Lines Beyond y = mx + b

What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then ohhhhhhhh.


Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for ohhhhhhh.

But there’s a common problem: too often, kids can beat the puzzles without feeling any of those things.

I find this, for example, with lines in the coordinate plane.

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The Island of Democrats and Republicans

On February 10th, the world lost Raymond Smullyan: logician, puzzlemaster, and blue-ribbon Gandalf lookalike.


Even if you don’t know his name, you’ve probably wrestled with his logic puzzles. They share a whimsical sense of rigor: “You come to an island where there are two types of people: knights, who always tell the truth, and knaves, who always lie…”

They’re silly and frustrating and fun; everything mathematics should be. I love this origin story for how Smullyan first got into such puzzles:

On 1 April 1925, I was sick in bed… In the morning my brother Emile (ten years my senior) came into my bedroom and said: “Well, Raymond, today is April Fool’s Day, and I will fool you as you have never been fooled before!” I waited all day for him to fool me, but he didn’t.

Or did he?

Young Ray had spent all day expecting to be fooled. But the fooling had never come. Didn’t this constitute the greatest fooling of all?

I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled.

In Smullyan’s honor, I wanted to offer up my own amateur variant on his knights-and-knaves puzzles.

I call it: the island of Democrats and Republicans.

Now, Republicans and Democrats look identical to an outsider like you. But they always recognize one another immediately. And because of their mutual antipathy, they follow this strange custom:


So, here comes your puzzle. Ten of them, really.

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1.2 Trillion Ways to Play the Same Sudoku

or, Group Theory on the Puzzle Page

Last week, I visited my dad, who still gets the newspaper.

(For my younger readers: that’s a stack of cheap paper printed with a detailed description of yesterday.)

Anyway, for an ungrateful millennial like me, a print newspaper means one thing: puzzles.

Like Sudoku.

You already know the rules: nine rows, nine columns, and nine medium squares, each containing the digits 1 through 9. You’re given some; you fill in the rest. It looks something like this (by which I mean, “here’s an example lifted from the Wikipedia page”):


Now, I’m not much of a Sudoku player. (Crossword guy, to be honest.) But glancing at the puzzle, my dad and I got to wondering: How do they generate these puzzles?

We weren’t sure.

So we found a more tractable question: What if you were a lazy Sudoku maker?


That is, suppose you managed to generate a single Sudoku puzzle. (Or steal it from the Wikipedia page.) And suppose you wanted to make a few bucks selling collections of puzzles in airport bookshops. But there’s a catch: You’re not sure how to make more.

How many “different” puzzles can you get from a single Sudoku?

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Why the Number Line Freaks Me Out

One of my favorite quotes about mathematics is from John von Neumann:

“In mathematics you don’t understand things. You just get used to them.”

On one level, this runs against everything I believe as a teacher. Mathematics should not be an intimidating collection of inscrutable methods! It should be a timidating collection of scrutable methods! We should accept nothing on authority. Everything in mathematics is there to be understood.

And I do believe that.

But I also know that mathematics is full of startles and shocks. I know that even the simplest objects can bury deep secrets.

Take the number line.


Boring, right? Nothing could be more prosaic.

Well, that’s only because you’ve gotten used to it. To my mind, the number line merits only one possible reaction:


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Imagine All the Numbers…



A fair question: how did “i” get the name of “imaginary number”?

It seems harsh. In some sense, all numbers are imaginary. After all, is there really such a thing as negative numbers? You can’t have -2 friends, no matter how alienating your Facebook posts are.


Or what about the irrationals? If you take a 1-meter stick and mark it up into equal segments, then no matter how tiny and minute the divisions, you’ll never get an irrational length. Even if you go down to the atomic level. That’s kind of weird. Continue reading

Math Exams with Only One Question


According to legend, this was once the actual  final exam at my high school. But according to legend, England chose kings by sword-yanking contests, so, you know.

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