Mazes, Puzzles, and Proofs

Many family restaurants offer paper placemats that entice children into solving puzzles as an alternative to kicking each other under the table, blowing bubbles in their beverages, and so on. I remember those placemats from my own childhood and I recall mazes in particular. The mazes weren’t large, but the designers, in their quest to keep us kids occupied as long as possible, would put long dead-ends near the start of the maze. I quickly hit on a strategy that the emphatically named Thomas T. Thomas also found, as he later recounted in his charming essay “Working Backward“: 

One of the most valuable techniques in problem solving I learned in the third grade. But it certainly wasn’t a lesson my teacher intended.

Flip Your Students, Flip Yourself

Collect content-summaries; return them at exams.

I’ll explain that prescription shortly. But first, a confession:

The title of this essay, in which the word “flip” refers to the pedagogical innovation called the “flipped classroom”, is misleading because I’m not going to tell you how to run a flipped classroom. For those of you who haven’t heard, the flipped classroom approach to education is based on moving lecture-delivery out of the classroom and into the dorm room. No, not by having the professor show up in everyone’s dorm room to deliver content on an individual basis; that would be both impractical and improper. Instead, the professor prepares a video, and the students watch it on their laptops. That way, when the students show up in the classroom, they can focus on teacher-supervised activities that take their knowledge to the next level.

Who Mourns the Tenth Heegner Number?

I’m the urban spaceman, baby; I’ve got speed.
I’ve got everything I need.

− Neil Innes, “I’m the Urban Spaceman” (Bonzo Dog Doo-Dah Band)

There’s an episode¹ of a science-fiction television series in which space travelers land on a planet peopled by their own descendants. The descendants explain that the travelers will try to leave the planet and fail, accidentally stranding themselves several centuries in the past. Armed with this knowledge, the travelers can try to thwart their destiny; but are they willing to do so, if their successful escape would doom their descendants, leaving the travelers with the memory of descendants who, thanks to their escape, never were?

This is science fiction, but it’s also math. More specifically, it’s proof by contradiction. As Ben Blum-Smith recently wrote on Twitter: “Sufficiently long contradiction proofs *make me sad*! When you stick with the mathematical characters long enough, you start to get attached, and then they disappear, never to have existed in the first place.”

Cartoon by Ben Orlin. You can buy his book “Math With Bad Drawings”.

This will be an essay about things that seem as if they exist but which, when you study them deeply enough, turn out not to exist after all.

The Magic of Nine

The kind of magic that grabs (and enchants) me is fantasy-literature magic, not stage-magic, but this month I’ll make an exception to talk about a new math trick inspired by Art Benjamin’s fun-packed book “The Magic of Math: Solving for x and Figuring Out Why”.

After you read this essay, you’ll be able to compute the answer to questions like “What’s the remainder when you divide 123456789 by 9?” or “What’s the sum of the digits when you multiply 123456789 by 9?” without setting pencil to paper, purely through the power of your mind. But before I explain that, I’d better address the question many of you are asking: why would anyone want to learn how to do this sort of thing? (Short answers: to stay sane, to balance their checkbook, and/or to become fame-ish.)

But first, a few words about numbers, rapidity, and mental math.

I HATE NUMBERS¹

Introducing “Thirdsday”

I’m pleased to announce that the greatest holiday in mathematics is almost upon us: the jubilant festival known as THIRDSDAY!

Huh?

Thirdsday is that magical day on which we celebrate the wonder and mystery of the fraction 1/3.

How come I haven’t heard of it before?

Don’t feel bad; I didn’t know about it either until it was discovered a couple of months ago. Or was it invented? Math can be so ambiguous that way.

Stance and distance in popular writing about math

Here’s something you’ll never see in popular writing about musicians:

“Music. For most of us, the mere word conjures up memories of metronomes and endless scales, the student’s never-ending fear of playing a wrong note (or the right note at the wrong time), and the frowns of teachers from whom a curt ‘Good’ was the highest expression of praise. And yet there are some people who just can’t get enough of making music; they practice hour after hour, honing their skills and punishing their bodies, long after the stage of life when there are parents and teachers forcing them to do it. What strange quirk of character compels this behavior?”

As I said, nobody writes about musicians that way. And yet, when the subject is a mathematician, writers sometimes come up with passages like this:

“You and math – one of the greatest love/hate relationships of all time. What is it about the subject that excites us yet sends a chilling tingle down our spine at the same time? How can it be so precise, yet so fickle? We may never know the answers to these questions, but we do know that math is ubiquitous, though some of us may try to hide from it.”

This is from the introduction to a 2010 profile of mathematics editor Vickie Kearn, which I recently saw on the blog-site of Princeton University Press, where Ms. Kearn has worked as senior editor for many years. The intro continues in a similar vein. “While math may sometimes cause us to cry tears of despair, it has caused Vickie to cry tears of joy.”

Between the World and the Mind

The wizard’s-cap graphic that appears at the top of my blog as part of the logo is a piece of an infinite mathematical surface called the pseudosphere.

I don’t study the pseudosphere in my research, and I can’t say I have a lot of intuition about it; in fact I don’t especially like the thing. So why did I choose it to visually represent what this blog is about?