Ultimate Tic-Tac-Toe

Updated 7/16/2013 – See Original Here

Once at a picnic, I saw mathematicians crowding around the last game I would have expected: Tic-tac-toe.

As you may have discovered yourself, tic-tac-toe is terminally dull. There’s no room for creativity or insight. Good players always tie. Games inevitably go something like this:

But the mathematicians at the picnic played a more sophisticated version. In each square of their tic-tac-toe board, they’d drawn a smaller board:

As I watched, the basic rules emerged quickly. Continue reading


How Fast is Exponential Growth? (Or, Yao Ming Confronts the Vastness of the Universe)

On this humble brown planet, we’re used to things growing at a steady pace. Trees add a ring every year. Families expand by one child or marriage at a time. Even in their most extreme months of food-gobbling growth spurts, teenagers will sprout at most a few inches. All of these are examples of linear growth (or something close to it). It’s modest, approachable – something the human brain has no trouble grasping.

But not all growth is like this. Take the old story of the sultan and the beggar. The beggar comes before the sultan, pleading for some rice to eat. When the sultan asks how much he needs, the beggar cleverly points to a nearby chessboard. He asks the sultan to put 1 grain on the first square, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number of grains for each successive square.

The sultan agrees, not realizing that on the 64th and final square, he’ll need to stack 600 trillion pounds of rice – enough to cover Rhode Island to a depth of 1400 feet. That’s exponential growth. It may start slow, but it quickly reaches dizzying heights.

This kind of growth occurs surprisingly often. Look at the human population, or the number of Facebook users, or even the amount of money you make on a brilliant investment. Most of these growth patterns eventually hit some kind of ceiling (the planet will hold only so many humans; Facebook has begun to max out its user base; and your investment can climb only so high), but as long as exponential growth is at work, change proceeds far faster than our linear minds are accustomed to.

To explain this idea to my students, I start with a graph of the function f(x) = 2x, using a one-inch scale. Continue reading

Three Sentiments (or, Ode to the School Year)

5 - gradingFast year, right? Summer is here. The seniors can be found draped across their desks, exploring stages of hibernation so deep that they are yet uncharted by the medical community. It’s all very festive.

And into this start-of-summer breeze, I’d like to offer three sentiments.

First, guys, let’s be honest. We didn’t always getting along. I assigned too much homework, gave too many quizzes, wrote tests that stumped you. When I relented even slightly, you celebrated like a labor union that had scored a victory against the cigar-smoking management. I pushed you too hard, expected too much, demanded the unfair or even the impossible. This brings me to my first sentiment: Continue reading

Explain It To Me

A nearly-verbatim dialogue, in honor of one of the hardest-working students I’ve ever taught.

Student: Can you explain this problem to me?

Me: Sure, the idea is… Wait. You got it right.

Student: But I don’t feel like I did. I feel like I guessed.

Me: Well, explain how you got your answer.

(He proceeds to give a perfect explanation of the problem, justifying every detail.)

Me: See? You understand it.

Student: No, I just guessed.

Me: You “guessed” a flawless, conceptually motivated solution to the problem?

Student: Yes. You have to explain it to me.

Me: I mean… okay…

(I proceed to parrot back his explanation to him, virtually word for word.)

Student: Ah, thanks. That helps.