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”):

sudoku-1

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?

20170104102812_00001

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?

Continue reading

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.

20161205084636_00018

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:

20161205084636_00019

Continue reading

Imagine All the Numbers…

 

20160718093148_00011

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.

20161205084636_00029

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

20160830090552_00005

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.

Continue reading