Math Games with Bad Drawings

DOTS AND BOXES

A GAME OF SQUARES

In the introduction to his 130-page book Dots and Boxes: Sophisticated Child’s Play, the mathematician Elwyn Berlekamp called this game “the mathematically richest popular child’s game in the world.” Whether he meant to call it a sophisticated game for popular children, a popular game for sophisticated children, or a sophisticated and worldly game for rich and popular children, the message is clear: This game slaps.

In this brief chapter, I can’t lay out a complete theory of Dots and Boxes. Instead, I’ll lay out something better: a complete theory of mathematical inquiry, straight from the scholar who first published the rules to this game.

Will reading these pages transform you into a rich, popular, and sophisticated child? Legally, I can’t promise that. So just look at my winking eye, and sally forth.

HOW TO PLAY

What do you need? Two players, a pen, and an array of dots. I recommend 6-by-6, but any rectangular array works.

What’s the goal? Claim more boxes than your opponent.

What are the rules?

1. Take turns drawing little vertical or horizontal lines to connect adjacent dots.

2. Whoever draws the fourth line to complete a small box gets to claim the box as their own (by writing their initial inside), and then immediately takes another turn.

This rule may allow you to claim several boxes in a row before your opponent has a chance to move again.

3. Play until the grid is full. Whoever claims more boxes is the winner.

TASTING NOTES

I first played this game in my childhood basement, amid shelves of VHS tapes and the rumbling footfalls of passing dinosaurs. My siblings and I lacked strategic sophistication: We moved pretty much at random, making sure not to draw the third line on any box (which would let your opponent draw the fourth), and otherwise scattering our marks willy-nilly.² Then, at some point, no safe moves remained. That’s when things got tense.

Sacrifice was now unavoidable—yet not all sacrifices were equal. Some moves might give your opponent just one or two boxes; others, practically the whole board. I always tried to give away the smallest region possible, hoping to reserve the larger ones for myself.

Years later, while working on this book, I learned a crucial stratagem. It’s simple to execute, yet sufficient to beat 99% of novices: the double cross. The idea is that, when your opponent is set up for a triumphant next turn, you don’t give it to them. Instead, cut your own turn short by skipping over the second-to-last move. You thus sacrifice two boxes, which your opponent will claim by drawing a single line (hence “double cross”). In exchange, you’ll receive the whole region that your opponent was eyeing.

Beyond this level of strategy, as you seek to control the sizes and structures of the regions formed, it all gets murky and complicated. For such details, consult the writings of the late, great Elwyn Berlekamp. He passed away while I was writing this book and shall always be remembered as a child of otherworldly sophistication.

WHERE IT COMES FROM

Today, you’ll find Dots and Boxes played just about everywhere: on blackboards, whiteboards, cardboard, legal pads, restaurant napkins, and, under desperate circumstances, bared arms.³ It was first published by the mathematician Édouard Lucas, in 1889, under the title La Pipopipette. Édouard credited its invention to several of his former students at Paris’s prestigious École Polytechnique.

This raises questions. Why would serious students spend their time crafting a game fit for children? And why would an esteemed scholar like Édouard choose to publish it?

Simple: Because serious mathematics is often born from childish play.

We see this pattern in Édouard’s own career. He is perhaps best known for his work on Fibonacci-like sequences, in which each number is the sum of the previous two. (The classic sequence begins 1, 1, 2, 3, 5, 8, and so on.) Fibonacci numbers seem like a silly game—that is, until you start counting the bumps on a pinecone, the petals on a daisy, or the fruitlets on a pineapple, and realize that this silly game is played not only by children (and by adults of dubious maturity) but by nature itself.

Or take the cannonball problem, another of Édouard’s favorites. It asks for a special number of cannonballs: one that you can arrange into a perfect square, and then rearrange into a perfect square pyramid. The puzzle is utterly frivolous. It is also devilishly hard. Édouard conjectured that the known solution (4,900 cannonballs) was the only solution.

Decades later, advanced work with elliptic functions finally proved him right.

Or consider Édouard’s most celebrated invention: the Tower of Hanoi. You may have seen one. It has three rods and a set of disks stacked from the largest on bottom to the smallest on top. Your goal is to transport the whole stack from one rod to another, moving one disk at a time, and never placing a larger disk on top of a smaller one.

In appearance and spirit, the tower is—how do I put this kindly—a baby toy. Yet it has found a variety of practical uses. Psychologists deploy it to test cognitive abilities; computer science professors, to teach recursive algorithms; and software engineers, as a rotation scheme for backing up data.

How does recreation blur so easily into research? Why is the boundary between work and play so porous and heavily trafficked?

I honestly don’t know. I don’t think Édouard did, either. All we can say is, time and again, simple mathematical premises deliver profound consequences. That’s what mathematics is, really: simple ideas in complex interplay. As Édouard wrote of Dots and Boxes: “Its practice, although easy, gives rise to continual surprises.”

WHY IT MATTERS

Because useless play often births the most useful insights.

In his initial publication on Dots and Boxes, Édouard Lucas indulged in a long tangent about the value of pure curiosity. Citing a litany of historical examples, he argued that we must pursue questions for their own sake, no matter how silly they seem, because we never know what deep truths we might uncover.

His writing is flowery enough to be mistaken for perfume, but it’s still worth quoting:⁴

All mathematicians strive to uncover deep connections between disparate ideas. The question is how to do it. Hard work? Maybe. Patient calculation? Can’t hurt. Looking up the answers in the back of the book? Sorry, you’re getting colder. Exuberant leaps of imagination?

Now we’re talking.

Édouard believed that profundity comes from play, science from silliness. He wasn’t alone in this conviction. Elwyn Berlekamp learned Dots and Boxes at age six, and 70 years later, he was still playing it. The game lasted him a lifetime. Somewhere in the middle, while studying electrical engineering at MIT, it dawned on him that he could use mathematics to transform the game into an equivalent “dual game,” which he called Strings and Coins.

How does this alternate version work? Picture a collection of coins, held together by bits of string. Each string has one end glued to a coin, and the other glued to another coin (or to the table). Players take turns cutting bits of string with the scissors. If your cut frees a coin, you pocket the coin, and cut again. When the last coin is freed, whoever has pocketed more coins is the winner.

No boxes, just coins. No drawing lines, just snipping strings. Yet the game is fundamentally the same. Without changing its core structure, Elwyn had turned Dots and Boxes inside out.

What’s the point? No point. It’s just cool. “Let the thinkers think and the dreamers dream,” wrote Édouard Lucas, “without worrying whether the object of their attention seems sometimes useful, sometimes frivolous, because, as the wise Anaxagoras said, everything is in everything.”

That philosophy has driven millennia of mathematical inquiry, and it will last us for millennia to come. Let the thinkers think. Let the dreamers dream. Let the students doodle during lecture. Don’t police imaginary borders between the practical and the impractical, the pointed and the pointless, the idle and the ideal. They all belong to the same vast continent, the same gorgeous wilderness that we have scarcely begun to explore.

VARIATIONS AND RELATED GAMES

SWEDISH BOARD: Begin with all lines along the outer rim of the board already drawn.

DOTS AND TRIANGLES: All rules remain the same, except you play on a pyramid of dots, vying for possession of little equilateral triangles. In my view, this freshens up the game beautifully (and the pyramids aren’t too tough to draw). Perfect for when you’re growing bored with the classic version and your meal still hasn’t arrived.

NAZARENO: In this clever variant from Andrea Angiolino’s Super Sharp Pencil and Paper Games, all rules remain the same, except two. First, on each turn, you can draw a straight line of any length that you wish, as long as it does not retrace any existing lines. (Thus, you may complete and claim multiple boxes with a single line.) And second, there is no bonus turn for completing a box.

Whereas Dots and Triangles’ new appearance disguises the same basic game, Nazareno is the opposite: a familiar appearance disguising a fundamentally different experience.

SQUARE POLYP: In his book 100 Strategic Games for Pen and Paper, the mad visionary Walter Joris offers several games reminiscent of Dots and Boxes. My favorite is 90: Square Polyp. To play, you need two players and two colors of pen.

1. Draw a 9-by-9 array of dots (or smaller for beginners/larger for experts), and take turns placing square polyps. These are squares with two adjacent lines poking out, like these:

2. Claim territory by enclosing it entirely in your color. Each polyp will automatically claim a 1-by-1 square, but with clever play, you can claim larger, more oddly shaped regions.

3. Overlapping lines are forbidden.⁵ This makes it possible to foil your opponent’s designs with a single stabbing tentacle (and for them to foil yours with the same ease).

4. Play until no legal moves remain. Whoever encloses a larger total territory wins.

Footnotes

2 Sometimes, a sneaky second player would mirror the first player’s moves, so that the board would look the same if rotated 180°. This guaranteed the second player would be able to claim a box first. But a savvy first player could exploit the strategy, sacrificing a single box to win the rest.

3 It is also played in many countries, under names ranging from the pedestrian (the US’s Dots; England’s Squares) to the melodious (France’s Pipopipette; Mexico’s Timbiriche) to the visionary (the Netherland’s Kamertje Vehuren, or “rent out a small room”; Germany’s Käsekästchen, or “little cheese boxes”).

4 By the way, if you ever feel like this book is beating around the bush, please consider that Édouard lays all of this out before even getting to the game.

5 If you want a variant on a variant, play tester Valkhiya suggested a neat twist on this rule: When placing polyps, you can overlap your own lines, but not your opponent’s.

SPROUTS

A GAME OF “CURIOUS TOPOLOGICAL FLAVOR”

School geometry teaches us an ugly lesson: Size matters. In fact, size is the essence of matter. Angles can be acute, right, or obtuse. Figures can have length, area, or volume. Salted caramel mochas can be tall, grande, or venti. All of these traits boil down to size. Heck, the very name of the subject—“geo” meaning earth and “metry” meaning measurement—is about sizing up the world itself.

Does this size-conscious philosophy offend you? If so, you’ll like topology. Its shapes stretch like rubber, squish like Play-Doh, and puff up like balloons. They’re not shapes, really, but shapeshifters. In this oozing, lava-lamp world, size doesn’t matter. In fact, “size” doesn’t even mean anything.

Topology seeks deeper truths.

There’s no better introduction to these truths than a game of Sprouts. Which spots can be connected? How many regions will form? What’s the difference between “inside” and “outside”? Hold on to your hat—or the topological equivalent thereof—and enjoy a game that any child can play, yet no supercomputer can solve.

HOW TO PLAY

What do you need? Two (or more) players, a pen, and paper. Start by drawing a few spots on the page. For your first few games, three or four spots are plenty.

What’s the goal? Make the final move, leaving your opponent with no viable options.

What are the rules?

1. On each turn, connect two spots (or connect a spot to itself) with a smooth line, and place a new spot somewhere along the line you just drew.

2. Just two restrictions: (1) Lines cannot cross themselves or each other, and (2) each spot can have at most three lines sprouting from it.

3. Eventually, you’ll run out of moves. The winner is whoever moves last.

TASTING NOTES

The delight of Sprouts is its flexibility. It doesn’t matter whether you draw short segments, lazy curves, or mazelike spirals; all that matters is which spots you connect. You might even sign your name. A sixth grader named Angela busted out this move in our play-testing, and though it technically violates the “no crossing” rule, it was too awesome to disallow.

This flexibility captures the spirit of topology: Things that look quite different might be, for functional purposes, the same.

Consider a one-spot game. The first player must connect the spot to itself. After that, the second player must connect the two spots. It seems there are two different ways: through the inside, or around the outside.

But wait. Imagine playing the game on the surface of a sphere. In this setting, nothing has changed, yet “inside” and “outside” have become arbitrary distinctions. The two moves are, for topological purposes, identical. The second player has no real freedom after all.

What about a game with two spots? Topologically, the opening move allows for just two choices: Connect the two spots, or connect one to itself. Whether you leave the other spot “outside” or “inside” is immaterial. In topology, it’s all the same.

So do topologists ignore all distinctions, treating all things as identical? Is “winning” the topological equivalent of “losing”? Is “good” just topology-speak for “bad”? Is a cat topologically equivalent to a fish, and if so, should we place tiny litter boxes in our aquariums?

Well, that’s up to you as a pet owner. But when it comes to Sprouts, you needn’t worry. Not all moves are alike. In fact, by the second move of the two-spot game, you already face six topologically distinct options. The freedom grows from there.

Dots and Boxes gave us rigid, rectilinear geometry, like a city laid out on a grid. Sprouts, by contrast, is an open-ended and free-form game, like the chaos of a traveling carnival.

WHERE IT COMES FROM

We can pinpoint the precise place and time of Sprouts’ birth: Cambridge, England, on the afternoon of Tuesday, February 21, 1967.

Its parents, computer scientist Mike Paterson and mathematician John Conway, were doodling on paper, trying to invent a new game. Mike proposed the “add a new dot” rule, John proposed the name, and Sprouts was born.⁶ The happy parents agreed to split the credit 60/40 in Mike’s favor, a division so amicable and precise that it’s almost more impressive than creating the game in the first place.

Sprouts is simple to play, yet almost impossible to solve. Mastering the six-spot game required a 47-page analysis from Denis Mollison. That was the state of the art until 1990, when a computer at Bell Labs solved games up to 11 dots. As of this writing, the largest solved game is over 40 dots—although Conway, before his death in 2020, questioned the legitimacy of this claim. “If someone says they’ve invented a machine that can write a play worthy of Shakespeare, would you believe them?” he asked. “It’s just too complicated.”

Has this dizzying complexity scared off casual gamers? Not at all.

“The day after sprouts sprouted,” Conway wrote, “it seemed that everyone was playing it. At coffee or tea times there were little groups of people peering over ridiculous to fantastic sprout positions… The secretarial staff was not immune… One found the remains of sprout games in the most unlikely places… Even my three- and four-year-old daughters play it,” he added, “though I can usually beat them.”

WHY IT MATTERS

Because, of all the branches of modern mathematics, topology is among the most (1) dynamic, (2) bizarre, (3) useful, and (4) beautiful.

That’s a lot of adjectives, so let’s rumble through them, one by one.

TOPOLOGY IS DYNAMIC. Topologists navigate a shapeshifting world of stretching fabric, molten metal, and swirling soft-serve ice cream. Everywhere they go, they seek invariants: traits and properties that somehow, through all the change and upheaval, remain the same.

The most famous invariant is the Euler characteristic. In the context of Sprouts, it boils down to a simple equation (this version courtesy of Eric Solomon): spots + regions = lines + parts.

This equation holds true for every possible Sprouts scenario, from the beginning of the game to the end, from the simplest to the most complex, whether you begin on two spots or 2 million. No matter what, the number of spots plus the number of closed regions will always equal the number of lines connecting spots plus the number of separate parts.⁷

This is typical of topology: Beneath wild flux, we find powerful regularities.

TOPOLOGY IS BIZARRE. Here’s a fun result from John Conway. If a game of Sprouts lasts the minimum number of moves, then it must always end (roughly speaking) in one of these shapes: