Wait, sorry, that’s Tagalog.

Still, mathematical words can feel like a foreign tongue. And they’re much harder to acquire than terms in Tagalog or English. To see why, consider how you might learn a new word. A word like, say, “cat.”

First, you might just pick up its meaning from context.

Or, second, someone can just tell you the meaning.

But there’s one situation where learning a definition demands extra effort: *when you’ve got no prior experience with the object being defined. *In this case, learning the definition is not simply a matter of affixing a name to something you already know.

You need a more elaborate introduction.

It’s this third scenario where mathematical words tend to belong. They name ideas that are invisible, abstract, and yet highly precise. The “derivative,” say, is something exotic and ethereal—even more so than cats.

Which brings me to my point: our mathematical culture gets this exactly backwards.

We tend to define a new term in the abstract, draping it in high-minded language like purple garments—all while nobody has any idea who’s under the robes. A much better method: explore motivating examples, and *then* give definitions.

How do you define “cat” without a cat? The fact is, you can’t.

]]>**Of a Literary Persuasion**

** Pride and Prejudice, by Jane Austen**. Okay, you can judge me for not reading this until my 30s. It’s a delight. I see why people turn to Austen not just for psychological insight and juicy plots, but for comfort, too: her writing has a lightness that’s so hard to find. A deft comic touch, a well-earned happy ending – these are rare things. Every year, there are like 17 gorgeous, bleak, midnight-dark drama films, and maybe, if you’re lucky, one decent comedy. But it’s often light stories touch us most deeply.

** The Remains of the Day, by Kazuo Ishuguro**. 200 pages of a butler philosophizing about what it means to be a Great Butler. But mesmerizing. Ishuguro understands avoidance, denial, the psychological mechanisms by which we protect ourselves. You can feel the narrator’s desperate efforts at self-preservation radiating off the page. I’m filled with dread and empathy, both at once.

** The Glass Hotel, by Emily St. John Mandel**. An oblique portrait of a Bernie Madoff figure, and the ripple effects of his crime. As in her last book (the post-apocalyptic

** Dear Life, by Alice Munro**. Lately, I’ve been mulling what makes a great short story, and what makes it different from a novel. Munro’s work is Exhibit A. Her stories pack novelistic twists and depth into just 20 or 30 pages, leaping across time and space en route to abrupt, devastating finales. So maybe that’s the secret to short stories: the agility, the maneuverability. A novel is a cargo ship; you can’t turn 180 degrees without signaling your intentions long beforehand. A short story is a kayak: Munro can lead us down waterfalls and land with a gentle splash.

** Circe, by Madeline Miller**. Wildly hyped. Worth the hype. The “retelling an ancient myth from the perspective of a relatively minor female character” things has been done, by authors as skillful as Ursula Le Guin (

**Of a Sci-Fi Persuasion**

** The Killing Moon, by N.K. Jemisin**. Apparently Jemisin borrowed a lot of the book’s hierarchical, dream-obsessed culture from ancient Egypt. I’m too dumb to have picked up on that while reading. Instead, it worked for me as a seamless, deeply imagined world. More conventional than the Broken Earth trilogy, but just as tightly woven.

** The Dispossessed, by Ursula Le Guin**. A scrappy anarchist society ekes out existence on a hardscrabble, barely habitable moon. From there, a brilliant scientist (with a peculiar mix of wisdom and naivete) visits the homeworld. Le Guin is the foremost anthropologist among sci-fi writers, and this is her foremost work of anthropology. It earns the book’s subtitle, “an ambiguous utopia.”

** The Eye of the Heron, by Ursula Le Guin**. A minor Le Guin work is like a Beatles album track; still better than 99% of stuff by mortal humans. The structure is a little funny, as it veers away from its protagonist midway through, but the left turn serves a structural purpose, widening the reader’s gaze from the traditional axis of “violence vs. nonviolence,” to encompass more feminine (and feminist) perspectives.

** The Starry Rift, by James Tiptree, Jr (penname of Alice Sheldon).** Though they deal with the pulpy fun of faster-than-light travel and journeys to alien planets, these stories are bittersweet. Some are downright sad. It’s like if Sufjan Stevens did the soundtrack to

** This Is How You Lose the Time War, by Amal El-Mohtar and Max Gladstone**. The epistolary love story of two time-traveling assassins. It somehow works both as a lovely fable (bright colors, lyrical prose, storybook structure) and a too-cool-for-school punk sci-fi experience (sly wit, wry futurism, LGBTQ romance). It’s tempting to credit those two disparate strains to the two authors, but I just read Amal El-Mohtar’s Hugo-winning short story Seasons of Glass and Iron, and though singly authored, it braids together the same two styles (with the same success).

**Axiomatic****, by Greg Egan**. Short stories with killer sci-fi premises: a mathematical theory of parallel universes; a rigorous vision of messages sent backward in time; a terrifying spatial anomaly that you can survive only by running into its dark center; a “jewel” implant that allows you to transfer your consciousness from mortal flesh to immortal silicon… Why haven’t lesser writers copied these ideas, making cliches of them? I suspect lesser writers can’t pull them off. Egan writes from deep knowledge and technical expertise; if you lack his skills, you can’t wield his tools.

**Of a Wildly Inventive and Basically Unclassifiable Persuasion**

** The Blind Assassin, by Margaret Atwood**. This book has a ridiculous structure. It’s a pulpy sci-fi novel, wrapped in an experimental piece of literary erotica, wrapped in a bleak memoir about aging, decrepitude, and alienation. Who could possibly pull

** Interior Chinatown, by Charles Yu**. A Chinese-American dude navigates life in a country that refuses to accept him as a protagonist, or even an individual. The whole book is structured as a screenplay: a shapeshifting, impossible-to-film, utterly brilliant screenplay. I love everything Yu has written, from the cerebral self-reflexivity of

** Someone Who Will Love You in All Your Damaged Glory, by Raphael Bob-Waksberg**. Absurd, fantastical short stories about the gritty realities of love. Each story is wildly individual, with its own distinctive voice. The bizarre blend (pathos + tragedy + bizarre comic premises + puns) is pretty much what you’d expect if you’ve seen Bob-Waksberg’s TV show

** The 13 1/2 Lives of Captain Bluebear, by Walter Moers**. Too silly for proper adults, too long for proper kids, but perfect for weird old improper kids like me. I don’t really get the gender structure of this magical world – around page 500 it occurred to me that, out of dozens of characters we’d met, all but one or two had been male – but I loved the descriptions (and evocative illustrations) of Bluebear’s bizarre fantasy continent. A gentle work of vivid imagination.

** Ladies and Gentlemen, the Bible!, by Jonathan Goldstein**. Famous passages from the bible, retold with a light and lovely narrative touch. It makes the old stories feel new again, bringing out their absurdity and (just as central to the book’s purpose) their insight. I smiled a lot.

** To Be Or Not To Be, by Ryan North**. A choose-your-own-adventure version of Hamlet, with hundreds of different endings. North has a genius for structure, and playing out every funny implication of his premise. My favorite: if you keep choosing the same “path” as the original Hamlet, the narrator eventually steps in to accuse you of making terrible character choice. Not wrong!

**Rereading Favorite Authors**

** Alice in Wonderland and Through the Looking Glass, by Lewis Carroll.** Much as I love these books, it struck me how plotless and meandering they are. They’re collections of clever scenes, memorable dreamlike exchanges. As such, they face the structural demands of stand-up comedy: you’ve got to land every joke, because one false moment can break the momentum.

** Invisible Cities and Cosmicomics, by Italo Calvino**. I enjoy Calvino’s short stories. I enjoy his novels. But his best work lies in between, with these “concept albums” of connected short stories.

** Mother Night, Cat’s Cradle, and God Bless You, Mr. Rosewater, by Kurt Vonnegut.** Vonnegut espouses a bleak form of hope: he thinks that humans are basically dull, broken, unlovable things, and that we can – and must – love them anyway. Compared to my first reading, I found myself less impressed by the philosophy contained the writing, and much more impressed by the craft. The climax of

Bonus: ** Slaughterhouse-Five: The Graphic Novel, by Ryan North and Albert Monteys.** One of the best graphic novel adaptations I’ve ever read.

**Of a Nonfiction Persuasion**

** The Chairs Are Where the People Go, by Misha Glouberman and Sheila Heti**. Glouberman is an original thinker, the kind of guy who has deeply explored how (and why) to get two dozen strangers to make silly noises together. His friend Heti elicited this book from him, and she kept all the looseness and sloppy phrasing (not in a bad way!) of the kitchen-table telling.

** Action vs. Contemplation: Why an Ancient Debate Still Matters, by Jennifer Summit and Blakey Vermeule**. A series of cultural probes into the ancient dichotomy between “action” (i.e., doing things in the world) and “contemplation” (i.e., reflecting on the world through philosophy, art, and research). There’s no overriding thesis (and some flux in the definitions of “action” and “contemplation”), but I enjoyed it as a linked collection of essays on contemporary cultural forms (from university syllabi to Pixar movies) tackling these issues of timeless import.

** Democracy in One Book or Less, by David Litt**. How the mechanics of our democracy have broken, and how to fix them. Written in rich detail and a conversational voice. More here.

** The View from Somewhere, by Lewis Raven Wallace**. A case against “objectivity” (i.e., the “View from Nowhere”) in journalism. Wallace argues that it is not so much an unattainable ideal as an incoherent one. More than just a theoretical case, he presents concrete alternatives from U.S. history, models of journalism that retain the best of objectivity (i.e., nonpartisanship and a grounding in facts) while ditching the worst (i.e., attempts at “balance”).

** Solutions and Other Problems, by Allie Brosh**. My own books live in the shadow of Brosh’s. Alongside her storytelling gifts, she has a rare knack for genuinely funny drawings. (I love the faces: embarrassed dogs, angry babies, stubborn adults.) She has also stepped up her game as an artist – the backgrounds and compositions are breathtaking. It’s a long book (at 500+ pages, with 1500+ illustrations, it amounts to a kind of reinvented graphic novel) and a great book: deeper, sadder, more unified, and more expertly crafted than her already-excellent first.

** Atlas of Poetic Zoology, by Emmanuelle Pouydebat **and

** A Mind at Play: How Claude Shannon Invented the Internet Age, by Jimmy Soni and Rob Goodman**. A compact and satisfying biography of an important (and very fun) mathematical thinker. The concepts are well-explained and well-contextualized.

** The End of Everything (Astrophysically Speaking), by Katie Mack**. A systematic explanation of five ways that the universe might end, each more horrifying and fascinating than the last. Though it’s more buttoned-down than her playful Twitter presence, Mack’s personality still shines through.

** Awakenings, by Oliver Sacks**. Haunting profiles of patients whose catatonic states were treated with the “miracle” drug L-DOPA. Only one or two case studies are wholly “good” or “bad”; the rest are complex narratives of recovery, adjustment, relapse, and compensation. Some say that no medical writer captured the whole humanity of each person better than Sacks did; I say you can delete the word “medical” from that sentence.

**And Finally, Math Books I Blurbed**

** How to Free Your Inner Mathematician, by Susan D’Agostino**. My blurb: “Think of this book as a series of eloquent postcards, sent from a wise and math-loving friend, each depicting a great mathematical idea and inviting you to join in her in the journey.”

** Supermath, by Anna Weltman**. My blurb: “From the search for aliens to the search for twin primes to the search for racial justice, Anna Weltman spins a lot of great yarns. Better yet, each tale is true: a story of how math serves good or evil, clarity or confusion, depending on the choices of the humans who wield it.”

** Mage Merlin’s Unsolved Mathematical Mysteries, by Satyan Devadoss and Matt Harvey**. My blurb: “An elegant, charming collection of 16 mathematical jewels: problems so simple they can be explained in a single page, yet so hard they can’t be solved in a single century.”

Critics of mathematics often call it “dry.” But if you ask me, they’re not going far enough. Mathematics is like Ben Stein reading an econ textbook in the Mojave desert: it exhibits several simultaneous kinds of dryness, and it’s worth unpacking what they are.

First, **cultural **dryness. Math often comes clothed in drab gray lectures and unmemorable texts. This is an artificial and undesirable dryness.

Second, **technical **dryness. Math is a highly technical discipline, and as such, tends to involve a certain level of fussy precision. Solving an equation is a kind of philosophical bookkeeping. That can be a little colorless, as careful bookkeeping is. This is a natural, neutral dryness.

Third, last, and most exciting (to me, anyway) is the dryness of **abstraction**.

As most practitioners of the subject will tell you, math has a Platonic purity. A simple equation or geometric form possesses an almost magical ability to refer to nothing in particular – and, thus, to everything.

In *How Not to Be Wrong, *Jordan Ellenberg describes math as a kind of x-ray vision. It allows you to see the conceptual skeleton of any situation, and thus to realize that totally different surfaces may disguise utterly similar structures. In my first book, I ape this insight with a metaphor of my own: mathematics is a series of Mario tubes, linking similar disparate quadrants of reality.

I think, when critics complain of math’s dryness, they mostly mean Dryness #1: the monotonous lessons of “open the book to page 127.” Sometimes, they mean Dryness #2: just as not everyone loves science or dance or U.S. history, not everyone loves the technicalities of math, and that’s okay.

But only on rare occasions do they mean Dryness #3.

This abstraction, I think, is a dryness to be celebrated and preserved. It’s a dryness that creates strange organisms – pricky pear cacti; desert pocket mice; topology – that could survive in no other climate. It’s the dryness that makes math “math.”

]]>But it’s not always true.

I’m a fan of James Surowiecki’s 2004 book *The Wisdom of Crowds. *He opens with the anecdote of the 1906 Plymouth county fair, where 800 people estimated the weight of an ox. Their guesses were all over the place – some too high, some too low. They averaged out to 1,207 pounds.

The ox’s actual weight? 1,198 pounds.

What makes for wise crowds? Not everyone needs to be knowledgeable. In fact, it’s okay if nobody knows much at all. What’s crucial is that **every person’s choice is independent of every other person’s**. When calamity strikes – and by “calamity” I mean “stock market crashes” – is when everyone in the crowd is herding around the same strategy.

By that logic, Twitter is the last place you’d look for wise crowds. What with retweets, favorites, blue check marks, and visible follower counts, Twitter users are quite sensitive to status. Herding is the name of the game.

And yet, check this out:

Not bad, right?

This flies against my prejudices about Twitter users. Also, against longstanding results in psychology. Researchers have found, again and again, that people are terrible at acting randomly.

Ask people to generate a sequence of 100 random coin flips, and they fail. Their patterns look totally nonrandom. Folks act as if a “heads” on this flip slightly raises the odds of “tails” on the next one. That’s a fallacy. (To be precise, the gambler’s fallacy.).

Evolutionary psychologists suggest that we evolved to seek patterns wherever they lurk. If that means falling for spurious patterns, then so be it. Perhaps that’s why we are so fond of the false patterns of astrology, and why we’re all convinced that our music shuffle algorithms are operating according to some secret logic, even though they aren’t.

In short, we don’t have mental dice. Our thinking is decidedly nonrandom. But again, that raises the question: how can we explain results like this?

To be fair, Twitter won’t show you the results of a poll until after you vote (or the poll closes). You are thus forced to give an independent answer. Still, I would never have trusted the wisdom of crowds in cases like these. With everyone reading the same question. I’d expect one of two results:

- Twitter users systematically think of themselves as “special,” and thus will overselect the “unlikely” option.
- Twitter users will default towards the likelier option, and thus will overselect it.

It seems strange, bordering on miraculous, that people spontaneously pursue these two strategies in just about the right proportions! It’s as if we can design and then shuffle mental decks of cards, and then report the results.

While I’m at it, here’s a related result from an old throwaway account.

The outcome (50.4% to 49.6%) is spookily close to 50/50. If instead of having 345 strangers vote, you instead flipped 345 coins, you’d get a result this close to even only about 20% of the time.

Yet, in peculiar contrast, consider the failure of the crowds to achieve this far easier task:

Here, the information is public. You just need to check the current ratio. If it’s too low, retweet. If it’s too high, favorite. And yet, as of this writing, the tweet has 405 retweets and 153 favorites, for a dismal pi approximation of about 2.65.

Forget pi; that’s not even a good approximation of e!

Somehow, making the information public – which should in theory allow for perfect coordination – led to a worse outcome. This is perhaps a parable for how Twitter works in general. There really is a surprising amount of wisdom on that social network – as long as we’re forced to be independent.

But give us the chance to herd, and we will herd ourselves off the cliff.

]]>Still, I was caught off-guard when HarperCollins mailed me this particular title:

Now, I have nothing against lapdog-sized unicorns. I also enjoy “recipes,” “literature,” and especially “lore” (though I’m lukewarm on “projects”). That said, I wondered what in my oeuvre distinguished me as the right guy to help promote *The Unicorn Handbook*.

Turns out it was mistake in the warehouse. The next day, I received the book that HarperCollins had *meant* to send:

Ah, there we are! A math book.

I’m not joking. Democracy is inevitably mathematical. It’s a process that channels the wills and wishes of more than 300 million people down into yes-or-no policy decisions. How could you attempt this, if not with some kind of math?

Of course, democracy is also personal, emotional, local – in a word, human. Citizens can get discouraged. Citizens can get confused. Citizens can have bulldogs named Mabel prowling their yards, ensuring no voter registration volunteer comes close.

And this is a book that understands both.

David has been a speechwriter for President Obama. He has been my friend (and one of the best, funniest writers I know) since before Obama got to him. He has even, while trying to register new voters, been chased by a bulldog named Mabel. And now he has written a book that is sensitive to both sides of democracy.

Math and emotion.

Disenfranchisement and disillusionment.

Structure and culture.

In the book’s first section, David divides the country into two halves: the **electorate** (who vote) and the **unelectorate** (who don’t). This gets mathematical fast. Which group is bigger? (The unelectorate.) How much bigger? (Depends on the election.) Is the electorate representative of the unelectorate? (Not even close.) By what mechanisms are voters excluded from the electorate? (Many, from the legal to the logistical to the psychological.)

The book’s second section, on congressional representation, is even more mathematical. This is where, I’m proud to say, David brought me in to draw a few cartoons. I’m humbled to have my stick figures flanking such important words.

And the final section, on the gears of government in Washington (from the legislature to the courts to the omnipresent lobbyists) is mathematical in its way, too. It reminds me a bit of the old image depicting “teachers,” “students,” and “parents” as three gears in the machine of a school. They’re supposed to work together. But in the design shown, the gears shown would be hopelessly gridlocked, unable to turn.

(A discouraging image of the three branches of government? Maybe so. Although mathematical problems, I like to believe, have mathematical solutions.)

The book also works as a travelogue. David ran with voter registration outlaws in Texas, staked out Mitch McConnell’s old frat house in Kentucky, and took a commiserating stroll with Shelly Simonds (who lost a Virginia legislature race on a coin flip.) These adventures provide the necessary complement to mathematics: a glimpse of the face-to-face, human side of democracy.

It may feel like a strange moment to meditate on the gerrymander and the filibuster. I mean, it’s 2020: the TV show of our lives has a terribly overstuffed plot, and keeps flitting from genre to genre. Who’s got time to meditate on the structure of our democracy?

We do, I think. We have to.

Jorge Luis Borges, in a moment of cynicism, described democracy as “an abuse of statistics.” But that’s not quite fair. Yes, there’s democracy as practiced today, tragically full of abuses. But then there’s democracy as a process, the messy and glorious process by which a society lurches toward something more perfect.

Anyway, the book comes out today. I admire what David has done as a writer: to the questions that frame our society, he brings a light touch, a deep acumen, and a hopeful heart. Here in 2020, that’s a gift as rare and precious as a lap-sized unicorn.

]]>You know what we need? A brain break. A quick game, from @_b_p. So here we are:

The title, you’ll notice, is missing. *Something* is growing over time, with an impressive boom around 1980, but we don’t know *what*.

Is it technological? Cultural? Demographic? *Star Wars*-related?

Think it over. Note the trends. Make a guess.

And then, when you’re ready, here’s the reveal.

Okay, that’s an unhappy graph. My apologies.

And of course, as you lovable pedants may note, it doesn’t tell the whole story. What else changed in the U.S. during this time? What if we adjust for population? How does trend compare to other countries?

So let’s try another game, from Connie Rivera. This one unfolds a bit more slowly.

What might the bars represent? Number of monkeys petted? Price required to pet various kinds of monkeys? Calories expended in petting various kinds of monkeys?

Well, here’s your next clue:

Okay, so monkeys are maybe not the secret here.

What do the U.S., Rwanda, and Russia have in common? Yes, yes, a shared love of the TV show *Friends*. But that applies to all countries. Why would India and Sweden be so low? What’s going on here?

Another quick clue. It’s there at the bottom, if you don’t see it on first glance.

Hmm. For the 1.5-billion person nation of India, a total of just 33.

But 33 *what*?

Something is odd here, if a much smaller country like Spain or Canada can punch so far above its weight class.

Anyway, a good game within this game: can you eyeball the values of the other bars? If India is at 33, what’s Germany at?

Brazil?

The U.S.?

Ready or not, here they are:

Wow! The U.S. is *crushing* India in this game, whatever it is. We’re beating them by a factor of 20. Take that, India!

And the game is…

Drat. Another game we didn’t want to win.

These images come from Slow Reveal Graphs, a site run by leading elementary math educator Jenna Laib. She’s got dozens of these, organized as slideshows, ready to be employed in classrooms.

(Full disclosure: Jenna is my sister.)

(Even fuller disclosure: Jenna is a champion.)

The logic behind this exercise, as I understand it, is simple and powerful. Graphs tell stories. But stories unfold in time, whereas graphs just splatter you in the face, with all of the information at once.

So, hold back part of the story. Leave the reader in suspense. Let them notice, wonder, ask, speculate.

Then, and only then, deliver the full truth.

To read a graph requires a host of skills, from specific technical matters (where are the axes? how is quantity represented?) to broader, softer virtues (patience, attention, a sense of context). Slow Reveal Graphs help students build those skills.

And they’re fun, too.

I’ve seen Jenna run this instructional routine, and it’s magic. (I’ve run it myself, too – less magic, but still a blast.) Students have sharp eyes. They’ll catch things you missed, interpret features in ways you would never have guessed. They’ll build on each other, quibble with each other, learn from each other.

Perhaps best of all, *there’s no shame in changing your mind*.

Every kid does it, and it happens naturally. They predict. They watch new information come in. And then they update their predictions.

We humans are usually such stubborn and prideful creatures, clinging to our views long after they’ve melted into mud. But with Slow Reveal Graphs, suddenly we become astute Bayesians, updating our priors on the regular.

I encourage every teacher to check out the Slow Reveal Graphs site. Jenna has curated an admirable resource. (I’ve chosen two that are heavy as a bag of flour, but some are silly and fun!) And if right now isn’t a good time to bring a dose of truth to math lessons, then I sure don’t know when that time will come.

(Also: happy early birthday, Jenna!)

]]>It is a book, I am pleased to say, about games. Mathematical games. Strategy games. Easy-to-learn, fun-to-play, hard-to-master games.

Some will be timeless classics.

Others will be fresh-faced originals.

And others will be glowing gems mined from the sooty depths of the gaming world.

But here’s the thing: Games need play-testers. And here’s the second thing: play-testers are hard to come by during a global pandemic. And here’s the third thing, which unlike the first two, is a question.

How would you like to help me play-test these games?

I’ll be sending out an illustrated rule set every Sunday, along with a Google Form for offering feedback. I ask only for: (a) Your email address; (b) Your unblinking honesty; and (c) Your not sharing the documents (other than as needed for play-testing).

Sign up for the email list here. I’d love to have you join me; you’ll both get a sneak preview of the book, and help it come to fruition.

While I’m at it, here are some frequently asked questions:

**I’m a teacher. Can I play these games with my students?**

Yes!

**I’m a parent. Can I play these games with my kids?**

Yes!

**I’m a soldier stationed in a nuclear silo, guarding a warhead, alongside only one other human being (with whom I am falling helplessly in love). Can I play these games with them?**

No. Go back to warhead-guarding.

**Really?**

Aww, who am I kidding? Play away. And good luck with the romance!

**What if I don’t have anyone at home to play with?**

The games should mostly work over Zoom/Skype/Hangouts/shouting to your neighbor across the street.

If you need a partner, reply to the first email to let me know you’d like to be matched, or leave a comment below.

**What makes these “math” games?**

I have defined “math game” as follows: *A game whose players are wont to remark, “Hmm, this game feels mathy.”*

For the record, this won’t be a book on combinatorial game theory. That book already exists: it was written by three eminent mathematicians over the course of a decade, and is called *Winning Ways for Your Mathematical Plays*. I give it five stars!

**Hey, I have a game you should consider!**

I’m delighted to hear it! Email it to me; I’m just the name of the blog at gmail.

Note that I’m looking for games which are easy to play with stuff at home. That means paper and pencil. Maybe dice. Perhaps a checker set, if you’re feeling fancy.

**What are you hoping to get from your play-testers?**

In no particular order:

- A sense of where the rules need clarifying and/or tweaking.
- A sense of camaraderie.
- A sense of which games are the most worthy of inclusion in the book.
- A sense of joie de vivre.
- A sense of how each game might look to a math-averse person.
- A sense of how each game might look to a mathematical expert.
- A sense of what “joie de vivre” actually means.
- A sense of this book being born not from a monologue (as my last two books were) but as part of a big, loopy, intellectually playful dialogue.

**What mathematical background should I have?**

I’m aiming for “a minute to learn, a lifetime to master” sorts of games. So, if I’m doing my job right, the games will be learnable by a 10-year-old, and still serve as a decent intellectual chew-toy for a 10th-year PhD student.

Also, let’s be honest: if you’re a 10th-year PhD student, then you must be both (a) easily distracted and (b) weirdly persistent. So this play-testing gig is perfect for you.

Long story short, I prize all feedback!

**I’m still on the fence.**

Sign up, my dude! Here’s the link again, for those too lazy to scroll up.

(No shame. I feel you, my lazy friends. I feel you.)

]]>Nowhere in all my research have I come across a mind quite like that of Walter Joris.

Walter generates games, puzzles, and pencil-and-paper experiments with such intensity and regularity that he must be a kind of pulsar: some heretofore unknown astronomical object, emitting what I admiringly call Joris Radiation.

You and I see right angles. Walter sees a doodle game.

You and I see paper. Walter sees the Incredible Paperman.

You and I see a cube. Walter sees… well, to be honest, I don’t know *what* Walter sees, but I can’t help wanting to see it too.

His book *100 Strategic Games for Pen and Paper* is the most bizarre and marvelous thing I’ve read this year. “Nearly all the games have been invented by me,” he writes in the introduction, and it’s true: his fingerprints are on every page.

From those hundred, I picked out half a dozen to share here. Each is for two players; each requires only pens and paper; and each has surprising strategic depths to plumb.

May the wondrous light of Joris Radiation shine upon you in these strange times!

(*See the bottom of the post for an interview with Walter.*)

Magical Squares

In an actual magic square, every row, column, and diagonal has the same sum. In Walter’s game, you won’t achieve that, but the goal is to get as close as possible.

Each player begins with a blank square, then secretly places numbers in the four corners. You may use whatever numbers you like (including repeats).

Then, you reveal your squares to each other. Now, whatever numbers your opponent put in her *corners*, you must put in your *edges (*in whatever order you like).

Finally, you can choose whatever number you like for the center.

The goal: have as many rows, columns, and diagonals as possible sharing the same sum.

A pair with the same sum scores 1 point; a trio scores 2 points; a quartet scores 3 points; and so on. (This sample game ended in a 3-3 tie.)

What happens if you start with identical numbers in your corners (e.g., 7, 7, 7, 7)? What if you pick radically different numbers (1, 10, 100, 1 million)? Is there a best strategy? If so, is it deterministic or probabilistic? The field is open for exploration!

This is, in one sense, a standard game of territory control. It’s like dozens of others I’ve encountered. Yet I can’t get it out of my head. It’s something about those juicy grapes, the silly theme, and the lovely drawings that result.

First, draw a bunch of grapes. Make it clear which grapes share a border.

Then, by turns, each player picks a grape on which their “fly” begins, and marks it with a colored dot.

Then, take turns moving. (Whoever placed their fly second should begin.)

On each move, your fly consumes the grape it’s on (shown by fully coloring in the grape), then moves to an adjacent grape.

Whoever winds up unable to move, because there are no adjacent grapes available, is the loser. Here, three more moves have been made:

The strategy seems straightforward, but the grapes can trick the eye, lending an element of suspense. (You may have less territory left than you think!) Also, whereas most pencil-and-paper games leave the paper coated with crisscrossing gibberish, this one ends up like a page from a coloring book.

Here, purple wins! (Green fly made some bad life choices.)

Ideally played while snacking on grapes.

This has more the flavor of a puzzle than a game. It’s a puzzle I’ve yet to solve.

You begin by drawing a pyramid of 21 circles. Draw six on the bottom row, five on top of that, four on top of that, and so on.

Then, take turns writing a 1 in a circle of your choice.

After that, take turns writing 2, 3, and so on, in order. (You must write your numbers in order; no skipping ahead.)

When you have each written your 10, there will be one circle left blank: the black hole.

The black hole destroys all its neighboring circles. Whoever has a greater sum of numbers left over – that is, whoever loses a smaller sum to the black hole – is the winner.

If this speedy game is still too slow for your overheated 21st-century attention span, then you can eliminate the bottom row, and play using just 15 circles, where each player writes the numbers from 1 to 7.

(Or if you want to kill an extra ten minutes, add three more rows, for a total of 45 squares, so that each player writes the numbers from 1 to 22.)

Anyway, after a few rounds, I still have very few strategic intuitions about this game, but I love the simplicity of the design.

You begin with a 6 by 6 grid. Players take turns filling in pairs of adjacent squares, as if covering them with a domino.

These covered squares belong to nobody. Rather, you are fighting for control of the *other* squares, which you claim **when your domino completes a fence closing off a region with odd number of squares**.

Close off 2, 4, 6, or 8 squares? That’s useless. Close off 1, 3, 5, 7, or 9? You can claim them. (More than that, though, and it doesn’t count.)

The winner is whoever claims more squares by the end.

Some games, like “Bunch of Grapes” and “Black Hole,” feel so simple that they must have existed all along, as if their designer merely “discovered” them. Other games, like this one, are so quirky that they can only have been invented.

Or perhaps I should say “bred.” To my eye, this game has genetic traces of Nim, Cram, Dots and Boxes, and more.

Many pencil-and-paper games feel airy and abstract. Why dots? Why boxes? Why tics, tacs, and toes? That’s why I love the cute and highly literal theme of this one; the game resides right there in the name.

Snake fight!

Begin with a 5-by-5 array of dots, and draw in the outline. The players begin their snakes in opposite corners.

The goal is to cross the enemy snake as many times as possible.

You take turns extending your snake via vertical, horizontal, or diagonal lines. Your snake can never cross or touch itself, and cannot trace over a segment that has already been drawn (by you, your opponent, or the border).

The game continues until neither player can move. Make sure to keep score as you go; otherwise, you won’t be able to tell who crossed whom!

(Fine print: Passing through the enemy’s “head” counts as a crossing. So does the last move above for orange, where the head reaches the enemy, but doesn’t pass through.)

I’ve saved for last the game that is perhaps the simplest – not to mention the deepest.

Begin with a 6 by 6 grid. On each turn, you mark any box you like, but you must also eliminate an empty neighboring box.

Eliminating a diagonal neighbor is allowed.

The winner is whoever creates the largest group of connected marks. (Diagonal connections count.)

Play until no more moves are possible.

I have no idea why this one is called “Collector.” Why not “Connector”? Or “Barrier”? No matter: a pleasant mystification is a natural byproduct of Joris Radiation.

The gameplay here reminds me of Amazons, a classic 1988 territorial game in which each move involves the annihilation of a square. This creates a “the world is falling away!” flavor of drama.

**How did you develop an interest in designing games?**

I always liked board games. And in my youth, there was a kind of a culture in pen paper games, long before computers were there. And then there was Martin Gardner and his math puzzles, which fascinated me.

**How do you go about designing a game? What’s the process like?**

The first ones are the hardest. But once you give your mind the task: “invent games”, it obeys. It starts to be creative. And then, whenever there is a kind of inspiration, a special pattern you see, a combination, your creativity will turn it into a game.

**Where do you farm for ideas? As in: what board games do you play? What books do you read?**

Well, mathematical puzzles of course, but also puzzles and pastimes for children. Also: existing board games in the world, and there is a lot of them. I myself play Go. And reading, I’ve read an enormous amount of books. Must be thousands. All kind of genres, philosophy, science fiction, novels, also the classical ones, from Dostoevsky to Rimbaud. Strange books likes the ones of Madame Blavatsky. Dali. Marinetti, Dada…Popular science also, but mostly in magazines. Art, history, cultural history…. I speak Dutch, French, English, German, and a bit of Spanish, In the first 4 languages I can read books rather easily. So, living in Belgium, in the middle of those cultures, you can imagine what an enormous amount of books and rare books you can find. With the fast train, you are from Brussels in about 1 hour in Paris, 1,5 in London, Cologne in Germany. Where I live, I can go by tramway from the Netherlands to France.

**Do you think of your games as fundamentally mathematical?**

Yes, I think of all games as fundamentally mathematical. I call myself a “matheist.”

**Lots of games in your book are adapted from board games. How do you decide if a game is suitable for adaptation?**

It has to be able to become a pen and paper game. And that is fundamentally different. In material board games, it’s all about the empty spaces; in a pen and paper game, about the occupied ones.

**What are your favorite games (not of your own making)?**

I played them all once. Now, I’ve limited myself to only Go.

**What are your favorite games (of your own making)?**

Since I invented them, I like them all. But Sequentium is definitely my favorite.

Thanks to Walter for answering my questions. You can find his work on Facebook, at Deviant Art, and of course in his book.

And at his request, here’s Walter’s explanation of his crown jewel, Sequencium:

]]>Gowers, a 1998 Fields medalist, has done breakthrough work in combinatorics. Dude’s a Royal Society Research Professor at University of Cambridge. His to-do list is no doubt a catalog of deep and important mathematical questions. So what is this meta-problem nagging at him?

“HOW DOES CATRIONA SHEARER DO IT???”

Catriona Shearer is a math teacher whose Twitter account features homemade geometry puzzles. But “puzzles” perhaps undersells them. These are puzzles that entice and entrance mathematicians of every stripe.

Puzzles that elicit caps-lock, triple-punctuated expressions of wonder.

“I don’t think I’ve ever seen anyone in my entire life,” says the mathematician Mike Lawler, “who has an eye for neat geometry problems like Catriona Shearer does.”

“These problems can’t just pop into her head,” insists Gowers. “Does she have a general theory? Or a nice bag of tricks? Or what?”

“Yesterday I read a tweet of hers,” chimed in John Carlos Baez, a leading category theorist, “where she said she’s not as creative as some people seem to think: she keeps using the same tricks over and over again.

“This,” Baez noted, “is also what Feynman said.”

In reply, Catriona shared a video snippet of her notebook. It was another tantalizing glimpse of her fertile thought processes:

Anyway, Catriona continues to decline my offers to put her in touch with publishers, but kindly picks out favorite puzzles to share here.

So, for your pleasure: eleven delightful excursions into geometry.

Transit Across a Purple Sun

“Easily my most popular tweet ever, this one,” says Catriona. “There are lots of very nice replies, but I particularly like this animation where the various possibilities seem to flow around the one fixed value.”

“This is a bit of a trick question,” says Catriona. Spoiler alert: “One of the lengths is a red herring – you only need [redacted] to be able to answer. It was actually based on an earlier puzzle that you featured in your first collection.”

“There are lots of ways to approach this,” says Catriona. “One of my favorites – which I would never have come up with myself – is this one, where the top part of puzzle is tessellated to create a [spoiler redacted].”

Also, please note: my solutions to Catriona’s puzzles are uniformly plodding, and usually devolve into calculation at the end. The ones on Twitter are always glorious, Olympic-gymnastic-level feats of symmetry. So it is.

“This one is a bit of a hangover from all those semicircles-within-rectangles puzzles I made back in December!” says Catriona.

Hex Hex Six

“Maths is a unique(?) international language,” commented one fan on this one. “See the word replies not in English, but the mathematical solutions are totally understandable.”

“My favourite thing about twitter,” Catriona agreed, “is being able to do maths with people all over the world.”

Four, Three, Two

“I sat down next to somebody at a training event in Birmingham,” says Catriona, “who recognised my name from twitter and proceeded to tell me his favourite geometry fact: the inscribed circle in a right-angled triangle with integer sides has an integer radius. We spent some of the quieter moments of the day trying to figure out all the patterns, and I made this puzzle on the train home.”

Funny coincidence! When I taught in Birmingham, one of our spectacular students taught this fact to me and other faculty.

The Trinity Quartet

“This one,” Catriona says, “is actually featured in Alex Bellos’s latest book, which I think is very cool. The design reminds me a lot of a stained-glass window in a chapel. My favourite solution was this one from Mike Lawler’s son, just because (unusual for social media) we get to see the entire thought process, including the bits where he gets stuck.”

(That is, in general, one of the delightful things about Mike’s blog; it’s a unique document of mathematical learning and teaching.)

The Falling Domino

“This one inspired a nice spin-off puzzle from Vincent Pantaloni of *Geometry Snack* fame,” says Catriona. “I mainly like it because it’s not often that you spot

Slices in a Sector

“The 13 isn’t a coincidence!” says Catriona, adding: “I liked this approach to the solution.”

Disorientation

“Less than a month after I posted this,” says Catriona, “I accidentally re-derived it. I thought I’d made a new puzzle until I noticed that the numbers were familiar, and realized it was exactly the same set-up, just in a different orientation!”

Sunny Smile Up

“Because,” says Catriona, “everyone needs a smile at the moment.”

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I find it troubling that the golden ratio has so little in common with the golden rule.

Like, if you did unto others 1.618 times what you’d have them do unto you, then we’d all wind up exhausted.

And if you’re only doing 1/1.618 times unto them, then isn’t that a bit lazy?

I’ve always enjoyed those puzzles like, “If 3 chickens can lay 3 eggs in 3 days, then how long will it take 100 chickens to lay 100 eggs?” They’re counter-intuitive (e.g., in my example, each chicken lays 1 egg per 3 days, so the answer is also 3 days), yet deal only with simple constant rates.

So what if the rates weren’t constant? Like in, say, a bureaucracy, where 20 times more people will accomplish only 1/20th as much?

(Sorry for putting the answer upside down. It reads: “Please complete the attached form (Z302: Aggregate Task Completion Rate Information Request) and we’ll process your inquiry in 4-6 weeks.”)

In this case, “a mathematician” refers specifically to Matt Parker, whose excellent book Humble Pi discusses the first two of these mistakes.

Lots of poets have found asymptotes a convenient literary symbol – the idea of eternal striving is a resonant one (even beyond the eternal striving of the struggling algebra student).

I love me some Raymond Smullyan.

Sorry again for putting the answer upside down. I dunno why I thought that was a clever idea. Mostly just forces you to turn off the auto-rotate setting on your phone.

Anyway, it reads: “Ask anything. You should already know not to buy lowfat yogurt.”)

Not that I’ve ever felt this myself. I’m just speculating.

What is parenting, if not a neat LARP?

(LARP = Live-Action Role-Playing Game, for those of you with less geeky acumen than I anticipate my audience to have.)

By the way, my friend Rayleen once described to me a brilliant comic, where one person asks, “When’s the baby due?” and the other person is drawn with a small horizontal stick figure emerging from their stick torso. (See? It’s such a good comic, I can just *describe* it.)

I think a lot about the different arguments for math, and the ways that they support or contradict each other. Is it a beautiful art? An urgent set of universal civic skills? Key preparation for technical professions?

The answer is yes to all three. But not for all math, and not all at once – and attempting to blend the purposes can lead to a muddle.

It’s always tickled me that the mathematician’s verb “let,” which sounds so chill and laissez-faire, is actually a binding command.

Every Unhappy Family is Unhappy In Its Own Way”

I wrote a bunch of these a few years ago. This one has the benefit of being true: all circles are geometrically similar, but not all ellipses are.

(The same is true, by the way, of parabolas and hyperbolas. The former are all the same basic shape, just zoomed in or zoomed out, whereas the latter constitute a whole family of different shapes.)

(Chew on that, Tolstoy.)

I wrote this little dialogue after listening to a great episode of The Allusionist, before it turned out that *Game of Thrones* would suffer the worst collapse in storytelling that I have ever experienced.

Oh well!

As my wife said, “At least this way we’ll never have to bargain with our daughter about when she’s old enough to watch *Game of Thrones*. The ending is so bad, in 10 or 15 years no one will be watching it anymore.”

This is a really dumb pun.

Also one of the more popular cartoons in this list.

Go figure.

This one is inspired by that time Malcolm Gladwell referred to eigenvectors as “igon vectors,” and Steven Pinker blasted him for it, at which point Gladwell blasted Pinker for something else, and eventually we all lost the thread and just went about our days.

And if you want more godawful matrix puns, I’ve got ’em.

I don’t know what day you’re reading this, but guess what? It’s also a bad approximation of pi! So go ahead and celebrate!

(Though if you want some very clever alternative pi days, check out Evelyn Lamb’s page-a-day calendar, which includes a Pi Day each month, and not where you’d expect!)

After I posted this, there was a bunch of discussion on Twitter about whether I’d mischaracterized the Axiom of Choice, which is totally possible, in which case, oops.

Also, some folks pointed out that it’s pretty greedy to wish for uncountably many wishes, when you could just as easily wish for countably many.

To which I say: What’s the point of a magic lamp, if not to have greed be your undoing?

For lots of optimization problems, maximizing makes sense, but minimizing doesn’t. (Or vice versa.) An example: What’s the largest rectangle you can make from 4 feet of wire?

It’s the 1-by-1 square, with an area of 1 square foot.

But what’s the *smallest* rectangle you can make (in terms of area)? Well, you could make the 1.9999 by 0.0001 rectangle, which has a very tiny area…

Or you could make the 1.999999 by 0.000001 rectangle, which has an even smaller area…

Or the 1.99999999999999 by 0.000000000000001 rectangle, whose area is microscopic…

…and so on.

I hope that was worth it! And I suspect it wasn’t! Anyway, moving on.

More thoughts here.

Clearly this villain should be assigning more group work.

Anyway, I for one am curious to know how a complex-valued currency might work. I’d pay a hefty fee for an accountant or tax attorney who can turn imaginary assets into real ones, or real debts into imaginary ones.

I found it very hard to draw a decent space-filling curve.

Also, to draw a decent cat.

This is how I feel about anyone who sleeps less than 7 hours in a given night.

This is my version of that xkcd about kitties.

Also pretty well summarizes parenthood. I still enjoy a cerebral geek-out, as I always have; but I also really enjoy holding my daughter in my arms and calling her the world’s best monkey over and over.

I would totally read a graphic novel about the dating life of Georg Cantor.

The problem is that no one is going to write this graphic novel except for me.

Oh well. I’m under contract for two more books at the moment, but after that will come TRANSFINITE LOVE: THE ROMANTIC ESCAPADES OF A SET THEORIST.

Drawn from an actual experience, in my first week teaching 7th grade. I hadn’t really figured out how to tee up a problem-solving experience yet.

Drew this one for a Jim Propp essay. Recommended as always!

A teaching friend of mine had a whole list of proofs that 1 = 0, which he busted out at various developmentally appropriate points in grades 6 through 12.

I love that. Curious how far you could get writing a book of proofs that 1 = 0, each introducing a key idea in mathematics…

Maybe that’ll be my next project after the George Cantor romance novel.

Philosophical question: Is this a pun?

The case against: “A pun is a joke that plays on words that sound similar but mean different things. This isn’t doing that.”

The case for: “A pun is a joke that plays on linguistic expressions with similar surface features, but different deep meanings. This is doing exactly that: the premise of the joke is that an exponent and a footnote are both denoted with a superscript, yet mean very different things.”

So I guess this has a deep resemblance to puns, but lacks a surface resemblance… which is itself, not very pun-like.

Ruling: Not a pun!

I guess you hear this inane phrase less often these days. But there was a time, kiddos, when people could hear a devastating counterexample to what they were arguing, and then blithely say “the exception proves the rule” with a straight face.

I’m pretty agnostic on the math sequence. But I have strong intuitions that Star Wars should be screened in the order: IV, V, I, II, III, VI, and so on. (I view the sequels as pretty optional. Prequels too, for that matter, but if you limit yourself to the original trilogy, it’s a boring problem.)

A lot of people on Facebook seemed to read this as though the right-hand character was creeping on Ariana Grande. Not my intention at all! I just wanted to pick a mid-20s celebrity. Could’ve just as easily been Bieber.

(My primary association with Ariana Grande, by the way, is her performance in the short-lived bar mitzvah-themed Broadway musical Thirteen.)

I’m not sure there’s a joke here.

I’m fond of this drawing anyway.

Michael Pershan, the internet’s most relentlessly analytical math educator, inexplicably loved this joke, so I call it a win.

Someone on social media speculated about the position by which this linear combination had been “conceived,” which I found quite vulgar and upsetting (but which I also sort of invited by drawing a comic about procreating vectors).

Where do we draw the line between logical succession, and outright stalking? I leave that to the courts.

Sometimes I just want to do a cute drawing that has no joke in it, okay?

I’m actually skeptical that the phrase “vertical line test” has any value. To me it feels like a fancy name for a fact that doesn’t need a fancy name. And, as in the two-column-proof version of geometry, giving fancy names to facts that students should be reasoning out for themselves can become obfuscatory rather than clarifying.

Please join me in making “Patricia gasket” a thing! E.g., “Did you know Copley Square in Boston is the approximate shape of the mathematical figure known as a Patricia Gasket?”

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