But don’t worry, there’s good news: You’ve narrowed the werewolf down to four suspects and you’ve taken a hair sample from each.

Alas, more bad news: your clumsy lab assistant has mixed up the hair samples. Each bag now contains hair from multiple suspects, hopelessly mingled together. If a bag tests positive, you’ll know one of those suspects is the werewolf, but not which one it is.

Further bad news: your assistant can only find two testing kits. It’s 11pm, and the tests take 50 minutes to develop, so you’ve got to choose two bags to test right now.

How can you use two tests to identify the werewolf before midnight?

***

Now, you may not believe it, but that werewolf business was a work of fiction. That’s right: the vivid details, the photorealistic illustrations, the highly plausible names… all deceptions. In truth, there is no werewolf, no clumsy assistant, and (unless you know something I don’t) no bags of hair. Rather, this scenario comes—and here’s the best news of all—from a new book I’ve been working on.

Ladies and gentlemen, assistants and werewolves: welcome to Puzzle Planet.

I am ludicrously excited about this one. I’ve spent two years gathering puzzles, polishing puzzles, theorizing about the nature of puzzles, and foisting puzzles upon friends, strangers, and unsuspecting spouses. (I find that after enough puzzle-foisting, unsuspecting spouses eventually become suspecting ones.)

Now, it is my sincere hope to foist these puzzles upon you. If you’re interested, you can sign up here to help me play-test material for this upcoming book.

You have questions, have you? Then I have answers, have I!

What kinds of puzzles?

Here are the ways I’ve described them in recent months to strangers (most of them friendly) and to friends (most of them strange):

• “Playable”
• “Game-like”
• “Mathy”
• “Not, like, school-mathy”
• “Hands-on”
• “Dive-in-to-able”
• “Accessible to middle schoolers”
• “Requiring nonzero thought from math professors”
• “Intended to give you—if only briefly—the maddening joy of being stuck”

Who do you want as play-testers?

Anyone willing! But I’m especially eager to recruit: (1) young’uns from ages seven to seventeen, (2) math teachers who want to try the puzzles with their classes, and (3) other puzzle fans of the sort who might buy the eventual book.

How will the playtesting work?

1. You fill out a form letting me know you’re interested.
2. I email you a PDF with puzzles to enjoy at your leisure.
3. You give feedback via a Google Form.
4. Your collective generosity and wisdom help me to clarify confusing bits, fine-tune difficulty levels, cull inferior puzzles, spotlight superior ones, and sprinkle play-testers’ insight and wit throughout the text. The final book glows with your contributions. Puzzle-love sweeps the world. Earth enters a new golden age. Cities smell of rose and cardamom. Human and dog develop a mutually intelligible language and pen joint works of philosophy. That sort of thing. All thanks to you.

When can I expect these puzzles?

Starting later this month. They’ll come in batches; I want to stagger play-testing so I can incorporate feedback and iterate my way to the best version of each puzzle.

What’s the answer to that werewolf one?

Have you tried it already?

Um… no.

Then no spoilers!

Okay, I’ve tried it now. What’s the answer?

So, when you test two bags, there are four possible outcomes: (1) both positive, (2) both negative, (3) only the first is positive, and (4) only the second is positive.

We need each of those outcomes to signify a different suspect. Which means we need our suspects’ hair to appear, respectively, (1) in both bags, (2) in neither bag, (3) only in the first bag, and (4) only in the second bag.

One solution: test AD and BD. The other solution, equally valid: test BC and BD.

Cool. Can I have another sample puzzle?

Sure! That’s what I’ve been saying! Just sign up here…

Fact #1: You have virtually infinite flexibility in writing an exam. You can make it as hard as diamond or as soft as chalk. You can offer copious partial credit or none at all. You can ask questions identical to the homework, or questions of baffling novelty. You can give the exam in-class, as a take-home, open-book, closed-book, group-style, or as an Instagram post on which students comment their answers and your score is the number of likes you accumulate.

Fact #2: Whatever the resulting exam, 90% means A, 80% means B, and 70% means C.

These two facts are the parents of an ancient and venerable custom in education: the curve. If the exam scores are not resulting in the grades you want, then what do you do? Turn them into new scores!

In that spirit, I give you…

(To answer a common question from Instagram: this only makes sense if you read the grades as percentages. Example: 60 is actually 60% = 0.6, which gets curved to 0.6^0.7, or roughly 70%.)

(Again, we’re reading these grades as percentages. And the sin(x) function takes them as radians, not degrees.)

The 100 books of the century.

The 100 movies of the century.

The 25 best pizzas in NYC (inexplicably updated six months later to “The 22 best pizzas in NYC,” portending a grim future where the category of “best pizza places” contracts toward some kind of pizza singularity).

Anyway, I say anyone can play at this game, and I hereby give you…

(And I’m sorry for the snub, 30-60-90-stans.)

EDIT: For future reference, a few other nominations from folks on social media…

• The 13-14-15, whose integer altitude of 12 divides the 14 into 5 and 9 (via Jutta Gut and David Williams)
• The 20-21-29, the smallest Pythagorean triple where c > b + 2 (via Dylan Rambow)
• The 80-80-20, which gives rise to the fabulous constructions of Langley’s Adventitious Angles (via Ng Boong Leong)
• The 3-5-7, which has a 120 degree angle, and can thus be attached to the 5-5-5 equilateral to create a 5-7-8 (via David Williams)
• The 1-i-0 (via lowcheeliang)

Before I get to the FAQs (frequently asked questions), FVOs (frequently voiced objections), and FRCBTWRVATTAUSSMDs (frequently repeated critiques, both those with real validity and those that are ultimately shallow social media dunks), let me give you the original argument:

Now, rather than belabor the specifics (can you specify the colleges? did you account for international students? are you aware of all the transatlantic differences? and hey dummy, isn’t this just a roundabout way of saying “Ivy League”?) I want to zoom out.

Education does many things. It animates old ideas. It stimulates new ones. It widens the apertures of the mind. It allows one to feel smart while watching Jeopardy!

And, skipping way down the list, it enables snap judgments based on your degree.

I do not think snap judgments are particularly noble, and I do not advise anyone to live life by snap judgments alone. But they happen. And, given the limitations of the human attention span, they probably always will. I plead not for an end to snap judgments, but merely for a recalibration of them, to reflect the unwieldy scope of the U.S. higher education system.

My home country is a charming and ungovernable sprawl of 340 million people, as big as five or ten of our “peer nations” combined. But our finest colleges are not proportionally bigger. Some are quite puny, just 2000 students or fewer, and any dramatic growth would change (and hence spoil) their wonderful character.

As a result, our system is hard to wrap your mind around. It’s like someone is reading you a grocery list, and you’re trying to commit it to memory, but the items just keep piling up, until you rue your choice not to start with pen and paper, because to memorize this whole list would take a real savant, or better yet, a top-level graduate from one of the many colleges whose names you have just heard and already forgotten.

The same isn’t true in Canada (population: similar to California) or Australia (population: similar to Texas) or even the U.K. (population: similar to California + Texas). In such countries, learning the top colleges by reputation doesn’t take much brain space. Here, it’s practically a college education unto itself.

So the point of the cartoon is not that we should draw the line precisely here (sorry, Penn; congrats, Northeastern!) and then treat that line as something deep and real.

The point is that the U.S. higher education is, like so much in my homeland, a glorious mess. The point is that we must eradicate the endemic disease I call “US News and World Report brain,” with its symptomatic effort to give power rankings to colleges as if they were football teams. (In reality, they are football teams with schools attached!)

If you want to draw crude lines, be my guest. But remember how crude they are. And yes, it’s silly to lump together 15 colleges as “indistinguishably super fancy.” It’s just that trying to separate them is even sillier.

Every baby is born the same way (namely, as a baby).

And every mathematician is born the same way, too: as a baby as a Platonist.

It usually begins something like this: What exactly are triangles? In a cosmos where no plane is perfectly flat, no segment perfectly straight, and no corner perfectly sharp, are Euclidean triangles in any sense “real”?

Yes, says the Platonist. Tangible? No. Physical? No. But the Platonist has tasted enough of math to know, in her bones, that math is more than just human whim. Math must, in some independent sense, exist. The Platonist believes (in the grand tradition of Seinfeld) that mathematical objects are real, and they’re spectacular.

Other philosophies may come later. Structuralists evolve, like birds from dinosaurs. Formalists are trained, like soldiers in boot camp. Intuitionists emerge, like neo-reactionaries from economic recessions.

But every mathematician begins in the same sweet swaddle of Platonism.

Then, one terrible day, some nosy would-be-Socrates starts poking his grubby fingers inside your thoughts.

When you say these mathematical objects exist, what do you mean by “exist”?

Well, it’s not a physical existence. It’s a conceptual existence.

I see. Where exactly do concepts exist?

There is no “where.” They’re not made of matter. They don’t inhabit space.

So, these immaterial, nonphysical, nonspatial concepts of yours… how do they come to influence our physical reality?

I don’t know. There must be some kind of causal mechanism.

Via the pituitary gland, no doubt?

Hey! Stop making fun of me!

And by the way, how many Platonic forms are there? Is there a Platonic form for the set of all sets that don’t contain themselves, or only for non-contradictory objects? By the way, is the Platonic form of a Gödel statement true or false? And what about the real numbers — what formalization is the Platonic one? Is it Dedekind cuts, or are the True Real Numbers equivalence classes of Cauchy sequences? Either way, isn’t it weird that the Platonic Form of the Reals is a complicated set-theoretic construction over the Platonic Form of the Rationals? And have you considered…

Stop! Stop!

This phase lasts somewhere between 30 seconds and 1 lifetime.

Sooner or later there comes a rebirth of energy. You stop caring about the metaphysics, and recommit yourself to the thing that always mattered most.

The math.

Maybe you read the actual capital-P pragmatists: James, Peirce, Dewey. Or maybe you come by your pragmatism honestly: a few trips around the sun are enough to teach you that the highest form of Truth is whatever gets you through the day, whatever helps you make progress and find meaning on this mixed-up planet of rocks and love and AI slop.

So you give up on the question of whether math is real.

But the question remains: Whether or not math is real, how should I think about it?

So you ask an expert.

Not an expert philosopher. Your mentor hasn’t read any more Wittgenstein than you have. Rather, this is an expert solver of math problems, an expert thinker of mathematical thoughts, an expert in precisely the kind of life you want to live.

This expert is a pragmatist, just like you. This expert has walked your path. This expert knows what kind of outlook will help get you through a day of epsilons.

This expert tells you. And finally, wisdom is yours.

You know how to think about mathematical truth.

Genre, they say, is a marketing gimmick. A needless box. The closing down of a conversation. Genre pre-chews your food, telling you how to read a text: what tropes, conventions, and emotions to prepare for. Ridiculous, the authors say. “There is only one genre in fiction,” writes Matt Haig, “and that genre is called book.”

Fair enough. But you know what desperately needs this treatment?

Mathematics.

Too many readers don’t know the tropes or conventions. Thus, they experience no emotions when reading mathematics. If every novel is unique, then every equation is the same: a sprawl of inscrutable symbols. Whereas genre impoverishes the reading of novels, it would (I argue) enrich the reading of equations.

In that spirit, allow me to tour the corner bookshop of equations…

Many definitions in mathematics operate as a kind of science fiction. They are first encounters with unknown beings. All the ensuing equations, spelling out strange and marvelous consequences, are simply playing out the logic of that stunning first contact.

A conditional equation is a mystery.

A good mystery? An interesting mystery? A mystery with a single culprit?

No, not necessarily… but a mystery nevertheless.

Much mathematics consists of the following activities, familiar to any mystery fan:

1. Laying out mysteries
2. Solving mysteries
3. Explaining why certain mysteries cannot be solved
4. Explaining why certain mysteries can be “solved,” but not by us, because the solutions are impossible to state within the bounds of our familiar language

Okay, maybe math is a weird form of literature. But that’s why we love it.

Not all equations are true.

(Especially not all equations written by students.)

In fact, some equations — if taken as general statements, and not as specific mysteries that are true only for a few values — are so world-shattering, so apocalyptic, that I shudder to read them.

The most romantic equations bring together characters in happy unions.

This is perhaps the most famous and heart-warming such romance, the When Harry Met Sally of the mathematical realm.

Some of the greatest equations in math are not about meeting a new character.

They’re about the moment that an old character becomes someone new altogether.

The moment that an old concept emerges transformed from a cocoon, spreads its wings, and takes off into the bright blue yonder.

Exponentiation is born as repeated multiplication. But once you realize that it obeys this rule — that it is, in a sense, the operation that turns products into sums — then life is never the same again. Thus begin new adventures of which the old exponential could scarcely have dreamt.

Okay, Hamlet maybe doesn’t make for great mathematics. What can I say; not all literature makes sense in translation.

But few will tell you how to write it with style and panache, so as to attract the oohs, aahs, and swiveling heads of passers-by.

In that spirit, allow me to channel the men’s fashion guy on Twitter. (Note to the men’s fashion guy on Twitter: please never look at me or my clothes.) Here are a few side-by-side case studies in how to make your mathematics look good:

Sure, some polynomials look fabulous when factored. Also, some athletic 23-year-olds look good in midriff-baring tops. This doesn’t mean we should all try it.

Better to leave something to the imagination; it’s a sign of maturity.

They’re called radicals for a reason, folks. Don’t conform to algebraic conventions. Give the people something to talk about.

I’m not against f^-1 notation in general. That would be like opposing casual wear at the office; no point shaking one’s fist at a ship that long ago sailed. (And anyway, why should x^-1 be reserved for reciprocals? Isn’t that just a clever and illuminating convention? Any use of negative exponents is already a high-fashion abstraction.)

Anyway, in this particular case, it’s madness to use the dainty superscript when there’s a robust and appealing alternative.

Okay, yes, if you’re actually calculating anything from the limit definition of a derivative, you should go with the more familiar h going to 0 definition.

But be honest. Are you working with the definition of a derivative? Is this the 19th century? Are you a yeoman farmer and/or a Cauchy-era analyst?

No?

Well, then, you’re not bringing this definition to work. You’re using it to make a point: namely, that the derivative is what happens to a slope as the two points draw closer together. And that point is best made with this stylish latter version.

I hesitate to wade into the long-simmering /phi vs. /varphi debates.

But c’mon, folks.

If we can’t agree on such an obvious matter of aesthetics, then I fear we may be approaching the end of our existence as a coherent civilization. Perhaps, in a few decades, the /phi advocates can be resettled on the surface of the moon, where they can build their own sorry little society, beyond the intimidating shadow of our superior fashion sense.

Ah, variance, you minxy concept.

I almost went the other way on this one. After all, is this not the opposite of my advice on the definition of a derivative? Here, am I not promoting easy manipulation over conceptual illumination?

Indeed I am. And that’s because we’re perpetually manipulating variance. The only thing you want to do with that first definition is change speedily into the second one.

I know I’ll ruffle some feathers with this one. Good. Those feathers look silly. They need ruffling.

Now, have I disrupted the beauty of an equation that “unites the five fundamental constants of mathematics”?

Or, have I just revealed that “-1 + 1 = 0” is not as profound a sentiment as some folks think?