And (spoiler alert) the solutions. (Click the image to enlarge it.)

Happy 2025! This is a special year, because — mathematically speaking — it is a year when you and I are alive, a year when lifeless matter has coalesced into living bodies on the wet skin of a rocky sphere in a vibrant corner of an inconceivable cosmos.

Also, it’s a special year because 2025 = 45² = (1+2+3+4+5+6+7+8+9)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³.

So, good times no matter how you slice it.

Finnished

I got to go to Helsinki this month, thanks to my translator/publisher/pal/Finnisher Juha Pietiläinen.

His company Terra Cognita publishes big-time science books by big-name authors on socially urgent topics, so that Finnish readers can join the global conversation in their native tongue. So, yes, my stick-figure jokes are a perfect fit. No further questions please.

Additional thanks to Rosebud Books, who hosted Juha and me in conversation on Saturday. I got to read aloud the Dr. Seuss parody from my new book, swapping verses with Juha’s Finnish version, so the audience could hear how he handled the impossible task.

(Juha got applause. I did not. This is fair.)

Signed Copies

This month, while making my rounds of the local bookstores, I took a few moments to sign the available copies of my books.

Sorry, did I use the first-person singular there? Bad habit. My daughters did most of the signing.

Autographed versions of Math for English Majors are now in abundant supply at Next Chapter.

Going Dutch

To my honor, the first pages from Math for English Majors (originally published at Lit Hub) were translated into Dutch and printed in Belgium’s paper of record, De Standaard.

At the editor’s invitation, I added a few paragraphs, which I offer in English here:

We all know the stereotype: there are math people, and “not math” people. The separation begins early in childhood. By adulthood, the two belong to wholly separate intellectual traditions—which might as well be different species, evolved on different planets, with no common language except for disapproving noises and warlike glares.

Why, then, has the union of math and literature been so fruitful?

One possibility is that the two are complements. Like sweet and salty, math and literature are opposite flavors that pair well together. “The union of the mathematician with the poet,” said William James, “fervor with measure, passion with correctness, this surely is the ideal.” Perhaps math’s clarity and coolness is the perfect balance for poetry’s intensity and heat.

Another possibility—the one I tend to favor—is just the opposite: that math and literature are not as different as they appear. “Mathematics,” said the mathematician Oswald Veblen, “is one of the essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry.” Mathematics and literature are both expressions of human imagination.

Sofia Kovalevskaya, the first woman to receive a PhD in mathematics, put it even more sharply. “It is impossible to be a mathematician,” she said, “without being a poet in the soul.”

Math Books I Enjoyed in 2024

For a while (2018, 2019, 2020, 2021) I compiled my favorite books of the year into a post. I cannot imagine where I got the energy for such an undertaking.

Instead, I offer a quick rundown of the mathier highlights from my 2024 reading.

First, if you’re looking for literature:

The MANIAC, by Benjamin Labatut. A 20th-century monster story. The monster is John von Neumann, and the world of artificial intelligence that he helped to create. The story is narrated in the voices of those who knew him.

Luminous, by Greg Egan. Egan’s sci-fi short stories are more mathematically sophisticated than a lot of math textbooks. (And don’t worry, the literary sophistication is higher, too.) A note for Ted Chiang fans: the title story makes a fascinating pairing with Chiang’s “Division By Zero.”

Second, if you’re looking not for literature per se, but good books about it:

Once Upon a Prime, Sarah Hart. A tour of connections between literature and mathematics. One of my new favorite works of math popularization. It’s a book I long intended to write, and I’m glad Hart got there first, because she did a better job than I would have.

Much Ado About Numbers, by Rob Eastaway. Best enjoyed as a kind of trivia book, with mathematician Eastaway guiding us through Shakespeare’s language, time, and world.

Third, if you’re looking for mathematical puzzles, I’ve already sung the praises of a few books, but for even more in the same vein, consider:

Seven Games, by Oliver Roeder. Those seven games: chess, checkers, go, bridge, poker, scrabble, and backgammon. Published a few years ago, but still a good backdoor exploration of artificial intelligence, in all its powers and limitations.

The Puzzler, by A.J. Jacobs. Cheeky, breezy nonfiction about the many worlds of puzzles. Ranges from crosswords to jigsaws to Smullyan-esque logical traps. Lots of good puzzles throughout.

And finally, if you’re looking for good ol’ nonfiction:

Trefethen’s Index Cards, by Lloyd N. Trefethen. Since youth, mathematician Trefethen has kept a sort of diary in the form of index cards. This book draws a few hundred from across the decades, with frank, wistful observations on math, life, and culture.

The Emergence of Probability, by Ian Hacking. A historical tour de force that I wrote about in a recent post.

Numbers Don’t Lie, by Vaclav Smil. A clear-eyed, iron-hearted, quantitatively-driven survey of the 21st century landscape.

Parting Puzzle(s).

These six come from a New Year’s email sent by my father Jim Orlin. A renowned researcher in network flows, he is widely considered one of the most adorable of all Orlins.

2025 is a special year in that it is the square of an integer.   

1.  What was the last year (prior to 2025) that was a square of an integer?

2.  What is the next year  (after 2025) that is a square of an integer?

Also, 2025 has 15 divisors. 

3.  What was the last year (prior to 2025) that has 15 divisors? 

4.   What is the next year (after 2025) that has 15 divisors?

The following is a bit surprising, I think.

5.  What was the last year (prior to 2025) that has 16 divisors?

6.  What is the next year  (after 2025) that has 16 divisors?

When my first book Math with Bad Drawings came out, I kept having a strange experience.

“Yes!” a stranger would say, pointing at the stick figure on the cover. “We need more of this.”

My enthusiastic supporter had, until 30 seconds prior, never heard of the book. They still hadn’t cracked the spine. I was honored by the support, but it was obvious that I hadn’t yet earned it. For all they knew, I had written 376 pages of vulgar limericks. What convinced them that I was on the right path?

I’ve come to see this as a basic dynamic in math education reform: an illusory spirit of consensus. Clearly math education needs more something. But more what?

One popular answer: “more data science.” Let’s renounce all of those fusty, old-fashioned trigonometric formulas. Instead, fill math class with 21^st-century virtues: data visualization, probabilistic thinking, and statistical literacy. This isn’t necessarily about new pedagogy; it’s about new content. It’s about re-centering math class on techniques that will cash out, on skills that students might actually apply.

Another popular answer: “more student-centered.” Let’s banish all tasks mechanical and rote. Instead, fill math class with open-ended puzzles, inquiry-based learning, and creative projects. This isn’t necessarily about new content; it’s about new pedagogy. It’s about re-centering math class on the voices, the ideas, and the ingenuity of the students themselves.

I see value in both of these approaches (and many others besides). My first book, Math with Bad Drawings, with its long sections on statistics and probability, was largely about data literacy. My second book, Change is the Only Constant, was a human-centered take on calculus.

But when it comes to systemic change, the two are orthogonal at best, and opposites at worst.

Should we dismantle our regime of standardized tests? Or are they a vital tool for measuring the success of a new curriculum?

Should math education impart deep experiences of beauty? Or practical, wage-raising skills?

Should students pursue open-ended, creative thinking? Or develop concrete knowledge with immediate applications?

Is math a liberal art, akin to art or music? Or is it a practical craft, like computer programming or home economics?

Should economic productivity be the guiding principle for secondary education? And if not, then what should be?

I wouldn’t endorse either vision in its purest form. Math education pursues a hodgepodge of goals: quantitative literacy, humanistic growth, preparation for STEM careers. The educators I admire most don’t subscribe to easy dichotomies. They value student voices and factual knowledge, algebraic fluency and open-ended exploration. Myself, I am comfortable with only two universal claims about math education: there are always tensions, and there are always tradeoffs.

That’s why I worry about the illusory consensus around reform. How can we make wise tradeoffs if we don’t acknowledge the tensions?

The solution, I think, is simple: show your work.

Show us what a good lesson (or unit, or year) looks like, and explain why.

I’m trying to do this in my own teaching now: to resolve all these contradictory possibilities in a way that works for my students. It’s not easy! There are days when I do an adequate job, and days when I do quite the opposite.

Anyway, take this as a pledge: in 2025, I’d like to show more of my work. I’m teaching two classes this term: Intro to Statistics, and Liberal Arts Mathematics (a kind of escape hatch for students who don’t want to take college algebra as their final required math class). I approach them both with a muddle of overlapping values, and a desire to do whatever works best for my students’ learning. More forthcoming.

“Oulipans are rats who build the labyrinth
from which they will try to escape.”
–Raymond Queneau

It brings me no joy to report the rebirth (or the renewed undeadness) of the zombie literary movement known as OuLiPo.

Oulipo’s first birth came in 1960, from the vibrant and idle minds of Raymond Queneau and François Le Lionnais. They dubbed it ouvroir de littérature potentielle (“workshop of potential literature”). A self-conscious experiment in applying strict mathematical constraints to art, its results (such as George Perec’s La Disparition, a novel that avoids the letter e) were spectacles of virtuosity, triumphs of ingenuity, and, at their very best, passable works of art. No coincidence, I say, that the name “potential literature” stands opposed to actual literature.

I believed this volcano had gone dormant. I was wrong. The last year witnessed four eruptions. I offer brief comments on each species of ash.

In Sarah Hart’s CENTIPEDES: 100-LEGGED FICTIONS (Flatiron Books, $24.95) the self-imposed constraints are punishingly tight. In each of the book’s 100 works of micro-fiction, you will find: (1) a single sentence, (2) precisely 100 words long, (3) with no words repeated.

The rules are so constrictive they threaten to cut off all blood flow. Fortunately, they do not. An example to illustrate the form:

After Amber’s dad left, her anger found tiny outlets everywhere: shoving in lunch lines, rude names flung at unsuspecting classmates, graffitied desks, smoldering glares when Miss Gillow pressed with “Are you feeling okay?” or “Is there anything wrong, Amber?” – but whenever his name came up, those hazel eyes suddenly pacified, the fighter’s stance softened, that mouth effected neutral smiles, and a demure young lady emerged, only vaguely curious about Father’s whereabouts; these reversals vexed counselors, threw them off-balance, like firefighters reaching an inferno’s site to find no smoke, nor heat, not even sparks, just houses collapsing amidst silent winter air.

Quibbles, of course, abound: I question whether “Amber” and “Amber’s” should count as distinct; I note the tonal awkwardness of “Father’s”; and I lament that the semicolon after “whereabouts” is writhing in existential pain, denied its rightful identity as a period.

But in such games, rule-bending is inevitable. And to Hart’s credit, these stories actually function as stories. (Never a guarantee in the labyrinths of Oulipo.)

Another of Hart’s centipedes:

First, Melissa left him, pledging love and reunion someday soon; second, she threw herself into the crucible of New York; third came two grueling years given over to worshipping deadlines, answering emails through lunch, hallucinating PowerPoint, day-dreaming Excel, squeezing selfish “errands” (like dental surgery) among more pressing corporate duties; fourth, there were occasional visits home, giddy weekends enjoyed mostly in bed, sleeping late together, pondering no future further than breakfast; fifth, finally, a permanent return, plagued by halting conversations, dodging topics that frightened them, such as whether their flame still smoldered, or if it had quietly spent its last fuel.

The constraints manage to serve the art, imbuing the story-sentences with nervous energy, as if they are navigating around some unutterable thought. The clauses search and twist, forbidden to return home, scrambling onward until an inevitable collapse.

One more illustration:

Aged eighteen, bored and restless, he took to sea: working odd jobs on ships, shunning routine, quitting when promotion beckoned; slipping between continents, self-medicating insomnia, pouring paychecks into liquor bottles; leaving no traces or friends among port cities, only half-concerned lovers, their faces scarcely more permanent in memory than Etch-a-Sketch portraits; until one day the trade-winds died, depositing his ship at a leeward harbor, whose piers were crowded with strange waterlogged fragments, forgotten decades earlier but improbably carried here, along fate’s currents, confronting him now as pale, bloated symbols of what neither sailor nor soul can ever really escape: oneself.

Great literature? Certainly not. But Hart’s work is never dull and only occasionally forced. Those merits alone place it in the higher ranks of Oulipo.

RANDOM INCIDENTS (MIT Press, $20), a collection of poems coauthored by Ben Orlin and Hendree Milward, takes up a different Oulipo tradition: randomness as the seed of creativity. Each cluster of poems is inspired by a random sample drawn from a different aspect of our digital world — for example, a randomly chosen pair of YouTube channels. The poets then seek to weave meaning from the fragments.

Note: I say that they seek to weave meaning. I do not say that they succeed.

Most indicative, and most abysmal, are the Wikipedia-inspired poems. The authors used Wikipedia’s “random article” button three times per poem, and then stitched the three disparate topics together. At best, the stitches are horrifyingly obvious. At worst, there are no stitches at all, just a loose Frankenstein of juxtaposed limbs, held together by inertia alone.

(One senses that Milward is the more capable poet, and that Orlin is responsible for the more severe embarrassments, but that is cold comfort. If you mix chicken soup with house paint, the result belongs neither on spoons nor on walls.)

I suppose I must inflict an example upon you. Here is one of their more valiant efforts, titled “Name-Giving.” (If only the whole book met this modest standard of quality.)

Wikipedia #6:
Name-Giving

1.

In the place we call Romania,
by the river they call Buhai,
there is a persistent little trickle
of water along the ground,
and since we cannot help
but give names to things
the trickle is called Pârâul Întors:
the inner creek.

Without the name, it’d still be inner.
Without the name, it’d still be a creek.
And the name will not keep it
from drying up.

2.

There are certain sunflowers
called “chamomile,”
and certain of these
called “mayweed.”
But this was not good enough
for scientists, who decided
the specimens were better suited
by the name Tripleurospermum.

Flowers must look different to them.

3.

In Washington DC,
on the 8^th of December, 2008,
film critics gathered to assign names.
Slumdog Millionaire they called “Best Film.”
Danny Boyle they called “Best Director.”
Mickey Rourke they called “Best Actor.”

Maybe such names really work,
like magic.
Maybe, for the next year,
Mickey Rourke really was
the Best Actor on Earth,
until they gave the name
to someone else.

Beyond Wikipedia, Orlin and Milward sample other digital universes: posts on Reddit, photos on Instagram, videos on TikTok. While these poems are no better, I admire the ingenuity of the sampling methods: in the absence of a “random post” button, the authors cleverly simulated random walks across the platform, tapping into the “dark matter” of unseen posts, those disfavored and concealed by the algorithms.

As sociology, the results are illuminating; as mathematics, effective; and as poetry, dire beyond redemption.

In short: a characteristic work of Oulipo.

Gizem Karaali’s AUTOBIOGRAPHY IN THE FORM OF INTEGER SEQUENCES (Milkweed, $17.99) is a poetic riff on the Online Encyclopedia of Integer Sequences. It probes the individual personalities (and the shared impersonality) of numbers, the extent to which they can (and cannot) be imbued with life.

Titles of some entries:

• “Apartment numbers at which I have lived, in chronological order”

• “Hours I spent on my phone each day for the last year”

• “Ages at which I felt a phase of my life had ended”

• “Ages at which, looking back now, I feel a phase of my life ended”

• “Page numbers of the poems that have changed me” (Karaali provides no information on the books or the poets, only the pages)

• “Number of my parents still alive on my birthday each year” (The sequence begins with a long string of 2’s, but reader and writer both know that the sequence cannot rise, only descend, sooner or later, in two terrible steps)

Some numbers are freighted with heavy-handed significance (“funerals attended each year of my adulthood”). Others are obscure and trivial (“number of times my friend Ruby said the word ‘anarchic’ in successive conversations last month”). Others seem to be saturated with hidden meanings, accessible only to the poet, while others must be inscrutable even to her, the inhuman data forming a kind of anti-autobiography (or, more properly, auto-anti-biography).

The collection works because Karaali herself does not seem settled on the basic questions. What can numbers do? What can they not do? Karaali offers tentative answers but no single thesis. This is, I suspect, the formula for successful Oulipo: careful calculation plus unyielding skepticism of the enterprise itself.

Fourth and finally, Rob Sleezer’s ON CONCISION (Princeton University Press, $14.95) is not necessarily a work of Oulipo proper, but rather, the kind of literary prank that Queneau et al might appreciate.

The book is structured as a sort of tedious page-a-day calendar, with each leaf imparting a famous person’s thoughts on concision. Invariably, our heroes are in favor (ardently in favor!) of expressing oneself concisely.

What are the proper proportions of a maxim? A minimum of sound to a maximum of sense.

Mark Twain

I have only made this letter longer because I have not had time to make it shorter.

Blaise Pascal

Just as a good editor of prose ruthlessly prunes out unnecessary words, so a designer of statistical graphics should prune out ink.

Edward Tufte

The quotations themselves are unfailingly concise. Sleezer is interested in form, and how it mirrors (or undercuts) meaning. The book’s epigraph (yet another quote, the only one not concision-themed) is from Jim Propp: “Form without meaning is incomplete, but even before meaning attaches itself to form, form can point the way towards meaning.”

Alas, I cannot stop yet. You must experience the full soporific effect of the repetition:

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

Carl Friedrich Gauss

Not that the story need be long, but it will take a long while to make it short.

Henry David Thoreau

Concision in style, precision in thought, decision in life.

Victor Hugo

Vigorous writing is concise. A sentence should contain no unnecessary words, a paragraph no unnecessary sentences, for the same reason that a drawing should have no unnecessary lines and a machine no unnecessary parts.

Strunk & White

I began as a naturalistic painter. Very quickly I felt the urgent need for a more concise form of expression.

Piet Mondrian

French parents teach their kids to be concise and amusing, to keep everyone listening.

Pamela Druckerman

Examine every word you put on paper. You’ll find a surprising number that don’t serve any purpose.

William Zinsser

The catalogue goes on. And on. And on. All this concision adds up to something quite bloated. I’m offering here only a sample of Sleezer’s onerous compendium—but it must be sizable sample, or else the punchline will fail to land.

In all pointed sentences, some degree of accuracy must be sacrificed to conciseness.

Samuel Johnson

Eliminate the unnecessary so that the necessary may speak.

Hans Hofmann

The most valuable of all talents is never using two words when one will do.

Thomas Jefferson

The concise man makes one think; the verbose bores.

Edouard Manet

There is no truth which cannot be given in fifty words; the truth is always concise.

Barry Malzberg

Most directors subscribe the principle that less is more, and the best direction is the most concise direction.

Joshua Marston

A story should, to please, at least seem true,
Be apropos, well told, concise, and new:
And whenso’er it deviates from these rules,
The wise will sleep, and leave applause to fools.

Benjamin Stillingfleet

It will be my earnest aim that The New York Times give the news, all the news, in concise and attractive form.

Adolph Ochs

The more alien and strange a world or situation is, the more concise you have to be if you want the reader to follow you.

Karin Tidbeck

By the book’s halfway point, the effect becomes almost unbearable. All these paeans to concision—none original, none deeper than the skin of an apple, none saying anything the others do not already encompass. A stack of paper squandered on the monotonous recitation of the same platitude.

Then, finally, you reach the last and longest quotation.

Concision means you have to be able to say things between two commercials. Now that’s a structural property of our media—a very important structural property which imposes conformism in a very deep way, because if you have to meet the condition of concision, you can only either repeat conventional platitudes or else sound like you are from Neptune. That is, if you say anything that’s not conventional, it’s going to sound very strange. For example, if I get up on television and say, “The Soviet invasion of Afghanistan is a horror,” that meets the condition of concision. I don’t have to back it up with any evidence; everyone believes it already so therefore it’s straightforward and now comes the commercial. Suppose I get up in the same two minutes and say, “The U.S. invasion of South Vietnam is a horror.” Well, people are very surprised. They never knew there was a U.S. invasion of South Vietnam, so how could it be a horror? They heard of something called the U.S. “defense” of South Vietnam, and maybe that it was wrong, but they never heard anybody talk about the U.S. “invasion” of South Vietnam. So, therefore, they have a right to ask what I’m talking about. Copy editors will ask me when I try to sneak something like this into an article what I mean. They’ll say, “I don t remember any such event.” They have a right to ask what I mean. This structural requirement of concision that’s imposed by our media disallows the possibility of explanation; in fact, that’s its propaganda function. It means that you can repeat conventional platitudes, but you can’t say anything out of the ordinary without sounding as if you’re from Neptune, a wacko, because to explain what you meant—and people have a right to ask if it’s an unconventional thought—would take a little bit of time…. Here, our media are constructed so you don’t have time; you have to meet the condition of concision. And whether anybody in the public relations industry thought this up or not, the fact is that it’s highly functional to impose thought control.

Noam Chomsky

Sleezer’s book, amusing as it is, suffers a limitation common to satiric works. It punctures the myth of concision, pops the overinflated balloon—but in the silence that ensues, it has nothing else to say. The straitjacket of concision is thrown off; but one wishes Sleezer (or Chomsky, or Twain, or anyone!) would chime in with the wisdom that concision had withheld from us.

Ah well. I suppose it’s fun to see balloons (and constraints) pop.

On Amazon, Sleezer’s book is listed (among other categories) as a work of “Inspirational Quotations.” Perhaps just a marketing blunder, or perhaps another layer to the satire, but I choose to read it in earnest. What better form of inspiration than to declare that all your prior inspiration was truisms and lies?