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.”

]]>

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?”

]]>Anyway, Celeste Ng is also the parent of a future math teacher.

I, for one, warmly welcome our 9-year-old hero to the time-honored profession of foisting equations upon the reluctant.

As a kid, I made similar efforts to stump my dad. Mine were mile-long arithmetic problems: “13406824360 times 78645103465” and the like. Such multiplication is certainly *hard.* But it’s also tedious and shallow (as my dad’s taxed patience can attest).

These equations, in contrast, run deep. Deeper than their inventor could have known.

That’s the way it is with math. When you put a shovel in the dirt, sometimes you get a shovelful of dirt; but sometimes, a 200-foot geyser springs forth. Beneath the innocuous patch of surface you’re exploring, there may lurk terrific geological forces.

This is one of those times. (Beware: spoilers for the problem follow.)

Our system involves four variables: **f**, **a**, **c**, and **e **(which pleasingly spell “face”). Before we solve algebraically, it’s worth investigating the matter geometrically.

Here’s the relationship between **a** and **e**:

Between **a** and **f**:

Between **f** and **c**:

And between **a** and **c **(which I derive from the final equation by replacing **e** with **11 – a**, an equivalence established by the first equation):

Since each equation relates just two variables, it can be visualized in two dimensions. The results are two lines, a parabola, and a hyperbola.

But what if we want to combine them all? With four variables in play, we need four dimensions. I don’t want to speak for you, but when my brain tries to visualize four dimensions, it mostly achieves something like this:

In context, each 2-variable relationship becomes a 3-dimensional solid, living in an ambient 4-dimensional space. The solution we seek is a point that belongs to all of the solids simultaneously.

Pretty heavy dinner-table conversation for a 9-year-old.

Thwarted by geometry, we turn to more algebraic methods. What do you do with 4 equations involving 4 unknowns?

Turn it into 3 equations involving 3 unknowns.

(In this case, I used the substitution mentioned above, then factored the left-hand-side.)

And what do you do from there? Why eliminate another variable, of course, boiling it down to 2 equations with 2 unknowns.

(In this case, I added the first and second equations from before.)

And from there, you can guess our next move: dessert.

Then, with our sweet tooth sated: boil our pair of equations down to a single one, in a single unknown.

(In this case, I solved the second equation for c, then substituted into the first equation.)

The result is a cubic. Potentially a nasty piece of work. Consult this table:

In this case, fortunately, our 9-year-old teacher has thrown us a bone. One of the solutions to this cubic can be guessed without too much trouble: **a = 4**.

(Ben Dickman, my arch-rival in being a mathy Ben on the internet, has a nice explanation of why, if you’re curious and also a traitor who is forever dead to me.)

Knowing this solution, we can factor the cubic into a linear and a quadratic:

Then, pick your favorite quadratic-solving method, and voila:

All of these solutions check out, giving us a total of three solutions to the original system of equations:

Back to the geometry: apparently those 3-dimensional surfaces, crisscrossing in 4-dimensional space, meet at exactly three points. To higher-dimensional aliens, this would perhaps be as obvious as noting that two lines cross at a single point. Or that a line and a parabola at a pair of points. It’d be elementary geometry, almost too trivial to speak of. But despite my siblings’ insistence, I am not an alien, and cannot see such things.

(Though if we retreat back to the step where we had two equations in two unknowns, Desmos can show us why three solutions emerge: the hyperbola and parabola intersect in three places.)

This is, if I may reiterate, some hefty dinner-table chitchat. Even my wife and I (together comprising one math professor, one author of math books, and two incurable nerds) don’t delve this deep on your typical weeknight. Hats off to the Ng household.

Anyway, this all got me thinking about the vast pointlessness of algebra.

I know I’m not supposed to say algebra is pointless. But I can’t help it. The other day, as we moved into factoring quadratics, my Algebra I students raised the timeless chorus: “Why do we have to learn this?”

I had no good reply. I never do.

It’s not for lack of thought. I’ve spent years meditating on this very question, which is exactly why I find the standard answers so empty and unsatisfying.

If algebra class is to be of daily use to the citizenry, why not focus on probability, statistical literacy, and personal finance?

If algebra class to impart valuable professional skills, why not focus on spreadsheets (plus maybe a little Python)?

If algebra class is to teach us “how to think” (which was once my preferred answer), then what do we make of research revealing that lessons in reasoning don’t generalize? That algebra teaches us to think, not about life or logic as a whole, but about equations only?

Finally, if algebra class is simply a platform upon which to compete for college admissions… then please excuse me while I scream into the void.

So. Why *do* I teach this stuff?

Here’s one reason. It’s an idiosyncratic reason, inspired by the exploits of the Ng household, offered with the caveat that your mileage may vary. It goes like this.

Arithmetic raises questions that it cannot answer.

Arithmetic poses riddles that it cannot solve.

Arithmetic uncovers patterns that it cannot explain.

But algebra can.

Arithmetic is the solid ground, the numerical surface of things. You figure it must be dirt all the way down. But if you stick your shovel in the right spot, as the Ng family did, you’ll be greeted with a geyser.

How can you not love those eruptions? How can you not want to know their source? How can you resist the urge to probe the depths?

So, three cheers for algebra, and for all the 9-year-olds at all the dinner tables tossing off questions so profound that, even on a well-caffeinated day, a half an answer is the very best I can muster.

*A NOTE ON THE TITLE OF THIS POST: Apparently the TV show is spelled “Kids Say the Darndest Things,” with only one “e” in “Darndest.” This is an abomination and I will have no part of it.*

EDIT: A cool animation from Adam in the comments, showing what the graphical approach looks like with three equations.

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The question: **What’s this equivalent to, for a typical person like you or me?**

As inevitably happens during brief radio interviews, I got a bit fumbly and mumbly, so to do greater justice to their excellent question, here are three different answers:

1. **As a percentage of wealth:**

Bloomberg spent a little under 1% of his wealth on the campaign. For a typical U.S. household, that amounts to perhaps $800. In this sense, Bloomberg’s campaign was the equivalent of, as Peter put it, buying a new laptop.

2. **As a sustainable daily expenditure:**

Bloomberg spent about $5 million per day on ads. If he cashed out his wealth, he could continue to do this, day after day, for another 35 years.

The proportional daily expenditure, for a typical U.S. household, would be $8 per day. In this sense, Bloomberg’s campaign was the equivalent of a daily sandwich.

3. **As an amount of labor:**

If I spend $800, I need to go earn $800 more, sooner or later, in order to keep financially afloat. That’s what $800 means to me, really: it’s the work I’ll need to do to earn it back.

The same isn’t true of Michael Bloomberg. On an ordinary day, he gains a passive income greater than my lifetime earnings will likely be. (Indeed, he ended his presidential campaign several billion dollars richer than he began it.)

Thus, Bloomberg does not need to go earn $500 million. The $500 million come to him. In this sense, the most closely analogous price in your or my experience is “free.”

*You can hear the interview at Here & Now. Thanks again to Peter and the team for having me on!*

- A mathematical topic arises.
- “You know,” I say, “someone has a great tweet about this… somewhere…”
- In order to find it, I am forced to read all of the tweets, ever.
- I am reminded that “all of the tweets ever” is rather too many tweets.

So about a year ago I started a compendium. Tweets, yes, but also videos, apps, memes… anything stimulating or arresting that I can use to embroider my lessons. For a while, this document lived where all important documents live: as a gmail draft. But now I share it as a blog post, and I intend to continue updating it as new ones cross my ken.

NOTE: I will, where convenient, use screenshots and links, because WordPress’s embedded tweets sometimes take ages to load.

A very strange pricing scheme:

A brilliant anagram from Colin Beveridge:

A gorgeous visualization of prime factors (from this Smithsonian blog post).

The timeless classic Powers of Ten, arguably the best film of 1977 (suck it, Annie Hall):

The mesmerizing interactive “Scale of the Universe” app (which requires you to enable Flash, but just do it).

Also, this black hole:

Four-story slides shaped like parabolas:

An ellipse as the maximum heights of a family of projectiles:

Throwing an object at the same speed but different angles defines an ellipse via its maximum height https://t.co/vQ8NMssCMf

—

〈 Berger | Dillon 〉 (@InertialObservr) July 22, 2019

And again, this time for figuring out the scoring system in Australian Rules Football:

Four place mats, arranged to make a quadratic identity at the dinner table:

Polar coordinates on pizza:

Putting sauce on a pizza. https://t.co/Oe9gsZaSjz

—

Machine Pix (@MachinePix) August 28, 2016

Volumes of a cylinder, a sphere, and a cone:

Volumes of earth, earth’s air, and earth’s water:

Animated visual proof that any polygon can be rearranged into any other polygon of equal area:

(You’ve just got to click here, it’s amazing.)

For your trigonometric Halloween, the blood function:

Defining a radian with a wooden model:

Tragic Tweet Delete! -- I thought I would at least add it back : ) We are interested in sending these to folks, es… twitter.com/i/web/status/1…

—

MathHappens (@MathHappensOrg) October 01, 2019

Simple harmonic motion:

Beautiful shapes created by simple harmonic motion 🧐 https://t.co/ifsFX4nfN9

—

Fermat's Library (@fermatslibrary) January 02, 2020

Riemann sums (comparing upper and lower sums as the grid is refined):

Concepts without words: Integration and Riemann Sums
bit.ly/2E7iNU3
#math #science #iteachmath #mtbos… twitter.com/i/web/status/1…

—

Tungsteno (@74WTungsteno) August 11, 2019

A professor solves an optimization problem (“smallest surface area for a given volume”), writes a company that makes cat food to ask why they don’t use this solution, and receives an incredibly thoughtful and interesting reply:

A real-life butterfly effect:

In office hours, sophomore @JackSillin showed me this real world example of the butterfly effect. An unexpectedly… twitter.com/i/web/status/1…

—

Steven Strogatz (@stevenstrogatz) September 10, 2019

The exquisite sensitivity of the double pendulum:

50 double pendulums, whose initial velocities differ only by 1 part in 1000 https://t.co/3b75BDkwF1

—

〈 Berger | Dillon 〉 (@InertialObservr) September 30, 2019

Independence is a delicate and rare phenomenon:

What do probabilistic words really mean?

I see you Anscombe’s Quartet, and I raise you the Datasaurus:

A delightful game aptly called Guess the Correlation:

The dangers of using r^2 as an effect size estimate:

The normal distribution in action:

Moore’s Law, and the glorious improbability of that exponential growth:

Fascinating: Moore’s Law predictions vs actual growth in transistor count.
by @datagrapha
reddit.com/r/dataisbeauti… https://t.co/ZwN1dBGE1n

—

Lionel Page (@page_eco) September 03, 2019

Quick sort, in an image:

Centrifugal force to restore a whiteboard marker:

I had no idea you could do this to take a dead whiteboard marker and give it life again!
Source:… twitter.com/i/web/status/1…

—

Robert Kaplinsky (@robertkaplinsky) July 06, 2019

Voronoi diagrams (i.e., which national park is closest to you?):

Set theory (specifically, the power set), where each rectangle is one of the possible sets of these 4 elements (ranging from the empty set in the middle, to the set of all four):

Mathematics in nature:

Stumbled into reading about hermit crabs and wut: https://t.co/ERdEs8TwUK

—

Derrick (@geekandahalf) November 29, 2019

Most are questions I am wholly ill-qualified to answer.

Take this doozy from high school student Charlie in Florida: **What makes a great teacher and what makes a bad teacher in your eyes?**

I fumbled over how to answer… then realized that I don’t have to. I can ask some great teachers to answer for me! From four hand-picked heroes of mine, I got a variety of insightful replies, to which I have added my feeble illustrations.

by Fawn Nguyen

School is inherently stressful for many. It’s mandatory. It’s an institution of compliance: take this class, follow this schedule, sit here, do this, and more often than not, shut up.

A great teacher makes school less stressful.

She understands that her students would rather be elsewhere, but they are here now and she’ll make the best of it for them. She will not waste her students’ time. She is pleased and excited that everyone is present because she has crafted the best lesson and together they will explore.

A great teacher does not take his class too seriously, he laughs with them, makes light of his vulnerability. He yearns to hear each student’s voice and works on inviting that voice to come out in whatever way it needs to.

A great teacher loses her mind because she’s so happy that one of her students has asked a wonderful question that leads the discussion down some glorious path that she had not intended.

A bad teacher does not care. About anything other than putting in the hours.

I’m reluctant to use the phrase “Bad Teacher.” Faced with hundreds of interactions and decisions every day, we all have good and bad moments. Those moments accumulate over a semester, a year, a career, and in most cases yield a net positive result I’d say.

But you can tell a lot about a teacher by how they respond when students don’t succeed. Some will say, “What’s wrong with you?” Others will ask, “What’s wrong with me?”

A wise man once told me that a great teachers know three things: they know their subject, they know how to explain their subject, and they know their students.

Subject knowledge is so often misunderstood by teachers. ‘I have degree in mathematics, of course I can teach it’. But it’s so much more than that. It’s knowing how to explain things, in multiple ways, in ways that make sense. It’s about making complex ideas crystal clear. It’s bringing topics alive, and that can only be done by teachers with a deep understanding of what they’re teaching. Knowing the history of a topic, the interesting problems, the common misconceptions, the multiple approaches, the links to other topics. And it’s about expert communication of all of that.

That’s not to say a great teacher needs to be an entertainer – an expert communicator does so with clarity, not with gimmicks. But passion goes a long way. A great teacher loves their subject, and can’t wait to tell their students about it.

No one sets out to be bad at teaching. But there are those that enter the field for reasons that don’t serve them (or anyone).

I believe that these teachers do not have a well-articulated “why.” Why are they teachers? Why their particular school? Why their grade band? They also don’t know the whys of their pedagogical moves, instructional questions, or mindset.

Since they are not adept at self-analysis, they also can not appreciate this in their students. They are unable to understand students (or colleagues) who need to know why. They see their incessant questions as unnecessary interrogation and even insubordination.

And when they grow up to become administrators…

Now, what about greatness? I see it as on par with wisdom. An educator can approach greatness only after much time (perhaps many years) honing their craft. Even then, it is never really achieved, but approached asymptotically.

Great educators have the humility to know that they do not know everything, and are forever seeking to know.

They seek this knowledge from unpopular sources.

They take risks daily, because what other way is there to live?

They know that the world is inequitable, and they seek their own place in creating lasting change.

They co-create educational experiences with their students because they realize that students hold key knowledge of their own that most don’t acknowledge or even recognize.

They do not expect perfection from their students or from themselves, but have unwavering faith in collective knowledge-making.

They hold each other accountable.

They are quick to apologize, quick to praise, yet do not hesitate to correct where needed.

They have a way of being that students respect.

They do not see education as subject-specific. They recognize the interconnectedness of disciplines and do their best to model this for students.

They are honest with themselves and with students.

They seek to be antiracist in belief and action; they value being racially fluent.

They understand that race is a topic that is hard to discuss openly, and take opportunities to do it well.

They understand that ultimately, the answers to the questions that plague us are found through communication across race.

They are courageous not because they don’t know fear, but because they don’t allow fear to keep them from questioning existing structures, challenging the opinions of their colleagues and administrators, or investigating their own participation and complicity in a system they find subpar.

Yet, they do more than just reflect. They push for change, within and without their classroom.

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