The Positive Side of Impossible

“I wish you hadn’t just told me not to touch it, because I don’t want to get into trouble and I didn’t even want to touch it, but your telling me not to makes me want to touch it!” my five-year-old exclaimed in frustration, apropos of something or other I’d asked him not to touch. Children are like that. Or, as the song “Never Say No” puts it: “Children, I guess, must get their own way the minute that you say no.”¹

Adults are like that too. Being told what we can’t do takes us back to the time when we were powerless children, and sometimes we grownups respond to prohibitions in childish ways. Consider how many supposedly grown-up people have tantrums when they’re told they can’t enter a certain establishment unless they’re wearing a face mask! I sometimes wonder whether I’ve really matured as much as my change in station over the past half-century (from snotty pre-teen to tenured professor) would indicate; maybe I only seem more mature because, in my present life circumstances, fewer people tell me what I can’t do. 

SQUARING THE CIRCLE

Among the adults who don’t like being told “You can’t do that” are many adults who enjoy math as a hobby, and the most common thing they’re told they can’t do is square the circle.

When 1+1 Equals 0

dedicated to the memory of Elwyn Berlekamp

The mistaken formula (x+y)² = x² + y² is sometimes called the First Year Student’s Dream, but I think that’s a bad name for three reasons. First, (x+y)² = x² + y²  is not exactly a rookie error; it’s more of a sophomoric mistake based on overgeneralizing the valid formula 2(x+y) = 2x + 2y. (See Endnote #1.) Second, most high-school and college first-year students’ nocturnal imaginings aren’t about equations. Third, the Dream is not a mere dream — it’s a visitor from a branch of mathematics that more people should know about. The First Year Student’s Dream is a formula that’s valid and useful in the study of fields of characteristic two.

How Can Math Be Wrong?

Let’s start with something uncontroversial: a valid mathematical assertion like 3+3+3=9 can be “wrong” if it’s been dragged into a situation in which it just doesn’t belong. Consider the Sufi tale¹ of Mullah Nasrudin and his wife.

Three months after Nasrudin married his new wife, she gave birth to a baby girl.

“Now, I’m no expert or anything,” said Nasrudin, “and please don’t take this the wrong way-but tell me this: Doesn’t it take nine months for a woman to go from child conception to childbirth?”

“You men are all alike,” she replied, “so ignorant of womanly matters. Tell me something: how long have I been married to you?”

“Three months,” replied Nasrudin.

“And how long have you been married to me?” she asked.

“Three months,” replied Nasrudin.

“And how long have I been pregnant?” she inquired.

“Three months,” replied Nasrudin.

“So,” she explained, “three plus three plus three equals nine. Are you satisfied now?”

“Yes,” replied Nasrudin, “please forgive me for bringing up the matter.”

The Muffin Curse

Here’s a small puzzle that opens the door to a surprisingly tricky general problem: How can a teacher divide 24 muffins among 25 students so that everyone gets the same amount to eat but nobody gets stuck with any tiny pieces?

To get a clearer sense of what counts as a good answer, let’s consider a bad answer. You could remove 1/25th of each muffin, give an almost-complete muffin to each of the first 24 students, and give the 24 slivers to the last student. Then everyone gets 96% of a muffin, but it‘s a pretty crumby scheme for the student who gets nothing but slivers. We’d like to do better. Can you find a scheme in which the smallest piece anyone gets stuck with is bigger than 1/25 of a muffin? Can you find a solution in which the smallest piece is a lot bigger? After you’ve found the best solution you can and you can’t improve it, how might you try to prove that it’s the best solution anyone could ever find? And how would you solve the problem if there were a different number of muffins and/or a different number of students trying to share them? Puzzles of this kind can be challenging and addictive, and the general solution wasn’t found until last year.

Math, Games, and Ronald Graham

In memory of Ron Graham, 1935-2020

Bear with me if I seem to be veering out of my lane (as they say nowadays), but let me ask: What is chess? If you play with a chess set in which a lost pawn has been replaced with a button, you’re violating tournament regulations but most people would say you’re still playing chess; the button, viewed from “inside” the game, is a pawn. Likewise, if you’re playing against your computer, the picture of a chessboard that you see on your screen is fake but the game itself is real. That’s because chess isn’t about what the pieces are made of, it’s about the rules that we follow while moving those pieces. Asking “Do pawns exist?”, meaning “Are there real-world objects that behave in accordance with the rules of chess?”, misses the point. If one of your pieces has been shoddily manufactured and spontaneously fractures, that doesn’t mean that your mental model of how chess pieces behave is flawed; it’s reality’s fault for failing to conform to your mental model.

You’ve probably already guessed the agenda behind my rambling about chess, but here it is explicitly: I claim that math (pure math, anyway) is as much a game as a science. The objects of mathematical thought, like the pieces in chess, are defined not by what they “are” but by the rules of play that govern them. The fact that in math the pieces exist only in our imaginations and the moves are mental events doesn’t make the rules any less binding. And even though the rules are human creations, once we’ve agreed to them, the answer to a question like “Is chess a win for the first player?” or “Is the Riemann Hypothesis true?” aren’t matters of individual opinion or group consensus; the answers to our questions are out of our hands, irrespective of whether we like those answers or even know what they are.¹

The Mathematics of Irony

The more you study, the more you know.
The more you know, the more you forget.
The more you forget, the less you know.
So why study?
The less you study, the less you know.
The less you know, the less you forget.
The less you forget, the more you know.
So why study?

— “Sophomoric Philosophy”

Poor Oedipus! The mythical Theban started out life with every advantage a royal lineage could offer but ended up as the poster child for IFS: Inexorable Fate Syndrome. His parents packed him off in infancy to evade a prophecy that he’d kill his father and marry his mother. He was found on a mountain and raised by a shepherd, so Oedipus didn’t know who his birth parents were. Once he learned about the prophecy he did everything he could to avoid fulfilling it (aside from not killing or marrying anyone, which in those times would have been an undue hardship), but he still ended up doing exactly what he was trying not to do.

If the story of Oedipus seems a bit removed from real life, listen to episode 3 of Tim Harford’s podcast “Cautionary Tales”, titled “LaLa Land: Galileo’s Warning”, to hear about systems that were designed by intelligent, well-meaning people to avert disasters but which ended up causing disasters instead.

Confessions of a Conway Groupie

To the memory of John Conway, 1937–2020

“So let me get this straight, Mr. Propp: you plan to go to England to work with a mathematician who doesn’t even know you exist?”

It was 1982, I was a college senior applying for a fellowship that I hoped would send me to Cambridge University for a year, and the interviewer was voicing justified incredulity at my half-baked plan to collaborate with John Conway.

Air from Archimedes

When I was ten, I read with astonishment that with each breath, I was inhaling molecules that were breathed by the mathematician Archimedes over two thousand years ago.

This sort of invocation of chemistry as a magic history-spanning bridge can be traced back to James Jeans, the English scientist and mathematician, who in his 1940 book “The Kinetic Theory of Gases” wrote: “If we assume that the last breath of, say, Julius Caesar has by now become thoroughly scattered through the atmosphere, then the chances are that each of us inhales one molecule of it with every breath we take.” The science writer Sam Kean recently wrote an entire book, “Caesar’s Last Breath”, that takes this proposition as its starting point.

In between Jeans and Kean, other writers making the same point have replaced Caesar by Archimedes or Jesus or da Vinci. I prefer Archimedes, because he was the first of the ancient Greek mathematicians to come to grips with really big numbers and to connect the macroscopic and microscopic realms; in “The Sand Reckoner” he calculated how many grains of sand would fill the universe as the Greeks understood it.

As I write this essay in April 2020, human society has been violently tipped on its side, and the eight billion or so people who share this planet have come to realize how small the world has become epidemiologically. We’ve also become fearfully conscious of the contents of the air we bring into our bodies. Perhaps now is a good time to take a deep and hopefully healthy breath and think a bit about how the content of our lungs connects us to people far away in space and time, situated in a past that, even at a remove of a few months, feels very distant.

The Square Root of Pi

Bella: You gotta give me some answers.

Edward: “Yes”; “No”; “To get to the other side”; “1.77245…”

Bella (interrupting): I don’t want to know what the square root of pi is.

Edward (surprised): You knew that?

— Twilight

March 14 (month 3, day 14 of each year) is the day math-nerds celebrate the number π (3.14…), and you might be one of them. But if you’re getting tired of your π served plain, why not spice things up by combining the world’s favorite nerdy number with the world’s favorite nerdy operation?

Chess with the Devil

“As far as God goes, I am a nonbeliever. Still am. But when it comes to a devil — well, that’s something else.”

— The Exorcist (William Peter Blatty)

Sometimes a key advance is embodied in an insight that in retrospect looks simple and even obvious, and when someone shares it with us our elation is mixed with a kind of bewildered embarrassment, as seen in T. H. Huxley’s reaction to learning about Darwin’s theory of evolution through natural selection: “How extremely stupid not to have thought of that.”

This phenomenon often arises as one learns math. Mathematician Piper H writes: “The weird thing about math is you can be struggling to climb this mountain, then conquer the mountain, and look out from the top only to find you’re only a few feet from where you started.” In the same vein, mathematician David Epstein has said that learning mathematics is like climbing a cliff that’s sheer vertical in front of you and horizontal behind you. And mathematician Jules Hedges writes: “Math is like flattening mountains. It’s really hard work, and after you’ve finished you show someone and they say “What mountain?””

Drawings by Ben Orlin. Check out his webpage: https://mathwithbaddrawings.com/