The statistician, meanwhile, is not going to draw any hasty conclusions about glass fullness from a data set where n=1.

**The Venn Diagram of Practicality and Fuzziness**

Did I reverse-engineer the “practical” and “fuzzy” labels to make the diagram work? YOU HAD BETTER BELIEVE IT MY BLOG-READING FRIEND

**Temperature/Pressure Diagrams**

In hindsight I would draw this one rather differently. The existing diagram suggests that increasing life pressure can drive me from coffee to iced coffee (maybe true) but also that increased ambient temperature can drive me from iced coffee to lemonade (definitely false).

**Devil’s Advocate**

“What, are you saying the devil doesn’t deserve due process?”

“Yes! And also, the devil’s not on trial! We really don’t need the devil to have a representative in every meeting!

**A Package for the Square Root of 2**

Far and away the most positive response I’ve gotten for a cartoon on Facebook! It goes to show, I have no idea what cartoons people will love.

**Step Count**

I’ve seen 7th-graders do this not for fitness-related reasons, just for the love of the game. Right on, 7th-graders. Take the scenic route to an answer.

**Platonic States of Matter**

Alas, it’s very hard to tell Platonic liquids from Archimedean liquids – or irregular liquids, for that matter.

**Data vs. Anecdotes**

The correct answer is C: self-referential cartoons about data and anecdotes!

**Prime Factorization**

G.H. Hardy confidently asserted that number theory would never serve any “warlike purpose,” but I think we can all agree that G.H. Hardy was a rube and a fool.

**Infimum Wage**

I mean, what if there’s an infinite sequence of workers making $8 + 1/n?

**Scooped**

Preliminary data points towards “probably not.”

“Are you finishing soon?*“*

*[bloodcurdling scream*]* “*THE AGONY!!! OH, THE AGONY…”

“Nice, only two years left! Good luck with your thesis project.”

**Time Moves Fast**

This joke never gets old! Well, it does, but only at a rate of 1 year per year.

**How Many Negatives Make a Positive?**

I think this phrase is more useful in describing language (where it’s sometimes true, and at least easy to interpret) than in math (where it just confuses).

**The Equation’s Story**

Really, this equation is a murder mystery. One of the variables is zero, and you’ve got to find out which one!

(Twist ending: it’s both.)

**Nuts and Bolts**

Sounds like… a breakfast cereal? Or something to do with why tables don’t fall down? I’ve always wondered about that…

**How Excited**

8.7! That is pretty darn excited.

]]>**Short version**: I spent the last two weeks visiting a baker’s dozen of delightful campuses, and meeting a baker’s infinity of delightful undergrads, faculty, PhD students, and other sundry charmers.

**Long version**…

**October 29th**: Kutztown University, in scenic Kutztown, PA.

Over dinner, we discuss the various kinds of mappings – isomorphic, diffeomorphic, holomorphic, and Francomorphic (a map whose image is contained in France).

I am also impressed with Kutztown’s pumpkin game:

**October 30th: **Dickinson College in beautiful Carlisle, PA.

I get to chill with my Twitter buddy Dave Richeson. Between the two of us, we have written a sweeping book about the history of topological ideas and several puns about “hyperbolic” geometry.

Please do not inquire who is responsible for which! Dave and I prefer to share credit.

**October 30th: **Gettysburg College in historic Gettysburg, PA.

Prof. Darren Glass shows me Mexican food better than anything I thought possible in the Northeast, and I get to visit the site of the Gettysburg Address, because when you’re doing a lot of public speaking, it’s good to be reminded that Lincoln said 10^15 times more than you ever could, and he did it in like 2 minutes.

(Also: the Gettysburg campus cookies live up to their lofty reputation.)

**October 31st: **Mount Saint Mary’s in lovely Emmitsburg, MD.

It’s a treat to meet Prof. Jonelle Hook (who shows me soap bubble frames shaped like Platonic Solids, from the “Mathematical Thinking” class she designed) and Prof. Scott Weiss (who appeared on Jeopardy and made history with a 3-way tie!)

My talk there is actually available to stream here!

**November 1st: **George Washington University in the needs-no-Homeric-epithet Washington, DC.

Strolling around campus with my pal Ian before the talk, I come across these posters promoting the event. They are psychedelic and strange and feature a humanoid frog in a Hawaiian shirt, who symbolizes everything I want to be in life.

**November 2nd: **College of Southern Maryland in delightful La Plata, MD.

A one-two punch: I get to share Ultimate Tic-Tac-Toe with the students, and then rap algebra with the faculty. Both are delicious. We conclude by unraveling an odd trigonometric fact: your age = arcsin(sin(“year you were born” – “years since 1980”)).

(Later I realize that this trick will fail if you’re older than 90, although perhaps telling a 94-year-old that they’re actually 86 is not a “failure” so much as a polite fiction.)

**November 5th: **University of Maryland in beloved College Park, MD.

Not only are there delightful people willing to chat math with me, but afterwards, there are* coconut macaroons in the rotunda. *Are there five sweeter, more deliriously happy words in any language than “coconut macaroons in the rotunda”?

**November 6th: **University of Mary Washington in timeless Fredericksburg, VA.

I cannot believe the VIP treatment I am getting. Check this out:

That’s me! Or else it’s the *other* Ben Orlin, who’s maybe 10 years younger than me and plays a pretty mean trumpet. Clearly he is the more talented Ben Orlin but don’t tell University of Mary Washington, because I had a great visit and don’t want them upgrading.

**November 6th: **College of William and Mary in unforgettable Williamsburg, VA.

Such a sharp and participatory audience it takes us 20 minutes to get past the first slide. For a while, I listen as two professors debate approaches to math education, which is about 600% more fun for me than talking. (And let’s be honest: I *love* talking.)

**November 8th: **A break from the travels to do a YouTube interview with Joanne Manaster and Jeff Shaumeyer of **Read Science**! You can watch below:

**November 8th: **University of North Carolina in happenin’ Chapel Hill, NC.

A bass ukulele left in the lecture leads to an impromptu performance from my host professor, Justin Sawon. The students are universally charming and brilliant.

**November 9th: **Appalachian State University in delightful Boone, NC.

The far point of my travels (about 800 miles from home) is well worth it, because they let me rap with them about p-values for an hour. By the end of the afternoon I am pretty ready to buy real estate here.

The projector decides to photobomb this shot with my Twitter pal Brad Warfield:

The brilliant faculty members also direct me to Woodlands for barbecue, where I have the finest chicken wings that any chicken has ever winged:

**November 12th: **Hofstra University in bustling Hempstead, NY.

It’s a wonderful visit – my host, Nick Bragman, is a senior that any sane grad program should totally accept if they’re reading this.

**November 12th: **Stony Brook University in scenic Stony Brook, NY.

Before the talk, I get a quick chat with Professor Moira Chas, a topologist who crochets stunning models of topological objects, like this:

Check out more at Prof. Chas’ site, which is 10,000% worth it.

And that brings us to the present moment, more or less. Thanks so, so much to all of my gracious hosts along the way, and to all those who came to chat math with me. When I started teaching 9.5 years ago, I could never have guessed it’d give me chances and experiences like this.

I leave you with one final image, from a weekend stop in Philly with my friends Ryan and Ana:

]]>It’s an old, familiar idea: *Truth is beauty. Beauty is truth*. They go together, inextricable, like friendship and laughter, or road work and traffic. Gazing at the universe will always satisfy your aesthetic itch—and if not, then you probably suck at gazing.

In her new book *Lost in Math*, physicist Sabine Hossenfelder lofts a very skeptical eyebrow at this orthodoxy.

Confession: I know about as much physics as the average squirrel. (Less, in fact; they, at least, reach birdfeeders via daring feats of applied mechanics.) Still, I found the book accessible, informative, and compellingly argued. Plus, the writing is great—a delicious mix of journalistic balance and iconoclastic snark.

Hossenfelder’s argument, in brief:

**There’s no reason to think nature cares what we find beautiful.**

“Why should the laws of nature care what I find beautiful?” she writes on page 3. “Such a connection between me and the universe seems very mystical, very romantic, very not me.” Later, on page 189, she elaborates:

mathematics is full of amazing and beautiful things, and most of them do not describe the world. I could belabor until the end of eternal inflation how unfortunate it is that we don’t live in a complex manifold of dimension six because calculus in such spaces is considerably more beautiful than in the real space we have to deal with, but it wouldn’t make any difference. Nature doesn’t care.

The “beauty = truth” dogma isn’t *a priori* obvious. It’s worth asking *why* we find such a connection, and if the link says more about the universe, or about our unreliable perceptions of beauty.

**A lot of the beauty narrative is just that—narrative.**

We’re storytelling creatures. And our “truth = beauty” conviction seems to rest on cherry-picked anecdotes. On p. 161, Hossenfelder relates a conversation:

“So why are you convinced that mathematics can describe everything?”

“All our successful theories are mathematical,” Garrett says.

“Even the unsuccessful ones,” I retort.

Another key factor: our notion of “beauty” shifts over time. Hossenfelder cites a historian named James McAllister, who argues that “every revolution in science necessitates overthrowing the concepts of beauty that scientists have developed.” Maybe it’s not so much that truth follows beauty, but that we rewrite the laws of beauty to match our latest understanding of truth.

**With new data so hard to come by, our beauty instinct runs amok.**

These days, fundamental physics is like San Francisco real estate: wildly and forbiddingly expensive. Experimental data is slow and hard to come by. This creates a danger—that in the absence of empirical data, scientists will get lost in dreamy speculation. Hossenfelder pulls no punches on page 108:

I can’t believe what this once-venerable profession has become. Theoretical physicists used to explain what was observed. Now they try to explain why they can’t explain what was not observed. And they’re not even good at that.

Because “beauty = truth” is such a pretty and pleasing idea, Hossenfelder worries that bad ideas can hide behind it, avoiding the scrutiny they deserve. “Where experimentalists go to great lengths to account for statistical biases,” she writes, “theoreticians proceed entirely undisturbed, happily believing it is possible to intuit the correct laws of nature.”

Hossenfelder knows that she’s one scientist, writing from a particular angle. Perhaps, she worries on page 96, “I shouldn’t psychoanalyze a community that neither needs nor wants my therapy.”

Still, I’m glad she did. I’m fond of mathematical beauty, and fascinated by its role in intellectual life—including the ways it might steer the car into the pond like a malevolent GPS. I leave physicists to judge the merits of Hossenfelder’s case; as an outsider, I enjoyed the window into a live scientific debate.

]]>The second law of desk dynamics is a ruthless master.

**“Formal”**

I wasn’t familiar with Usage #2 until my mid-twenties. It comes from “form,” as in, “you’re manipulating the symbolic forms on the page, without necessarily ensuring that the manipulations make conceptual sense.”

**The Mach 4**

This sounds like a very painful razor to me. Like, there’s the direction of *down* my cheek, and the direction of *across*, but the next direction after that will consist of burrowing *into* my cheek… and I neither can nor wish to imagine the fourth direction.

**Zorro vs. Zero**

Harder to slash with your sword, but a much cooler conversation starter.

**Boole’s Periodic Table**

I thought there wasn’t suppose to be any memorization in math!

**Social Circles**

“The Social Network” needs a topology-themed sequel.

**Ant Alarmism**

C’mon, right ant! Put it in perspective!

For what it’s worth, it can be tricky to identify, in any given scenario, whether you should be thinking about the percentage or the actual value. E.g., at the grocery store, should you buy more of an object than you planned because of a sale that lowers the unit price? Or should you stick to your original plan, which will have a lower absolute cost?

**Social Networks**

I mean connections like this conversation. But they’re meaningless, also like this conversation.

**Intuition and Rigor**

As with many things in mathematical mastery (skill vs. concept, technique vs. insight, standard procedure vs. creative solution) it’s not either-or. It’s both-and.

**Evolution of Square**

Mathematical archaeologists are just now coming across another transitional fossil called the *rectangle*. The phylogeny of squares is tricky!

**CMB**

Inspired by a true story. A horror story.

(For the record, 13 million seconds is just under 5 months. ACCURATE.)

**Weather Forecasting**

I think about this a lot and have come to no definitive solution.

Maybe I should try to look it up instead of drawing comics about it, you say?

Pish-posh.

“Oh, a comic about hotdogs!” -Literally everyone’s reaction to seeing this picture

**Phi Facts**

I find continued fractions very cool and rather mind-bending. They’d make for a fun slightly-off-the-beaten-path exploratory project for a high-schooler or undergrad.

I’m probably revealing too much of my own psychology here. As they say, Freud feeds on friends’ feuds.

**Teaching vs. Comedy**

I have since been informed that my teaching involves too few celebrity impressions.

It seems like a weird job to think professionally about things no one understands.

An astute commenter pointed out that this bizarre creature, although self-similar, is not really fractal for most definitions of fractal (e.g., shapes of non-integer dimension).

**A Random Walk**

Random walks in 2D are really weird. They return to their starting point with probability 1; the number of expected returns is infinite; but the expected time until the starting point is infinite.

**Tuition Prices**

I wanted to draw the y-axis with little arms and legs instead of numerals, but I’ll leave that to someone whose blog is “Math with Competent Drawings.”

Those who have seen me dance acknowledge that I’m getting pretty close.

]]>

You must not attempt this approach to parallels. I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone… Learn from my example.

*Farkas Bolyai, 1820*

I came over the dunes a little before dawn. Already waiting for me were three Examiners: robed, masked, inhumanly tall. Their silhouettes loomed against the dark ocean; their silence loomed, too, against the dull grinding of the waves.

“I’m ready for the final test,” I told them. I wasn’t sure I wanted to pass, but I knew I didn’t want to fail.

The first Examiner spoke in a thunderous rasp: “Then your task is simple.”

The second Examiner explained: “Draw two parallel lines in the sand.”

I looked to the third Examiner, who said nothing. No further instructions came.

Was that all, then?

I grabbed a dry stick of driftwood, and scraped two parallels, a few feet in length, into the damp earth.

“Longer,” said the first Examiner.

I extended the lines.

“Longer,” said the second Examiner.

I did it again.

The third Examiner shook its head, and I began to sense the nature of the task.

It was arbitrary, a measure only of blind obedience.

(Maybe all tests are, in the end.)

I fetched a second stick, similar in length to the first. Then I walked backwards along the beach, dragging the two sticks behind me, one in each hand. The Examiners followed, seeming to glide, leaving no footprints. I extended the parallel lines until they were a quarter-mile, a half-mile, a full mile long.

“How much further?” I asked, perhaps an hour after sunrise.

No one spoke. I counted a dozen waves before the first Examiner replied: “Do you know the meaning of ‘parallel’?”

“It means the lines never meet,” I said.

The second Examiner nodded. “Prove that you know what this signifies.”

I shivered, despite the building heat of day. “Why? I know what ‘never’ is.”

“Do you?” the first Examiner asked. “You scratch symbols, but symbols of what? You mutter sounds, but are there thoughts behind them? A mortal mouth can mimic immortal speech. It does not mean you understand.”

Ah. So this was the test.

I picked up the sticks and began to scrape again. The sun continued to climb. I watched my shadow shorten, and couldn’t help noticing that the Examiners cast no shadows of their own.

By the early afternoon, I was sweating, stumbling, panting with thirst. I stopped to rest, though I knew this displeased the Examiners.

“I know what parallel lines are,” I told them again.

“Do you?” the second Examiner asked. “You watched the Archer of Time notch his first arrow? You observed the Sculptor of Space, when this universe was mere potter’s clay on a spinning wheel? You, a mortal creature, flinching at pain—you claim to know eternity?”

There was nothing to say to that.

I continued to scratch lines in the wet sand, step by backwards step, my body fraying with thirst, my worthless work trailing into the distance.

Hours later, as the sun set over the dunes, I fell to my knees. “I can’t go on,” I told them. My hoarse voice scarcely sounded like my own.

“Then you don’t understand,” concluded the first Examiner.

“Parallel lines never meet,” whispered the second. “Never, never, never.”

The third Examiner said nothing.

What, I wondered, was the purpose of this test? Did it sift the wise from the rash, the patient from the arrogant? Or did it exist only to separate test-givers from test-takers? Did it merely flaunt their power to demand—and my powerlessness to refuse?

I felt a surge of impotent anger, and lashed out the only way I could: I drew the two sticks together, crossing the lines in the sand. “There,” I said. “There are your parallel lines—at least, the closest you’ll have from my hands.”

I braced myself for their final judgment, for the fatal pressure on the back of my neck.

But when I looked up, two of the Examiners had vanished. Only the third remained. I saw its outstretched fingers, elongated and skeletal, and I sensed that it was smiling behind its mask.

“Congratulations,” the Examiner said, with a sinister calm. “You do understand.”

]]>I came across Catriona Shearer‘s math puzzles on Twitter a few months ago. I was immediately drawn in: they’re so tactile, so handcrafted, so ripe for solving. Each of her gorgeously tricksy problems can swallow an hour in a single bite.

She agreed to let me brainfish you folks by sharing 20 of her favorites. She even indulged my curiosity and admiration with an interview (see the bottom of the post).

Enjoy. And don’t say the Surgeon General didn’t warn you.

**1.
The Garden of Clocks**

“Unfortunately, my favourite one of the six is the only one I didn’t come up with myself,” says Catriona, “the dark blue one.”

**2.
The Toppled Square**

(This one feels like an instant classic to me.)

**3.
It’s a Trap**

“A ‘second attempt’ puzzle that was nicer than the first one I came up with.”

**4.
Three Square Meals**

“I quite like this one – I drew lots of pretty patterns based on it.”

**5.
Shear Beauty**

“Probably my all-time favourite. It just looks impossible! Apparently the solution method used here is called shearing (unfortunately, not in my honour).”

**6.
All Men are Created Equilateral**

“Another corollary that I much prefer to the original.”

**7.
Semicircle Turducken**

“I find angles puzzles much harder to write. My students told me this one was quite easy, but my parents found it really hard. I think you need to ‘know’ more to do this one, but the problem-solving aspect is easier.”

Power Chords

“I never learnt the intersecting chords theorem at school, so I love anything where I get to use it!”

**9.
Tale of Two Circles**

“This was a corollary to a different puzzle, but I like it more than the original!”

Doc Oct

“I think this one’s quite neat, although it looks like a massive rip-off of Ed Southall’s puzzles.”

**11.
All in the Square**

“I like the fact that although you can work out all the dimensions of the orange triangle from the information here (and I did when I solved it), you don’t actually need to – using the area and the hypotenuse is enough.”

**12.
Spike in the Hive**

“This one’s quite neat – I like the fact that you don’t need to work out any of the actual side lengths, which are almost certainly horrible.”

**13.**

**Isosceles I Saw**

“I think the wording of this one is my favourite. Lots of people missed the important information and concluded there were infintely many solutions!

**14.
Green vs. Blue**

“Another one of my favourites.”

**15.
Jewel Cutters**

“The best thing about this one: the really nice dissection solutions that were posted.”

**16.
Going, Going, ‘gon**

“This one isn’t so neat, but the answer really surprised me. I think because it’s harder it didn’t get so much traction on twitter!

**17.
Just One Fact**

“This is one of my favourites, as it just doesn’t look like there’s enough information.”

**18.
The Tumble Dryer**

“I like the higgledy-piggledy squares here, like they’re rattling around in a tumble dryer. And the answer is surprisingly neat too.”

**19.
Fly the Flags**

“This one’s quite simple, once you see it – but I didn’t straight away so the simplicity of the answer surprised me.”

**20.
The Tiger-gon**

“This one I nearly didn’t post. But the picture reminded me of Tony the Tiger.”

**BONUS:
Sunset Over Square City**

“I like this one because it reminds me of a sunrise over a city of squares.”

In case you’ve made it this far down the post – in which case, it’s probably 6 months after you started, and your desk is surrounded by crumpled papers and empty Chinese food containers – then here are some questions I had for Catriona.

**How did you get into designing these puzzles?**

*I went on holiday to the Scottish Highlands, but forgot to take a coat with me, so I ended up spending more time inside than my friends did! I kept doodling along the lines of “I wonder if I could work out…”*

*I wasn’t expecting it to turn into a hobby, but it gets a bit addictive – especially when people reply with their solutions, which I love. There’s almost always a neat shortcut that I’ve missed.*

**What’s your creative process like?**

*It just starts with doodling. I’ll end up with a whole page of overlapping squares at different angles, or regular(ish) pentagons with different parts shaded in, and then see if there’s any nice Maths hiding there – relationships between lengths or areas or angles.*

**Lots of your images are marker on paper. Why the low-tech approach?**

*I did try using Desmos and Geogebra, but I’m not very good. I found it way quicker to draw an inscribed circle by getting my compass out and doing a bit of trial and error than by constructing it nicely in geometry software.*

*Also, with felt tips you can fudge things because the lines are so thick. It’s a nice compromise between it looking ‘right’ but also knowing you can’t just get your ruler out and measure it.*

*One of the nice things about geometry is it’s very forgiving – I can show you a hopeless picture of a square or a circle, but it’s enough to communicate the concept because they’re so well defined.*

**Several of your puzzles provide just enough information. How do you find that boundary, where a diagram is just barely determined?**

*Sometimes giving the bare minimum is actually a giveaway, because it only leaves one avenue. My preference is for giving slightly too much information, so there are a couple of decoy routes. This also means I get to see more variety when people reply with their solutions!*

*I’ve posted a couple of puzzles that were impossible – luckily someone usually points it out quite quickly!*

*I’ve also posted puzzles that I’ve massively over-specified, because I didn’t see a nice shortcut that would only use half the information.*

**Advice for would-be puzzle makers?**

*Ok, my imposter syndrome has fully kicked in here. I’m definitely still a novice – I’ve only been doing this since August! On the other hand, I’ve discovered I enjoy making puzzles and reading solutions even more than I like solving them myself.*

*A puzzle’s primary purpose should be amusement – that’s what marks it out from a standard Maths problem. So you need at least two of:*

**A neat set up**. Perhaps just enough information, so that the reader is wondering how on earth this is possible. Or several tantalising pieces of information that each feel like they offer a way in. Regular polygons and circles are a fantastic two-birds-one-stone here, because they disguise a wealth of information, without the specifically useful bits being marked on the diagram.**A neat method**. A trick, or a shortcut, or an insight that simplifies the whole thing. This might not be the most obvious method – I can think of lots of puzzles I’ve solved with lines of algebra, or surds, or horrible expressions with pi, only for it all to cancel out at the end and I realise there must have been an easier way.**A neat answer**. It’s a bit unsatisfying to work through a puzzle to get to a messy answer.

*Basically, get drawing – find a puzzle you enjoyed solving and see what happens if you extend it, or change some elements of it. If you find a relationship that surprises you, chances are it will also surprise the rest of us, so put it out there. Twitter’s a great platform as people can post their own diagrams in reply.*

**Also, while you’re here: check out Math with Bad Drawings: The Book of All New and Wildly Enjoyable Stuff!**

The only caveat: Words are not necessarily fungible.

For example, many of my pictures are worth only 500 “Huh”s and 500 “Nope”s.

**Professional Development**

Silly folks! It’s much more complicated than that. Teaching = Explanation + Problem Sets + Office Hours!

**The Interesting Part of Mathematics**

I mean, equality is just an edge case. Hardly the meat of the subject!

**Applied Probability**

I considered several other punch lines for this, including:

- “Ugh. Physicists.”
- “Ugh. Empiricists.”
- “Ugh. I hate reality.”

**Run a Regression**

I wonder how many of my cartoons make fun of misapplied statistical techniques?

I know, LET’S RUN A REGRESSION ANALYSIS

**Got It, Thanks**

Some folks really didn’t identify with this one. For me, it’s the first 2 months of teaching any class.

**What Mathematicians Do**

“Watching mathematicians try to explain their work to laypeople” really ought to be a televised sport. It’s a uniquely inexplicable job.

**Bacterial Inspiration**

The nice thing about drawing bacteria is that it’s okay when they change shape from frame to frame. They’re squooshy bags of water!

It’s less credible when it happens with my people/desks/chalkboards.

**Having Three Kids**

I’ve heard this from lots of quarters: the marginal challenge of adding a third child is *huge*, and the marginal life satisfaction isn’t necessarily positive. But past 3, the marginal challenge decreases, and marginal satisfaction grows.

Takeaway? Babies are super cute and wicked hard to raise. The end.

**Intellectual History**

Newton: “Hey, did you guys notice how stuff falls to the ground?”

Everybody: “Whoa, now that you mention it, yeah!”

Anyone who has read Dickens has encountered an argument for why we shouldn’t pay fiction writers by the word, either.

**Contra That One xkcd Comic**

I asked my niece about this. “Why are you stabbing Elmo’s eye?”

“I stab his eye!” she reported.

**Carpet Pricing**

I’m proud of this cartoon because the joke consists entirely of an exponent.

**Time Travel Commute**

I just read *Timescape* and found it both (A) a pretty rich portrait of academic life, and (B) a weirdly self-contradictory vision of time travel. Anyone want to explain to me why it actually makes more sense than I believe?

To be clear: no, nobody actually knows almost half of mathematics! This is a not-to-scale pie chart. The to-scale version would just look entirely red.

**Agree to Disagree**

I think the last panel describes a surprising number of math fans/teachers/professionals/bloggers.

Most conjectures are a bargain at that price, if you ask me!

**Pi Chart**

That’s right, a pi vs. pie pun! That’s the sort of high-quality content you can expect here at Math With Bad Drawings Dot Com.

**Infinity War**

#1:

Is your spirit crushed yet? No? Then try #2:

And if any of your neurons are still functioning, please extinguish them with #3:

With that urgent business out of the way, let me say a stunned and bumbling thank you.

As you know if you have stood within earshot of me during the past few months, I wrote a book. It is blue, and heavy, and (in spite of my illustrations) beautiful. It came out yesterday, and the reception has been overwhelming. Everyone is so kind and supportive. The necessity of thanking far exceeds my petty capacity to thank.

Although it’s exhilarating to watch the book climb the Amazon sales charts (thus far it has peaked at #101! Like, among all books!), there’s no doubt that my favorite part of it all is seeing folks on Twitter showing off their copies.

(This is true whether or not they happen to be a brilliant YouTuber whose gorgeous animations of topological proofs I was admiring on Sunday.)

If you want a taste of the book, you can check out the wonderful Jennifer Ouellette’s piece at *Ars Technica*, The Math of Why It’s So Hard to Build a Spherical Death Star in Space, or the excerpt that *Popular Science* very kindly published: What does math look like to mathematicians?

I wish to close with a self-indulgent story:

Last month, I climbed Mount Kilimanjaro. (Further proof that I cannot say no to my friend Roz.) On Day #6, we donned our headlamps and started climbing a little before midnight. Eight hours later, we were at the 19,000-foot summit, where one of our guides (great guys!) snapped this picture:

It was hard to climb the mountain. It was hard to write the book. But in each case, my own efforts were little molehills compared to a far larger factor: a sky-high pile of outrageous luck. I relied on so many kind and encouraging people, who helped me up the (literal or figurative) mountain, and who showed faith despite a total absence of evidence I merited it.

So: thanks to family, friends, agents, blurbers, Black Dog & Leventhal team, and (since you’re the one reading this) to you. Yes, you. You know who you ARRRRRRRRRRRRRR

]]>What is this vaunted rule? It asserts that there are no whole-number solutions to this equation (where n is at least 3):

On the surface, it’s a statement about numbers. Yet Wiles proved it using the esoteric geometry of “elliptic forms.”

That’s typical of math: many great breakthroughs are acts of translation. Up against a brain-melting algebraic equation? Turn it into geometry, and watch the intuition bloom. Facing a vexing geometric riddle? Turn it into algebra, and watch it become a clean and easy paint-by-numbers computation.

How do you solve a tough problem? Find a language where it’s easy.

I say all this because I recently came across an out-of-left-field paraphrase of Fermat’s Last Theorem. This statement about numbers—or, if you ask Wiles, geometry—becomes a statement about *combinatorics*, the mathematics of enumerating combinations. How?

*Scenario*: You’ve got *n* objects. Congratulations, collector!

*Task*: You’re placing your objects into a row of bins. Some bins are red; some are blue; and the rest are colorless, left unpainted.

Now, with enough bins and objects, you’re going to have a *lot* of choices. All in the first bin; or all in the second; or one in the first and the rest in the second; or two in the first and the rest in the third; and so on, and so on, and so on…

Now, among those arrangements, I want to pick out two kinds. First, there are **no-color combos**: these “shun” both red and blue, relying only on colorless bins.

Second, there are **two-color combos**: these “shun” neither color, using at least one red and at least one blue. (They may or may not use some colorless bins, too. Don’t care.)

And what does Fermat’s Last Theorem say? Quite simply, that the number of no-color combos and two-color combos can never, ever be equal.

How’s *that* for an act of translation? From the austere realm of nonphysical numbers raised to immaterial powers, we have been transported to the kindergarten classroom, where we must put our toys into bins.

And yet nothing has been lost in translation. The heady Platonic question corresponds precisely to this Lego-sorting puzzle.

Here, I’ll show you why.

Let’s call the number of red bins R, the number of blue bins B, and the total number of bins T. Thus, the number of colorless bins is T – R – B.

So, how many no-color combinations are there? Well, we’ve got T – R – B choices for the first object; and T – R – B choices for the second; and T – R – B for the third… so for all *n* objects, we’ve got a total of (T – R – B)^{n} combinations.

Next up: how many two-color combinations are there? Well, I think of it like this:

Now, suppose that our two values were equal. We’d have:

Now, let **c** be the total number of bins, let **a** be the number of non-red bins, and let **b** be the number of non-blue bins. We are left with:

In other words: Fermat’s Last Theorem!

Now, is this a useful paraphrase? I’m sure it isn’t! We’ve rendered a super-clean statement about numbers as a slightly complicated claim about toy-sorting. It feels a little like translating a haiku into the language of a technical manual.

But something sparkles and catches my eye in this act of translation.

Something special about mathematics.

When we translate works of human art from one language to another, something is always lost. Connotations decay. Idiomatic sense vanishes. New and unintended implications arise. Such translations are deformations; in changing the form, we can’t help changing the content.

Mathematical translations aren’t like that. Nothing is lost. No information is destroyed. Anyone can translate the new statement back into the old one, with no corrosion or corruption of the data.

The logic is perfectly preserved.

And yet, the translated statement *feels* so different! It sits suddenly in a new context, a different body of literature, from which it can soak up connotations and implications, without losing an ounce of its original sense.

That’s the magic I love. Human translation alters. Mathematical translation only adds.

**ADDENDUM: ***My father, in the comments below, makes a further translation. **If you let a = red bins, b = blue bins, and c = colorless bins, then Fermat’s Last Theorem can be translated “the number of no-color combos will never equal the number of monochrome combos,” where a monochrome combo has either all objects in blue bins, or all objects in red bins.*

*I wish I’d thought of that simplification! (It doesn’t require any of the messy algebra above to verify.) I’m sure W.V. Quine, from whom I learned the original, would feel the same way.*

I know Twitter’s reputation: it’s a cross between a skull, a mushroom cloud, and a swamp fire. But dang if I don’t love math teacher Twitter! So many friendly people. So many great questions. So many array photos.

I got a lasting reminder of this awesomeness in June, when I tweeted this:

I expected to hear two dominant kinds of memory: happy recollections of a teacher’s validation, and traumatic ones of a teacher’s censure. The actual flood of memories that filled my notifications were richer, weirder, and far more diverse.

Backseat revelations on car trips.

Games with parents and grandparents.

The endlessness of counting.

And—coolest of all—strong emotions excited not by a teacher, not by a peer, not by a parent, but by the mathematics itself. More than I’d have guessed, people remember early encounters with mathematical truth, with the deep persistence of pattern. It made them feel angry, joyous, fearful, awestruck.

I offer up these stories with a minimum of curation. Each is a vivid little window into one of the strangest symbioses on planet earth: the relationship of a human being to mathematical reality.

**“I Knew the Best Secret in the World, Because I Could Count Forever”:
The Endlessness of Number**

Several folks recalled glimpses of infinity, or (equivalently) the idea that there’s always a next number. I’m still reeling from this, and I am objectively growing old and dusty.

** **

**“We’ll Get There in One Episode of Sesame Street”:
Telling Time**

The quantification of time is one of the more peculiar achievements of civilization – and one with immediate importance to kids, who are always being made to wait for stuff.

(Interestingly, this “multiply by 10 then divide by 2” trick showed up in several people’s earliest memories!)

**“For a Six-Year-Old, It Was a Big Revelation”:
Making Discoveries**

It will surprise few of my math teacher colleagues that people remember the intellectual breakthroughs that felt like their own.

Your teacher tells you a thing? Fine.

You uncover a universal truth with *nothing but your mind*? More than fine!

**“Minus Three Blocks North”:
The Allure of Negatives**

Several folks remembered early encounters with negative numbers. Makes sense! They’re as weird as anything in Dr. Seuss or Lewis Carroll.

**“They Seemed Arbitrary to Me”:
Mathematical Symbols**

Those conversant in mathematics don’t think much about the symbols. They take a backseat to the thing symbolized.

But kids – just learning to shape symbols with their hands and to discern them with their eyes – put a bigger emphasis on the marks on paper. What shape is an 8? A 9?

**“I Still Think I Was Right”:
Conflict with a Teacher**

As a teacher, I half-dreaded and fully expected these answers. I know that disagreeing with a teacher burns a moment in memory far better than any mnemonic.

(As the follow-up tweet explains, Erick’s teacher couldn’t find one, and the vindicated Erick repeated the exercise the next day – this time with *five* colors.)

(I’m with the student on this one! 2-1 games at halftime rarely end 4-2.)

** **

**“Feeling Ecstatic That I Could Stack Two Odds”:
Odds and Evens
**

These recollections bolster my sense that “odd” and “even” are the best conceptual playground for the young mathematician. They’re abstract yet easy to visualize, rich yet elementary, not to mention a natural place to make the leap from specific (*what’s 6 + 7?*) to general (*what’s an odd plus an even?*).

** **

**“A Half of What?”:
Fractions**

Given how fractions dominate the upper-elementary curriculum (and the laments of secondary teachers), it’s no surprise that they surfaced often in folks’ memories.

These memories also point towards a meaningful context that stuck with people: coinage. “A quarter” = 1/4.

** **

**“Just Couldn’t Figure Out How My Dad Was Younger Than Me”:
Confusion & Clicking**

Two emotions predominated: the feeling of “WHAT THE HECK IS GOING ON” and its successor, the sweet relief of “oh THAT’S what the heck is going on.”

** **

**“We Were Now Properly Going to Start Mathematics”:
Visions of Things to Come**

Given the hierarchical structure of math education (year-by-year advancement, a sense of cumulative build-up, the sorting of topics as “beginning” or “advanced”), I’m not surprised that many folks remembered occasions when they got to glimpse around the corner, to see what lay ahead on their educational path.

**“This Makes the Patterns in My House Better”:
**

I have a conjecture for why there weren’t *more* memories like this: because people don’t necessarily think of this work as mathematical!

It is, of course – pattern and classification are deep, essential mathematical tasks. But I wouldn’t be surprised if some folks’ minds’ skipped past similar memories, and latched onto something arithmetical, i.e., something more quintessentially “mathematical.”

** **

**“I Could Break the Ice and Scrape the Fragments into Triangles”:
Shape & Geometry**

I was surprised to find so little geometry among these memories. The few that did surface were notably beautiful.

**“What Did That Little x Do?”:
From Addition to Multiplication**

Most folks have a sense of autobiographical memory that kicks into high gear around age 7 – right when they’re starting to encounter multiplication.

Apparently, meditating on the meaning of “2 x 3” is a very common activity for mathematicians at this age!

**“I Am Still So Satisfied and Delighted By This”:
Memorable Tricks**

Whether self-discovered or not, clever numerical tricks and patterns have a way of sticking in the brain.

**“By the Teens I Was Struggling”:
Place Value**

Some memories concerned not just the endlessness of number, but the peculiar nature of our numeral system itself, with place value and positional notation.

This is, after all, a pretty fancy technology, one that took centuries to develop, and which then swept the globe. The early experience of students often recapitulates historical development, in one fashion or another.

**“I Am Five! How Should I Know?”:
Memorization and Drill**

These early memories of drill and memorization aren’t all negative. But it’s striking, I think, how *few* of the memories overall concern drill!

Perhaps it’s because drill isn’t as common as we think.

Perhaps it’s because repeated events, by nature, are hard to pluck out as a “first” memory.

Or perhaps it’s because drill doesn’t stick in the mind quite the way other experiences do.

** **

**“Using This Little Guy”:
Technology**

Only three of the memories dealt with calculators, and none with computers. A generational thing? Or – as I suspect – do technological experiences (e.g., playing with a calculator) stick in memory less than social (e.g., conversing with a parent) or interior experiences (e.g., sitting in the back of a car, dreaming mathematical thoughts) do?

** **

**“A Puzzle To Be Solved:
Learning Through Games**

Again, given how much kids love games, I was surprised how few adults remembered playing them!

** **

**“I Always Wanted the Book with the Counting Monkey”:
Falling in Love with Math**

No further comment necessary.

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