Math with Bad Drawings

Chapter 1

ULTIMATE TIC-TAC-TOE

WHAT IS MATHEMATICS?

Once at a picnic in Berkeley, I saw a group of mathematicians abandon their Frisbees to crowd around the last game I would have expected: tic-tac-toe.

As you may have discovered yourself, tic-tac-toe is terminally dull. (That’s a medical term.) Because there are so few possible moves, experienced players soon memorize the optimal strategy. Here’s how all my games play out:

If both players have mastered the rules, every game will end in a draw, forever and ever. The result is rote gameplay, with no room for creativity.

But at that picnic in Berkeley, the mathematicians weren’t playing ordinary tic-tac-toe. Their board looked like this, with each square of the original nine turned into a mini-board of its own:

As I watched, the basic rules became clear:

But it took a while longer for the most important rule of the game to emerge:

You don’t get to pick which of the nine mini-boards to play on. Instead, that’s determined by your opponent’s previous move. Whichever square she picks on the mini-board, you must play next in the corresponding square on the big board.

(And whichever square you pick on that mini-board will determine which mini-board she plays on next.)

This lends the game its strategic element. You can’t just focus on each mini-board in turn. You’ve got to consider where your move will send your opponent, and where her next move will send you—and so on, and so on.

(There’s only one exception to this rule: If your opponent sends you to a mini-board that’s already been won, then congratulations—you can go anywhere you like, on any of the other mini-boards.)

The resulting scenarios look bizarre, with players missing easy two-and three-in-a-rows. It’s like watching a basketball star bypass an open layup and fling the ball into the crowd. But there’s a method to the madness. They’re thinking ahead, wary of setting up their foe on prime real estate. A clever attack on a mini-board leaves you vulnerable on the big board, and vice versa—it’s a tension woven into the game’s fabric.

From time to time, I play Ultimate Tic-Tac-Toe with my students; they enjoy the strategy, the chance to defeat a teacher, and, most of all, the absence of trigonometric functions. But every so often, one of them will ask a sheepish and natural question: “So, I like the game,” they’ll say, “but what does any of this have to do with math?”

I know how the world sees my chosen occupation: a dreary tyranny of inflexible rules and formulaic procedures, no more tension-filled than, say, insurance enrollment, or filling out your taxes. Here’s the sort of tedious task we associate with mathematics:

This problem can hold your attention for a few minutes, I guess, but before long the actual concepts slip from mind. “Perimeter” no longer means the length of a stroll around the rectangle’s border; it’s just “the two numbers, doubled and added.” “Area” doesn’t refer to the number of one-by-one squares it takes to cover the rectangle; it’s just “the two numbers, multiplied.” As if competing in a game of garden-variety tic-tac-toe, you fall into mindless computation. There’s no creativity, no challenge.

But, as with the revved-up game of Ultimate Tic-Tac-Toe, math has far greater potential than this busywork suggests. Math can be bold and exploratory, demanding a balance of patience and risk. Just trade in the rote problem above for one like this:

This problem embodies a natural tension, pitting two notions of size (area and perimeter) against one another. More than just applying a formula, solving it requires deeper insight into the nature of rectangles. (See the endnotes for spoilers.)

Or, how about this:

A little spicy, right?

In two quick steps, we’ve traveled from sleepwalking drudgery to a rather formidable little puzzle, one that stumped a whole school’s worth of bright-eyed 6th graders when I tossed it as a bonus on their final exam. (Again, see the endnotes for solutions.)

As this progression suggests, creativity requires freedom, but freedom alone is not enough. The pseudopuzzle “Draw two rectangles” provides loads of freedom but no more spark than a damp match. To spur real creativity, a puzzle needs constraints.

Take Ultimate Tic-Tac-Toe. Each turn, you’ve got only a few moves to choose from—perhaps three or four. That’s enough to get your imagination flowing, but not so many that you’re left drowning in a sea of incalculable possibilities. The game supplies just enough rules, just enough constraints, to spur our ingenuity.

That pretty well summarizes the pleasure of mathematics: creativity born from constraints. If regular tic-tac-toe is math as people know it, then Ultimate Tic-Tac-Toe is math as it ought to be.

You can make the case that all creative endeavors are about pushing against constraints. In the words of physicist Richard Feynman, “Creativity is imagination in a straitjacket.” Take the sonnet, whose tight formal restrictions—Follow this rhythm! Adhere to this length! Make sure these words rhyme! Okay… now express your love, lil’ Shakespeare!—don’t undercut the artistry but heighten it. Or look at sports. Humans strain to achieve goals (kick the ball in the net) while obeying rigid limitations (don’t use your hands). In the process, they create bicycle kicks and diving headers. If you ditch the rulebook, you lose the grace. Even the wacky, avant-garde, convention-defying arts—experimental film, expressionist painting, professional wrestling—draw their power from playing against the limitations of the chosen medium.

Creativity is what happens when a mind encounters an obstacle. It’s the human process of finding a way through, over, around, or beneath. No obstacle, no creativity.

But mathematics takes this concept one step further. In math, we don’t just follow rules. We invent them. We tweak them. We propose a possible constraint, play out its logical consequences, and then, if that way leads to oblivion—or worse, to boredom—we seek a new and more fruitful path.

For example, what happens if I challenge one little assumption about parallel lines?

Euclid laid out this rule regarding parallel lines in 300 BCE; he took it for granted, calling it a fundamental assumption (a “postulate”). This struck his successors as a bit funny. Do you really have to assume it? Shouldn’t it be provable? For two millennia, scholars poked and prodded at the rule, like a piece of food caught between their teeth. At last they realized: Oh! It is an assumption. You can assume otherwise. And if you do, traditional geometry collapses, revealing strange alternative geometries, where the words “parallel” and “line” come to mean something else entirely.

New rule, new game.

As it turns out, the same holds true of Ultimate Tic-Tac-Toe. Soon after I began sharing the game around, I learned that there’s a single technicality upon which everything hinges. It comes down to a question I confined to a parenthetical earlier: What happens if my opponent sends me to a mini-board that’s already been won?

These days, my answer is the one I gave above: Since that mini-board is already “closed,” you can go wherever you want.

But originally, I had a different answer: As long as there’s an empty space on that mini-board, you have to go there—even though it’s a wasted move.

This sounds minor—just a single thread in the tapestry of the game. But watch how it all unravels when we give this thread a pull.

I illustrated the nature of the original rule with an opening strategy that I dubbed (in a triumph of modesty) “The Orlin Gambit.” It goes like this:

When all is said and done, X has sacrificed the center mini-board in exchange for superior position on the other eight. I considered this stratagem pretty clever until readers pointed out its profound uncleverness. The Orlin Gambit wasn’t just a minor advantage. It could be extended into a guaranteed winning strategy. Rather than sacrifice one mini-board, you can sacrifice two, giving you two-in-a-rows on each of the other seven. From there, you can ensure victory in just a handful of moves.

Embarrassed, I updated my explanation to give the rule in its current version—a small but crucial tweak that restored Ultimate Tic-Tac-Toe to health.

New rule, new game.

This is exactly how mathematics proceeds. You throw down some rules and begin to play. When the game grows stale, you change it. You pose new constraints. You relax old ones. Each tweak creates new puzzles, fresh challenges. Most mathematics is less about solving someone else’s riddles than about devising your own, exploring which constraints produce interesting games and which produce drab ones. Eventually, this process of rule tweaking, of moving from game to game, comes to feel like a grand never-ending game in itself.

Mathematics is the logic game of inventing logic games.

The history of mathematics is the unfolding of this story, time and time again. Logic games are invented, solved, and reinvented. For example: what happens if I tweak this simple equation, changing the exponents from 2 into another number, like 3, or 5, or 797?

Oh! I’ve turned an ancient and elementary formula, easily satisfied by plugging in whole numbers (such as 3, 4, and 5), into perhaps the most vexing equation that humans have ever encountered: Fermat’s last theorem. It tormented scholars for 350 years, until the 1990s, when a clever Brit locked himself in an attic and emerged nearly a decade later, blinking at the sunlight, with a proof showing that whole number solutions can never work.

Or, what happens if I take two variables—say, x and y—and create a grid that lets me see how they relate?

Oh! I’ve invented graphing, and thereby revolutionized the visualization of mathematical ideas, because my name is Descartes and that’s why they pay me the big bucks.

Or, consider that squaring a number always gives a positive answer. Well, what if we invented an exception: a number that, when squared, is negative? What happens then?

Oh! We’ve discovered imaginary numbers, thereby enabling the exploration of electromagnetism and unlocking a mathematical truth called “the fundamental theorem of algebra,” all of which will sound pretty good when we add it to our résumés.

In each of these cases, mathematicians at first underestimated the transformative power of the rule change. In the first case, Fermat thought the theorem would yield to a simple proof; as centuries of his frustrated successors will attest, it didn’t. In the second case, Descartes’s graphing idea (now called “the Cartesian plane” in his honor) was originally confined to the appendix of a philosophical text; it was often dropped from later printings. And in the third case, imaginary numbers faced centuries of snubs and insults (“as subtle as they are useless,” said the great Italian mathematician Cardano) before being embraced as true and useful numbers. Even their name is pejorative, originating from a dis first uttered by none other than Descartes.

It’s easy to underestimate new ideas when they emerge not from somber reflection but from a kind of play. Who would guess that a little tweak to the rules (new exponent; new visualization; new number) could transform a game beyond imagination and recognition?

I don’t think the mathematicians at that picnic had this in mind as they huddled over a game of Ultimate Tic-Tac-Toe. But they didn’t need to. Whether we’re cognizant of it or not, the logic game of inventing logic games exerts a pull on us all.