The Roots of Unity

Two primal pleasures from my years of childhood (and maybe my years of infancy too, though how would I know?) are feelings I’ll call coziness and spaciousness. The first is the feeling I’d get at night pulling the blankets up over my head; the second is the feeling I’d get standing on a beach staring out at the ocean. These pleasures seem like opposites, but sometimes you can have both at once — for instance, when beneath those blankets you have a flashlight and a book full of magic and adventure. I’ve learned that the adventure called mathematics has its own ways of combining the love of the large with the delight of the small, and I’m going to tell you about one of my favorite combinations of these two opposite pleasures: the roots of unity. Continue reading

Impaled on a Fencepost

Customer: “But this receipt proves that I bought the phone less than two weeks ago!”

Manager: “I understand, sir. But you can only get a full refund if you return it within fourteen days, and you’re one day late.”

This surreal exchange isn’t from The Twilight Zone. It took place, with me starring as the hapless Customer, in a perfectly ordinary suburb called Watertown, Massachusetts, and this insidious mixture of social and mathematical ills could visit your town too. The social sickness is a familiar one: big companies like Verizon Wireless can screw you over any way they like, and if they screw over enough people in enough different ways, then the people who got screwed over in any particular way will be too dispersed to find one another and take action. The mathematical malaise? Fencepost error.

Cartoon by Ben Orlin ( Follow him on Twitter @benorlin.

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The Global Roots of Exploding Dots

[This is the text of a presentation I made on October 7, 2017 at the kick-off event for Global Math Week, held at the Courant Institute of Mathematical Sciences in New York City. Earlier in the day, James Tanton gave his usual brilliant presentation on Exploding Dots, so in my talk I was able to assume that the audience knew what Exploding Dots is about; they also recognized my riff on Tanton’s signature line “I’m going to tell you a story that isn’t true”, as well as the significance of the word “Kapow!” (and its variants) in the Exploding Dots story. You might want to visit YouTube and sample Tanton’s Exploding Dots videos to get a feel for what it’s all about. For the full Exploding Dots spiel, try There are two videos of this talk: there’s (the one made by the Global Math Week folks back on October 7) and there’s (my PowerPoint slides, narrated by me). The latter has better visuals, and it has captions, but it’s missing the audience reactions.]

I’m going to tell you a story that’s as true as I know how to make it. I don’t have a background in ethnomathematics, or the history of mathematics, or the history of math education, so please forgive any mistakes, omissions, distortions, or mispronunciations (and let me know about them, but not now!). The story I’m going to tell is as old as civilization, or at least as old as money — because as long as currency has existed in different denominations, there’s always been a need to make change, and to find systems for making change efficiently and accurately. The story I’m about to tell involves many parts of the world over the course of many centuries. And in many ways it’s a story about sand.

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How Do You Write One Hundred in Base 3/2?

You and your computer have a fundamental disagreement about how to represent numbers. Your computer was designed to calculate in base two (binary), while you use base ten (decimal). But there is something that your decimal self and your binary computer can agree on: representing numbers in base three-halves is a damn fool thing to do. I mean, I haven’t even told you yet what “base three-halves” is, but you probably already guessed it’s one of those things mathematicians came up with not because anyone asked them to but simply because they can and because they think it’s fun.
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Prof. Engel’s Marvelously Improbable Machines

When the path from a simple question to a simple answer leads you through swamps of computation, you can accept that some amount of tromping through swamps is unavoidable in math and in life, or you can think harder and try to find a different route. This is a story of someone who thought harder.

His name is Arthur Engel. Back in the 1970s this German mathematician was in Illinois, teaching probability theory and other topics to middle-school and high-school students. He taught kids in grades 7 and up how to answer questions like “If you roll a fair die, how long on average should you expect to wait until the die shows a three?” The questions are simple, and the answers also tend to be simple: whole numbers, or fractions with fairly small numerators and denominators. You can solve these problems using fraction arithmetic (in the simpler cases) or small systems of linear equations (for more complicated problems), and those are the methods that Engel taught his students up through the end of 1973. Continue reading

Swine in a Line

Last month I launched a venture similar to Mathematical Enchantments: a YouTube channel called Barefoot Math. The first few videos are about a game I invented called Swine in a Line. The rules are easy to state but the winning strategy is not easy to find, and the challenge I posed is to find that strategy. In the videos, and in this essay, I explain the strategy and I explain how a person might figure it out. It’s a story about how numbers, and the ways we represent them, can turn out to be relevant in surprising ways. I’ll also sketch how the game relates to a hot topic called the abelian sandpile model. Finally, I’ll connect the Swine in a Line game to James Tanton’s exciting way to make pre-college math vivid through his device of “Exploding Dots”, which will be the subject of Global Math Week later this year. Continue reading

Reading, Writing, and Rigor

My career as a serial extortionist was triggered by an act of theft — more specifically, by an honor student’s appropriation of another student’s words on a homework assignment.

To tell this story properly, I should back up a bit and describe my earlier, non-extorting self.  When I was a naive young assistant professor, I was convinced that if I wrote up detailed solutions to the homework problems I assigned, students would eagerly read them and absorb not only habits of effective problem-solving but also habits of clear writing.  I know for a fact that the students appreciated the extra effort I went to in writing up the solutions; they praised me for it in their end-of-term evaluations.  There was only one problem: over the course of years, it became clear to me that hardly any of them actually read my solutions.

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