Thomas Mann once said, “A writer is someone for whom writing is more difficult than it is for other people.”
I believe the same applies to mathematicians doing arithmetic.
It’s a running joke among mathematicians that they’re bad with numbers. This confuses outsiders, like hearing surgeons plead clumsiness, or poets claim illiteracy, or Rick Astley confess that actually he is going to give you up and let you down, maybe even run around and desert you.
Does it come from some false modesty? A skewed sense of humor?
No, some mathematicians insist: it’s really true, we’re bad at arithmetic.
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.
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?
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:
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
My 6th- and 7th-grade students are pretty effective at calculating with negative numbers. They all know, for example, that 5 – (-2) = 7. Ask them why, and you’ll hear this:
“Because two negatives make a positive!”
Then, if you listen carefully, you will hear something else: the low rumble of my teeth grinding together with tectonic force.
“Two negatives make a positive” is one of those math slogans that drives me crazy, because it is so pithy, so memorable, so easy to apply… while also being so vague and non-mathematical that I’m amazed students find it useful at all.