Wynne started off by describing how the seed for this book was initially planted–through her friendship with Amie Wilkinson and Benson Farb of the University of Chicago: “One day, Benson is working at his dining room table. For several hours he sits, thinking, jotting down the occasional note. He is creating something beautiful and expansive beyond words.” When asked to explain his work, Farb replies that he can’t.

Later, Wynne traveled to an elementary school in Jaipur, India with her photography students. The school’s chalkboards caught her eye–covered in lessons written in Hindi, they were as incomprehensible as Farb’s mathematical notes. “The writing on these boards reminds me of the symbols in Benson’s notebook. There my project begins to take shape.”

Wynne’s story of her experience leading up to and throughout the project was followed up with a Q&A with the audience. Some highlights–edited for length, clarity, and my inability to transcribe quickly enough to keep up:

Q: What was the most surprising to you after seeing all the blackboards?
JW: Initially, when I started working on the project, the most surprising thing to me was how creative math at this level is, and the connections I saw to the artistic process. One of the things I loved about doing this project was spending time with mathematicians and having conversations with them.

Q: What is the aesthetic difference between math on a chalkboard vs a whiteboard?
JW: One of the reasons I restricted to chalkboards was that these boards could have been from 100 or 200 years ago. There’s no sense of time and I liked that. From a visual artist perspective, I liked the way chalk on a board looks and how you can see the different layers through eraser marks and the like. And as a photographer, I think about light, and the way light reflects on a chalkboard has a very different quality.

Q: I’ve always found the eraser blurs as a distraction from the black-and-white contrast. Any thoughts on the smudges?
JW: I think they’re very beautiful, I don’t find them distracting. I’ve shot over 150 boards all over the world, and when I see the eraser marks I can sense the energy and time and frustration of the work.

Q: How did you go about choosing who to include in the book?
JW: A lot of it came about organically. I started by getting to know Amie and Benson and they introduced me to some of their friends. I also was at the common room at Columbia quite a bit and met people that way. Traveling, it was different because I would email people in advance. But there wasn’t a specific agenda in terms of reaching out to particular people. There were also mathematicians that I met who referred me to other people, they were aware of one another’s boardwork which was exciting.

Q: Were people nervous about the content they were writing on the board?
JW: There were a couple of people who panicked when they noticed there was a mistake in their photograph and I reshot the photograph later. I didn’t have anyone that seemed particularly nervous in the moment, though.

Q: Are there enough pictures for a sequel?
JW: I want to keep shooting. I did get a lot of the work done before everything was shut down but there are a lot of places and countries I still want to go and photograph.

Q: Are there any photos you know have mistakes?
JW: I don’t know, but I’m sure there are. That’s part of the artistic process; it’s not interesting if everything is too perfect.