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.
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 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. Continue reading
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.
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.
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?
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?
“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?””
“You don’t have to believe in God, but you should believe in The Book.”
— Paul Erdős
Creating gods in our own image is a human tendency mathematicians aren’t immune to. The famed 20th century mathematician Paul “Uncle Paul” Erdős, although a nonbeliever, liked to imagine a deity who possessed a special Book containing the best proof of every mathematical truth. If you found a proof whose elegance pleased Erdős, he’d exclaim “That’s one from The Book!”
I’m a fan of Erdős, but today I’ll argue that the belief that every theorem has a best proof is misguided.1
“If you have arugula, basil, celery, dandelion, and endive leaves, how many different tossed salads can you make?” That question, or something like it, was asked in a Math Bowl that I participated in back in high school, during my halcyon days as a mathlete.1 Actually, “halcyon days” are supposed to be calm days, and quiz-show-style math-smackdowns aren’t known for being calm. It was certainly an un-halcyon moment when my Math Bowl teammates were urgently saying we should buzz in with the answer 32 to that salad question, and I was saying we needed to figure out whether the judges would think that a bowl containing no ingredients at all was a valid salad. While we were debating the issue, the other team buzzed in with the answer 32, only to be told “That is incorrect.” Our team immediately buzzed in with the answer 31, which seemed likely to be the answer the judges were looking for.
We got the points, but I liked the other team’s answer better. The idea of an empty salad might seem like a purely mathematical fancy, but half a dozen years later I saw a restaurant menu that offered the null salad, or rather “Nowt, served with a subtle hint of sod all” (for the unbeatable price of 0 pounds and 0 pence).2
Cartoon by Ben Orlin. Follow him on Twitter @benorlin! Read his blog at https://mathwithbaddrawings.com! Buy his books from your local independent bookseller!
It was a truism of mid-twentieth century popular intellectual culture that many disagreements were “merely semantic” and could be resolved if only people would agree on the meanings of the words they used, or at least were more clear about the different ways they used words so that they could focus on substantive issues rather than language.
Cartoon by Jules Feiffer. Permission pending.
It’s not hard to see that this idea has serious limitations. For instance, even though many legal issues surrounding abortion hinge on different definitions of the word “life”, when it comes to the moral side of the debate, definitions don’t change anyone’s mind. Usually we each choose the definition that matches an outcome we’ve decided on, not the other way around. But in mathematics (thank goodness for the consolations of math!), things are different. Continue reading