First, the **cartoons** (which you can see early by liking the MWBD Facebook page, or following me on Twitter):

To clarify: no integer base gives the conclusion 8 x 7 = 54, because 52 is not a multiple of 5. You can change the meanings of the symbols, but at that point, you might as well just claim to live in a base-10 world where 6 and 4 have swapped places.

That said, I find this is a good tactic, because the modular arithmetic slows people down enough for you to make your getaway.

This one generated lots of controversy on Facebook.

On reflection, it’s weird to label John as the second least-insufferable Beatle. (There’s a case for him being the most insufferable.) On the other hand, I’m underrating John by excluding the z-axis of “coolness” (on which he wins by a mile) so it balances out.

Also, I love George. He was many things: a brilliant guitarist, a sharp lyricist, a next-level songwriter. But he was not sufferable. The man wrote *Piggies*. Even his best stuff is tinged with insufferability. (“I look at the floor and I see it needs sweeping”?!)

But please, feel free to rip me to shreds in the comments.

This is pretty much how it goes when you teach “Theory of Knowledge.”

*Step 1*: We know lots of things.

*Step 2*: But how do we *knoooooow*? We don’t *know *anything!

*Step 3*: Okay, maybe we don’t *italicized **know* anything, but we know lots of things.

Don’t think of this joke as hopelessly dated. Think of it as a 20th anniversary!

And now, the **links**:

Evelyn Lamb assigns household chores to various mathematicians. For example, number theorists:

Contrary to popular belief, you don’t necessarily want them to be balancing your budget or calculating a tip, but they are excellent at dusting. They are no strangers to time-consuming, frustrating tasks, so they won’t even bat an eye at dusting the baseboards and crown molding.

Patrick Honner revisits a controversy over whether it’s good algebraic hygiene perform the same operation to both sides of an algebraic identity:

While it’s nice to see mathematically valid work finally receiving full credit on this type of problem, it’s no consolation to the many students who lost points for doing the same thing the year before. What’s especially frustrating is that, as usual, those responsible for creating these exams will admit no error nor accept any responsibility for it.

Three years old now, but well worth reading, is Shecky’s interview with Fawn Nguyen, a beloved figure on math teacher Twitter, because she says things like this:

Consensus in education – that’s like finding tofu in bouillabaisse.

And this:

Mathematics is my passion, and kids are my love – one fuels my head, the other expands my heart. That’s grace. My school feels like home, sans bathrobe and slippers.

And heck, the interview opens with this stop-everything line:

When Saigon fell in 1975, my family made a failed attempt to flee the country by sea.

FiveThirtyEight has a great, no-punches-held piece titled The Supreme Court is Allergic to Math:

But maybe this allergy to statistical evidence is really a smoke screen — a convenient way to make a decision based on ideology while couching it in terms of practicality.

Michael Pershan takes a generous but unflinching look at a sloppy (i.e., false) research claim in education:

What strikes me about YouCubed is that the errors just seem so unnecessary. The message is a familiar one, and I’m OK with a lot of it: don’t obsess over speed, think about mindset, don’t be afraid of mistakes. But there’s this sloppy science that gets duct taped on to the message. What purpose does that serve?

And saving the best for last, Jenna Laib (full disclosure: my sister) has a great essay about her trials as a first-year teacher:

I reworked my classroom supply routines to impede access to scissors, and allowed fear to root in me. I am sure the students could tell; ten-year-olds can smell fear like dogs.

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“Everybody’s talking about big data, the data deluge—all this data being rained down on us,” Efros says. “But I think a lot of people don’t appreciate that most of the data is actually visual…. YouTube claims to have 500 hours of data uploaded every single minute. The earth has something like 3.5 trillion images, and half of that has been captured in the last year or so.”

Today, teams of computer scientists like Efros are working to understand that data via “deep learning” algorithms. First, you prepare a network of connections. Then, as a training regimen, you show it vast quantities of photographs. With time, it learns to accomplish extraordinary tasks—writing captions, colorizing black-and-white photos, recognizing animal species.

Unless, of course, you troll it.

Pranking your algorithm is not just fun and games. There’s a research principle here. By tracking a system’s errors, you can see how it functions. That’s why psychologists love optical illusions: not just to see people flail and sputter, but to reveal what shortcuts the brain takes in processing images.

By analogy, recent work has turned up “optical illusions for deep learning”—errors that, by frustrating and confusing the algorithms, hint at their inner nature.

Here are five:

**1. Defy its expectations.**

“It’s very easy to fool yourself and think that the network is doing more than it is actually doing,” Efros says. He gives an example from a neural network that was trained to caption images:

Impressive, right? Not so fast, says Efros. “If you go and look for cars on the internet,” he points out, “that description applies to pretty much all of those images.”

To give the network a real test, he dangled some bait – and got a giant bite:

“It’s kind of true,” says Efros. After all, it *is* by the side of the road. “But what about this?”

“Well, there is a car; there is a road; and probably that’s all it’s getting,” he says. “It is wishful thinking [that] it’s doing more than finding a few texture patterns.”

Deep learning creates programs with extraordinary results but mysterious inner workings. By showing the computer something fresh and strange, you can begin to lift the lid on that black box. In this case, we learn a stark lesson: the computer can’t recognize semantic categories, just lower-level visual features.

**2. Show it what it wants to see.**

Computer scientist John Hopcroft, in the talk that preceded Efros’s, showed how some researchers have taken this principle even further.

“People can take an image of a cat, and change a few pixels, so that you can’t even see that it’s a different image,” Hopcroft explained. “All of a sudden the deep network says it’s an automobile.”

“This worried people at first, but I don’t think you have to worry about it,” Hopcroft said. “If you take the modified image of the cat, it doesn’t look like an image to a deep network, because there’s a pixel which is not correlated with the adjacent pixels. If you simply take the modified image and pass it through a filter, it will get reclassified correctly.”

Still, it’s a powerful demonstration of the susceptibility of deep learning to optical illusions – in this case, ones that the human eye can’t even detect. Although neural networks were born from an analogy with the human brain, it’s clear that they have long since parted ways.

**3. Lull it into a sense of security with your cute puppy face.**

In another project, Efros and his colleagues separated pictures into a grayscale component and a color component, then asked neural networks to predict the latter from the former. In other words: they trained the computer to colorize black-and-white pictures.

It produced some cool results, including this version of the famous photo “Migrant Mother”:

And yet, like most denizens of the internet, it found its intelligence diminished by the presence of an adorable puppy, making a strange error in colorizing the face:

Why the pink under the chin? Efros explained: “Because the training data has the [dogs with their] tongues out.” Accustomed to panting dogs, it colors a phantom tongue on the chin of a closed mouth.

For his part, Efros finds this kind of error encouraging. “Whatever it’s learning is not some low-level signal,” he said. “It’s actually recognizing that it’s a poodle, and then saying, ‘Well, all the poodles I have seen before had their tongues out’…. This actually suggests it’s learning something higher-level and semantic about this problem.”

**4. Trap it between two alternatives.**

As Efros explained it, visual data presents a fundamental challenge for computers: defining how “far apart” two images are. With text, it’s much easier—you can, for example, count the ratio of letters that two text strings share in common. But what does it mean for one image to be “close to” another?

“Close in what sense?” asks Efros. “In high-dimensional data like images, [the standard approaches] don’t work very well.”

“Imagine this bird could be blue, or could be red,” Efros explained. “[The neural network] is going to try to make both of those happy. It’s going to split down the middle.” The result: a muddled gray, instead of the bright reality. Equidistant from two plausible colorings, it couldn’t choose a direction to go. “It’s not very good when you have multiple modes in the data.”

Doing some “fancy things” mathematically, Efros and his team coaxed the network into committing to one colorization option:

Once modified, however, the network began to “overcolorize” images—for example, turning the wall below from white to a pixelated blue:

It remains hard for the colorizing network to find the right level of aggression, a middle ground between leaving colored things in gray-scale and turning gray things bright colors.

**5. Show it Vladimir Putin on a horse.**

In another project, Efros’s team trained a neural network to convert horses into zebras, and vice versa. It produced some notable successes—and one notable failure, which earned the biggest laugh of the conference:

Unable to distinguish man from horse, the neural network gave a surreal, sci-fi vision of a zebra centaur. “I showed this in my talk in Moscow,” Efros laughed. “I thought, ‘They are not going to let me out.’”

Still, there’s reason to take pride in this neural network’s achievements. “There is no supervision. Nobody told the network what a horse looks like, what a zebra looks like…. It’s like two visual languages without a dictionary.” The neural network learned to translate between them – albeit imperfectly.

This image led Efros to his conclusion: “It’s time that visual big data be treated as a first-class citizen,” he said. Visual problems loom large in computer science, and neural networks offer tremendous potential—as long as we remember how to keep them honest.

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The noodly mathematics of gerrymandering remains a hot topic in the news. I recommend the More Perfect episode, as well as Patrick Honner’s explanation for Quanta Magazine. (Speaking of which, Quanta is like the Beatles of popular math writing. The quantity of quality stuff is unfathomable.)

Quick summary: The aim of gerrymandering is to redraw legislative districts so that your party wins more seats. This is best accomplished by winning each seat with a tiny majority, so you don’t “waste” any votes. What does it mean to “waste” a vote? Well, any vote in a district you lose is wasted, ’cause you didn’t win. And any vote above the winning threshold (50%) is wasted, ’cause you didn’t need it. (This means that, by definition, half of all votes in any district are wasted.)

The efficiency gap is pretty simple: it’s the difference between the two parties’ wasted votes (expressed as a percentage of all votes).

I’ve always got way too many tabs open. Quick screen grab:

Thus, I’m just now getting around to reading math writer Evelyn Lamb’s interview with the AMS, and therefore just learning about the podcast My Favorite Theorem.

The podcast is what it says on the tin: Mathematicians recounting their favorite theorems. I just downloaded all five episodes. Eager to start listening.

Here’s a bizarre one: apparently how you eat corn is super predictive of what kind of math you like, analysis or algebra.

I bristle a little at the writer’s descriptions of the fields, but the correlation is kind of uncanny. My wife (an analyst) and my dad (who prefers algebra) both fit the pattern. I’ve always felt pretty agnostic between the two (if you’re batting .000 from both sides of the plate, you might as well call yourself a switch hitter!) but the corn test pegs me as an algebraist. Who am I to disagree?

My niece (age 2) sometimes asks me to draw “happy triangles” for her. Then, as soon as I finish, she scribbles them out. I’m sure it’s a metaphor for something.

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Back in the 1970s, Hellman and Diffie couldn’t have known that their work would lead to this stage. In fact, there was a likelier destination.

Federal prison.

“It’s July 1977,” Hellman tells the audience. “Whit and I are involved in a major fight with NSA over the data encryption standard.”

American law banned the unlicensed export of weapons. Makes sense: the government doesn’t want civilians wandering into Moscow with a trenchcoat full of fighter jet parts. The question is: Does this law apply to abstract mathematical ideas? By developing new approaches to cryptography, are Hellman, Diffie, and their collaborators *de facto* arms traffickers? If so, Hellman says, “then by publishing our papers in international journals, we are in some sense exporting plans for implements of war.”

“I think the penalty,” Hellman recalls, “was something like five years in jail.”

As a Stanford professor, Hellman sought the advice of the university’s general counsel, John Schwartz. Schwartz told him that, in his view, prosecuting a computer scientist for cryptography research was unconstitutional—a violation of the first amendment. But he also warned that only a court of law could settle the matter.

His next words remain burned in Hellman’s memory.

“If you’re prosecuted,” Schwartz said, “we will defend you. If you’re convicted, we will appeal. But I have to warn you… if all appeals are exhausted, we can’t go to jail for you.”

It’s a line straight out of a Hollywood thriller, which cannot be said of most conversations in the faculty lounge.

That October, Hellman planned to present two cryptography papers, co-written with students, at a conference. He intended to shine a spotlight on the students by having them give the talks.

Schwartz advised against it. “From a practical point of view,” Hellman says, “I was a tenured professor, and the students were just starting out.” That left them more vulnerable. “A multi-year court case could totally ruin a new PhD.”

The students courageously insisted on taking the risk—until their parents intervened. Hard to blame them: an academic career unfolds slowly enough, even without taking a five-year federal-prison hiatus.

“We came up with a very good system,” says Hellman. “When it was time for the papers, both of us went up… I explained to the audience, who already knew what was going on, that on the advice of Stanford’s general counsel, I would be giving the paper instead of the students. But from every perspective except legally, I wanted them to consider the words I was saying as if they were coming from the student.”

“And so,” Hellman explains, “the students stood there, not saying anything.” This bizarre visual amount to the best sales pitch a PhD could hope for: you as the ventriloquist, and a star professor as your cheerful dummy.

The conflict simmered until 1978. Then, out of the blue, Hellman received a call from the office of NSA director Admiral Bobby Inman to schedule a meeting.

“Whit and I had been fighting this out with the NSA in the press, and never actually talking to them,” remembers Hellman. “It’s a bad way to have a disagreement, and so I jumped at the opportunity.”

Weeks later, he found himself face to face with his adversary. Inman leaned over and said wryly, “Nice to see that you don’t have horns.”

Hellman looked back and said, “Same here.”

That broke the tension. Hellman soon learned that Inman had scheduled the meeting against the advice of all his senior colleagues at the NSA. But he, like Hellman, saw no harm in talking things out. “Out of that very cautious initial meeting grew, eventually, a friendship,” says Hellman. “That’s something we should keep in mind today, and not just in the cryptographic community.”

Hellman began his talk by joking that Diffie was his “partner in crime.” It was an offhand bit of humor; for the moment, he was not reflecting on how close they came to making that phrase literal.

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I’m awaiting the day when the *New York Times* becomes a full-time math-only publication. This week brought us a step closer.

First, Manil Suri meditates on the social impact of mathematical discovery, by asking who invented zero.

And second, Jordan Ellenberg describes the state of gerrymandering in Wisconsin, where new computational techniques have elevated the old practice from an art to a science. “As a mathematician, I’m impressed,” writes Ellenberg. “As a Wisconsin voter, I feel a little ill.”

A gem from ArXiV: Marvel Universe Looks Almost Like a Real Social Network, applying graph theory to the Marvel comics universe. Each character is a node; appearing together in a comic book is an arc.

Perhaps unsurprisingly, 99.4% of all characters belong to a single connected component of the graph.

Last thought: the Best Mathematics Writing of 2017 looks sharp.

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Alas; it turns out that Alfred Nobel didn’t much appreciate the icy abstractions of mathematics. And before you ask, there’s no prize for computer science, either.

Faced with this emphatic snub, what’s the best response for the math/CS community? Should we pout? Throw tantrums? Pen angry, tear-stained letters to the King of Sweden? Gossip that Nobel was just jealous because his fiancée slept with a mathematician (fact-check: probably false)?

Though not above juicy gossip, the math/CS world has mostly pursued a different solution: Give fancy prizes of our own. Today, these five prizes rank among the highest in math and computer science:

Given to a quartet of mathematicians every four years since 1950, math’s most famous prize honors exceptional work done before the age of forty. The prize money: $15,000 Canadian.

If the age cutoff strikes you as arbitrary, and the low prize as peculiar, then congratulations! You’re quite right. The Fields began life as a way to honor early-career research that showed the promise of future breakthroughs. Only later did it take on the role of “top prize in mathematics.”

According to math historian Michael J. Barany, the trope that the Fields is the “Nobel of Math” dates to 1966, when Stephen Smale traveled to Moscow to collect his medal. The spectacle of a vocally anti-war academic traveling to Russia in 1966 didn’t sit too well with the American establishment. So, in an attempt to explain to the wider public the importance of Smale’s prize, mathematicians across the country began referring to the Fields as the “Nobel of Math.” The shorthand stuck.

First awarded in 2003, the Abel honors “contributions of extraordinary depth and influence” in mathematics. By eschewing an age requirement (and boosting the prize to Nobel-level money), it fills the gap created by the Fields.

That makes “Abel” a peculiar choice of name; after all, Niels Abel made his algebraic breakthroughs in his early twenties. He died at 26. But the prize name goes back to 1899, when Norwegian mathematician Sophus Lie (a legendary figure himself) advocated for such an award. Reviving the idea a century later, Norway saw fit to keep the name.

First awarded in 1966, the prize honors contributions of “lasting and major technical importance to the computer field.”

You might imagine “Turing” was a controversial choice of name at the time; Turing had died a decade earlier after a humiliating prosecution for the crime of homosexuality. But as ACM president Vicki Hanson recounts, records from the time show little controversy. It’s testament to Turing’s extraordinary influence: whatever the anti-gay sentiments of the time, everyone knew his name was a worthy one.

In crude shorthand, it’s a Fields Medal for computer science: awarded by the same body (the International Mathematical Union) on the same timescale (every 4 years) with the same age requirement (a maximum of 40) to a researcher for “outstanding contributions in mathematical aspects of information science.”

However, whereas the Fields goes to 2 to 4 recipients, the Nevanlinna goes to just one.

A bit like a Nevanlinna prize, except yearly, this award honors “an early to mid-career fundamental innovative contribution in computing that, through its depth, impact and broad implications, exemplifies the greatest achievements in the discipline.”

**

In all, you can see that in math and computer science, no single award does the work of the Nobel. To earn an honored spot at the scientific Lindau Laureate Meetings, you need a Nobel. At the math/CS-focused Heidelberg Laureate Forum, any of these prizes will do.

And, if you ask me, that’s for the best.

Prizes are an awkward fit for the world of research, where there’s no playoff season, no championship game, nobody voted off the island, no obvious “winner.” By definition, great discoveries lack precedent. No two breakthroughs are alike, no two researchers interchangeable, and to assert their equivalence is just plain silly.

But just because prizes are arbitrary doesn’t mean that they’re worthless. They shine bright (if uneven) lights on work of exceptional value, punctuating the grind of research with moments of warmth and ceremony. The multiplicity of math and computer science awards serves (if only accidentally) to honor the sprawling truth of human inquiry, better than the singular, Oscar-esque Nobel.

Case in point: In 1994, when Andrew Wiles capped a centuries-long search for the solution to Fermat’s Last Theorem, he was 41. Too old for the Fields, too young for the not-yet-existent Abel. Would the official organizations let the century’s most dramatic mathematical discovery go unrecognized? What could the IMU do?

Simple: they gave Wiles an ad hoc prize, a silver plaque. It stands alone in the annals of achievement.

The same goes for all research, not just prize-worthy breakthroughs. Every act of discovery, no matter how humble, leaves its signature on the document of human thought. By nudging forward the human understanding of reality, it makes winners of us all.

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I just got back from the most exciting and undeserved week of my year: the Heidelberg Laureate Forum.

It gathers, in an adorable German city, 25 laureates of math and computer scientists (winners of the Fields Medal, Abel Prize, Turing Award, etc.) along with 200 young researchers (students and postdocs), for a week of lectures, discussions, and fancy dinners at museums like this:

Among my favorite activities of the week is ambushing the young researchers and asking them to draw cartoons for me. You can find three posts on the HLF blog:

If Your Research Were a One-Page Cartoon

It’s the Fifth HLF. What Will the Next Five Years Bring?

What Does Good Mentoring Look Like?

Since it’s drawn by people other than me, this stuff already surpasses my usual fare, but for even better art, you can check out the work of my pal Coni Rojas-Molina, who did some great posts for the HLF blog:

On the mentoring side of things, Alaina Levine did a two-part interview with Vinton Cerf, one of the creators of the internet. (Pretty great line on a C.V., if you ask me.)

You should also check out the coverage by Katie Steckles and Paul Taylor, a delightful (read: English) couple with two math PhD’s between them. I especially recommend Leslie Lamport Thinks Your Proofs are Bad and What is a rotogon?

I enjoyed hearing Nana Liu plan out her piece on deep learning – another good read.

A necessary conversation-starter comes from Marcus Strom, who asks whether the HLF is doing all it can to support young women in the male-dominated fields of math and CS.

And if you want to relive the action moment by moment, check out the HLF Twitter feed, which was expertly run by Tobias Maier.

Lastly, for completeness’s sake, here are the cartoons I posted on Facebook this week:

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*© Heidelberg Laureate Forum Foundation / Kreutzer – 2017*

Last year, when I asked a few of them about this grand entrance, they shrugged and laughed. Yes, they’re icons of academia, titans of scholarship. But also, they’re human beings, blinking in the megawatt spotlight, frowning as they scan for their reserved seats.

The HLF gathers these 25 laureates alongside 200 young researchers to talk about what comes between those two phases: A career spent solving problems. A lifetime of chasing your curiosities and your demons through those strange borderlands, from the cold realm of mathematical abstraction to the messy world of human reality, and back again.

In short: research.

The crowd rises for a moment of silence, to honor three laureates who died in the past year. Two are legends of computer science, both named Charles: Thacker and Bachman. The third picture breaks my heart.

It’s Maryam Mirzakhani.

Born in 1977 in Tehran, Maryam was three years old when war broke out between Iran and Iraq. “I was a teenager when things became more stable,” she said in her Fields medal video, calling herself part of a “lucky generation.” Gifted from her youth, Mirzakhani won back-to-back gold medals for Iran at the International Mathematical Olympiad. She earned her doctorate at Harvard, became a professor at Stanford, and carried out groundbreaking research in the burgeoning field of dynamics, prompting one colleague to call her “a master of curved spaces.” In 2014, she won math’s greatest honor, the Fields Medal—the first woman to do so, and the first Iranian.

Doctors found her cancer in 2013. This July, a few months after her fortieth birthday, it killed her.

In her absence, it’s easy to freight Mirzakhani’s life with our external meanings. The audience rises; doors are thrown open; orchestral music blasts. A woman of warmth and texture is lost, replaced by something larger in stature and smaller in meaning: an icon, a symbol, a portrait on a coin.

But that’s not what the grieving mathematical community needs, nor is it what the HLF is about. It’s about how the ideals of scholarship manifest in individual lives. How a single woman, in her irreducible, irreproducible way, can embody the spirit of inquiry.

Mirzakhani labored for her breakthroughs, as all researchers must. She pondered, wandered, wondered. Her working style, in her own description, was “slow.” A colleague, Amie Wilkinson, remembers her “just pushing and pushing and pushing, completely optimistic the whole time.”

Mirzakhani doodled. Her daughter, watching Maryam sprawled on the floor, filling vast sheets of poster-sized paper, dubbed the work “painting.” Indeed, her work was shot through with geometrical insight: strange surfaces, elliptic paths, visions of billiard balls bouncing around tables.

“I wasn’t always very excited about math,” she said once of her youth. “I was more excited about reading novels, and I thought I would become a writer one day.” She laughed. “I got excited about it just as a challenge. Then I realized that it’s really nice, and that I enjoy it.”

In 2014, when she won her Fields, the presenters mixed up the four medals, not realizing that each had a name engraved on it. The story appeared in *The New Yorker *(in a piece by Siobhan Roberts) after Mirzakhani’s death:

“I received Martin [Hairer]’s, who received Maryam’s, who received Artur [Avila]’s, who received mine,” [Manjul] Bhargava said. “An unlikely scenario, even if the medals were distributed randomly.” The mathematicians had a real-life combinatorial problem in their hands. “After the ceremony, it was very busy, and there was little chance for all four of us, or even say three of us, to be in the same place simultaneously,” Bhargava explained. “Also, due to constant photo shoots, we each needed a medal with us at all times so that we could fulfill our duties and pose with one when asked.” When Mirzakhani and Bhargava ran into each other, they laughed and tried to figure out the optimal path toward a solution. What to do, standing there, Bhargava with Hairer’s medal, and Mirzakhani with Avila’s?

She got her medal in the end. And now, an inevitable process has begun. Time turns leaders into heroes, and heroes into statues. Mirzakhani shoulders the weight of so many hopes: for women in math, for transcending politics, for the unifying power of research. Those pressures will harden her life’s story into a kind of diamond: something shiny and durable, if not particularly human.

For now, though, I’ll hold on to the memory of this woman I never met. Her ambition. Her humility. Her optimism. Her doodles. Here in Heidelberg, you can feel her absence so acutely that it almost – but only almost – becomes a presence.

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Like the rest of the math internet, I recently fell in love with this rascal of a problem:

- I roll a die until I get a 6. What is the expected number of throws?
- When I tried this, I happened to roll only even numbers prior to getting the 6. Knowing this, what is the expected number of throws now?

I’ll let you think. See the end of the post for a solution.

By the way, I’m back in Heidelberg this week, working on the official blog of the illustrious Heidelberg Laureate Forum, which brings together Nobel-level scholars in math and computer science with young researchers just starting this career.

It’s the coolest thing I do all year. Let me know if you’ve got questions I should ask the honored laureates and/or German locals.

Martin Hellman – a pioneer of public key encryption, and one of the laureates here in Heidelberg – poses a fun problem about n-dimensional spheres.

*Spoiler*: n-dimensional objects are super counter-intuitive.

Speaking of linguistic precision, Rogers George spotted an interesting linguistic error in a recent roundup post: I referred to a “four-millennia-old Babylonian tablet,” when I should have called it a “four-*millennium*-old Babylonian tablet.”

Put them side by side, and my ear prefers the corrected version, even if I can’t explain why or elucidate the underlying rule. I find that kind of instinct fascinating. I suspect it’s more common in one’s native language than in any other domain of thought (since language is a complex rule-governed area in which we all have copious daily experience from a young age) but I feel like I have similar experiences in math from time to time. Sometimes a student only needs their eye drawn to an error to recognize it as wrong, even without knowing why.

Okay, solution to the dice problem:

* I roll a die until I get a 6. What is the expected number of throws? *This is a pretty standard question about the “geometric distribution.” When you’re waiting for an event with probability p, it takes 1/p attempts on average. So the answer is 6.

** When I tried this, I happened to roll only even numbers prior to getting the 6. Knowing this, what is the expected number of throws now? **A tempting approach: assume that this is like rolling a 3-sided die, and so the answer is 3.

Tempting, but quite wrong.

Here’s an intuitive illustration. Imagine that the three odd sides of the die have been labeled “RESET.” You count the rolls it takes until you get a 6, but landing on “RESET” sends you back to 0. This is equivalent to the original problem, and clearly *not* equivalent to a 3-sided die, because your count keeps resetting. You expect a lower value than 3.

But what value? Well, a truly elegant solution comes via the original post (well worth checking out):

*Rephrase the problem as “Roll a die until the first time you get a 1, 3, 5, or 6.” You’re waiting for an event with probability 2/3, so on average it will take you 3/2 attempts. And the expected number of rolls remains the same no matter which terminating value (1, 3, 5, or 6) happened to come up. Knowing that the terminating value was a 6, your answer is still 3/2, or 1.5 rolls.*

I find probability a fertile source of intuition-busting puzzles like this. (See also: Monty Hall, the two daughters problem, and others). It’s a circus trick that no other branch of math can quite match: elementary, easy-to-state problems; elegant, accessible solutions; and still, the ability to stump a mathematically sophisticated thinker.

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Andrew Wiles gained his fame by solving a nearly 400-year-old problem: Fermat’s Last Theorem. The same puzzle had captivated Wiles as a child and inspired him to pursue mathematics. His solution touched off a mathematical craze in a culture where “mathematical craze” is an oxymoron. Wiles found himself the subject of books, radio programs, TV documentaries—the biggest mathematical celebrity of the last half-century.

And so, having lucked into attending a press conference at the Heidelberg Laureate Forum in Germany, where Wiles was an honored guest, I asked him:

The essence of Wiles’ answer can be boiled down to just six words: “Accepting the state of being stuck.”

For Wiles, this is more than just a vague moral, an offhand suggestion. It’s the essence of his work. It’s an experience at once excruciating, joyful, and utterly unavoidable. And it’s something desperately misunderstood by the public.

“Accepting the state of being stuck”: that’s the keystone in the archway of mathematics. Without it, we’re left with nothing but a pile of fallen bricks.

***

Wiles began his answer, like any good mathematician, with a premise everyone can accept: “Many people have been put off mathematics,” he said. “They’ve had some adverse experience.”

It’s hard to argue with that.

“But what you find with children,” he continued, “is that they really enjoy it.”

In my experience, it’s true. Kids love games, puzzles, learning to count, playing with shapes, discovering patterns—in short, they love math. So how does Wiles account for our alienation from mathematics, our loss of innocence?

“What you have to handle when you start doing mathematics as an older child or as an adult is accepting the state of being stuck,” Wiles said. “People don’t get used to that. They find it very stressful.”

He used another word, too: “afraid.” “Even people who are very good at mathematics sometimes find this hard to get used to. They feel they’re failing.”

But being stuck, Wiles said, isn’t failure. “It’s part of the process. It’s not something to be frightened of.”

Catch me and my teacher colleagues any afternoon, and—if you can get past the “sine” puns and fraction jokes—you’ll likely find us griping about precisely this phenomenon. Our students lack persistence. Give them a recipe, and they settle into monotonous productivity; give them an open-ended puzzle, and they panic.

Students want the Method, the panacea, the answer key. Accustomed to automaticity, they can’t accept being stuck.

Wiles recognizes this fear, and knows that it’s misplaced. “For people who carry on,” he said, “it’s really an enjoyable experience. It’s exciting.”

Wiles explained the process of research mathematics like this: “You absorb everything about the problem. You think about it a great deal—all the techniques that are used for these things. [But] usually, it needs something else.” Few problems worth your attention will yield under the standard attacks.

“So,” he said, “you get stuck.”

“Then you have to stop,” Wiles said. “Let your mind relax a bit…. Your subconscious is making connections. And you start again—the next afternoon, the next day, the next week.”

Patience, perseverance, acceptance—this is what defines a mathematician.

“What I fight against most,” said Wiles, naming an unlikely enemy, “is the kind of message put out by—for example—the film Good Will Hunting.”

When it comes to math, Wiles said, people tend to believe “that there is something you’re born with, and either you have it or you don’t. But that’s not really the experience of mathematicians. We all find it difficult. It’s not that we’re any different from someone who struggles with maths problems in third grade…. We’re just prepared to handle that struggle on a much larger scale. We’ve built up resistance to those setbacks.”

Of course, Wiles isn’t the first to name perseverance as the key to mathematical progress. Others have analyzed the same challenge—albeit through different conceptual lenses.

One prevailing framework is *grit*. Under this approach, perseverance is a partly a matter of personality, of exhibiting the right characteristics: tenacity, determination, a sort of healthy native stubbornness. When the going gets tough, grit-less kids bail, whereas gritty kids keep working—and thus prosper.

But recently, the currency of “grit” has fallen among teachers. It’s not that the idea lacks psychological validity. It’s more the weight of its educational connotations. Grit has become an excuse to romanticize poverty as “character-building.” It has devolved into a vague catch-all at best, and at its paradoxical worst, a reason to write kids off as lost causes.

These days, the educational conversation revolves instead around Carol Dweck’s grit-related concept of *mindsets*.

Some people exhibit a *fixed mindset*. They believe that one’s intelligence and abilities are unchanging, stable traits. Success, to them, is not about effort; it’s about raw ability. To struggle is to reveal your intellectual shortcomings. They can accept the state of being stuck only insofar as they accept the state of being visibly and irrefutably stupid—which is to say, not very far.

By contrast, those with a *growth mindset* believe that effort fuels progress. The harder you work, the more you’ll learn. To be stuck is a transient state, which you overcome with patience and persistence.

Wiles is no educational theorist, of course, but I find that he offers a resonant and compelling third path. For him, perseverance is neither about personality (as with grit) nor belief (as with mindset).

Rather, it’s about emotion.

Fears and anxieties come to us all. You can be a nimble mathematician, a model of grit, and a fervent believer in the human potential for growth—but still, getting stuck on a math problem may leave you deflated and disheartened.

Wiles knows that the mathematician’s battle is emotional as much as intellectual. You need to quiet your fear, harness your joy, and cope effectively with the doubt we all feel when stuck on a problem.

Perhaps it’s only a folk psychology of perseverance. But I’m drawn by its potential to explain how students behave—and to motivate them to strive for more.

For example, take Wiles’ musings on the value of forgetfulness. “I think it’s bad to have too good a memory if you want to be a mathematician,” Wiles said. “You need to forget the way you approached [the problem] the previous time.”

It goes like this. You try one strategy on a problem. It fails. You retreat, dispirited. Later, having forgotten your bitter defeat, you try the same strategy again. Perhaps the process repeats. But eventually—again, thanks to your forgetfulness—you commit a slight error, a tiny deviation from the path you’ve tried several times. And suddenly, you succeed.

Wiles has a nifty analogy for this: it’s like a chance mutation in a strand of DNA that yields surprising evolutionary success.

“If you remember all the false, failed attempts before,” said Wiles, “you wouldn’t try. But because I have a slightly bad memory, I’ll try essentially the same thing again, and then I’ll realize I was just missing this one little thing.”

Wiles’ forgetfulness is a shield against discouragement. It neutralizes the emotions that would push him away from productive work.

Of course, immunity to fear isn’t enough. You need a positive incentive, something to strive for. And here, Wiles understands the delicate emotions of discovery better than anyone. He knows the immense release, the inner fireworks, of solving a problem at last. His problem, after all, took seven years of daily grind. Centuries, if you count the generations of mathematicians who tried and failed before.

“You find this *thing*,” Wiles said. “Suddenly you see the beauty of this landscape.” Before, “when it’s still some kind of conjecture, it seems really far away.” But now, with a solution in hand, “it’s like your eyes are open.”

For Wiles, doing mathematics is not merely the flexing of an intellectual muscle. It is a long and harrowing journey, so rich and involving that it becomes tactile, sensory, literal.

Listening to Wiles, you feel this. Beneath his gentle poise, you can sense the ten-year-old boy, pouring hours into Fermat’s Last Theorem, undeterred by the centuries of failure that have come before, unafraid of the decades of work ahead.

If you hold one mental image of Wiles, he wants it to be this: not the triumphant scholar with the medal around his neck, but the child learning to glory in the state of being stuck.

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