How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843)

by Ellenberg, Jordan

  In this spirit, one might try to get some ideas about Buffon’s needle via the solution of the easier franc-carreau problem: “Consider a circular needle . . .” But it’s not clear what useful information one can draw from a coin, whose rotational symmetry robs it of the very feature that makes the needle problem interesting.

  Instead, we turn to the other strategy, which is the one Barbier used: make the problem harder. That doesn’t sound promising. But when it works, it works like a charm.

  Let’s start small. What if we ask, more generally, about the expected number of crack crossings by a needle that’s two slats wide? That sounds like a more complicated question, because now there are three possible outcomes instead of two. The needle could land entirely within one slat, it could cross one crack, or it could cross two. So to compute the expected number of crossings it seems we’d have to compute the probability of three separate events instead of just two.

  But thanks to additivity, this harder problem is easier than you think. Draw a dot in the center of the long needle and label the two halves “1” and “2,” like so:

  Then the expected number of crossings on the long needle is just the sum of the expected number of crossings by half-needle 1 and the expected number of crossings by half-needle 2. In algebraic terms, if X is the number of cracks crossed by half-needle 1, and Y the number of cracks crossed by half-needle 2, then the total number of cracks the long needle crosses is X + Y. But each of the two pieces is a needle of the length originally considered by Buffon; so each of those needles, on average, crosses the cracks p times; that is, E(X) and E(Y) are both equal to p. Thus the expected number of crossings of the whole needle, E(X+Y), is just E(X) + E(Y), which is p + p, which is 2p.

  And the same reasoning applies to a needle of length three, or four, or a hundred times the width of a slat. If a needle has length N (where we now take the width of a slat to be our unit of measure) its expected number of crossings is Np.

  This works for short needles as well as long ones. Suppose I throw a needle whose length is 1/2—that is, it’s just half as long as a slat is wide. Since Buffon’s length-1 needle can be split into two length-1/2 needles, his expected value p must be the twice the expected number of crossings on the length-1/2 needle. So the length-1/2 needle has (1/2)p expected crossings. In fact, the formula

  Expected number of crossings of a length-N needle = Np

  holds for any positive real number N, large or small.

  (At this point, we’ve left rigorous proof behind—some technical argument is necessary to justify why the statement above is okay when N is some hideous irrational quantity like the square root of 2. But I promise you that the essential ideas of Barbier’s proof are the ones I’m putting on the page.)

  Now comes a new angle, so to speak—bend the needle:

  This needle is the longest yet, length 5 in all. But it’s bent in two places, and I’ve closed the ends to form a triangle. The straight segments have length 1, 2, and 2; so the expected number of crossings on each segment is p, 2p, and 2p respectively. The number of crossings on the whole needle is the sum of the number of crossings on each segment. So additivity tells us that the expected number of crossings on the whole needle is

  p + 2p + 2p = 5p.

  In other words, the formula

  Expected number of crossings of a length-N needle = Np

  holds for bent needles, too.

  Here’s one such needle:

  And another:

  And another:

  We’ve seen those pictures before. They’re the same ones Archimedes and Eudoxus used two millennia ago, when they were developing the method of exhaustion. That last picture looks like a circle with diameter 1. But it’s really a polygon made out of 65,536 tiny little needles. Your eye can’t tell the difference—and neither can the floor. Which means the expected number of crossings of the diameter-1 circle is just about exactly the same as the expected number of crossings of the 65,536-gon. And by our bent-needle rule, that’s Np, where N is the perimeter of the polygon. What’s that perimeter? It must be almost exactly that of the circle; the circle has radius 1/2, so its circumference is π. So the expected number of times the circle crosses a crack is πp.

  How’s making the problem harder working out for you? Doesn’t it seem like we’re making the problem more and more abstract, and more and more general, without ever addressing the fundamental issue: what is p?

  Well, guess what: we just computed it.

  Because how many crossings does the circle have? All of a sudden, what looked like a hard problem becomes easy. The symmetry we lost when we went from coin to needle has now been restored by bending the needle into a circular hoop. And this simplifies matters tremendously. It doesn’t matter where the circle falls—it crosses the lines in the floor exactly twice.

  So the expected number of crossings is 2; and it is also πp; and so we have discovered that p = 2 / π, just as Buffon said. In fact, the argument above applies to any needle, however polygonal and curvy it might be; the expected number of crossings is Lp, where L is the length of the needle in slat-width units. Throw a mass of spaghetti on the tile floor and I can tell you exactly how many times to expect a strand to cross a line. This generalized version of the problem is called, by mathematical wags, Buffon’s noodle problem.

  THE SEA AND THE STONE

  Barbier’s proof reminds me of what the algebraic geometer Pierre Deligne wrote of his teacher, Alexander Grothendieck: “Nothing seems to happen, and yet at the end a highly nontrivial theorem is there.”

  Outsiders sometimes have an impression that mathematics consists of applying more and more powerful tools to dig deeper and deeper into the unknown, like tunnelers blasting through the rock with ever more powerful explosives. And that’s one way to do it. But Grothendieck, who remade much of pure mathematics in his own image in the 1960s and ’70s, had a different view: “The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration . . . the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it . . . yet it finally surrounds the resistant substance.”

  The unknown is a stone in the sea, which obstructs our progress. We can try to pack dynamite in the crevices of rock, detonate it, and repeat until the rock breaks apart, as Buffon did with his complicated computations in calculus. Or you can take a more contemplative approach, allowing your level of understanding gradually and gently to rise, until after a time what appeared as an obstacle is overtopped by the calm water, and is gone.

  Mathematics as currently practiced is a delicate interplay between monastic contemplation and blowing stuff up with dynamite.

  AN ASIDE ON MATHEMATICIANS AND INSANITY

  Barbier published his proof of Buffon’s theorem in 1860, when he was just twenty-one, a promising student at the École Normale Supérieure in Paris. By 1865, troubled by a nervous condition, he’d left town with no forwarding address. No mathematician saw him again until an old teacher of his, Joseph Bertrand, located him in a mental asylum in 1880. As for Grothendieck, he too left academic mathematics, in the 1980s; he now lives in Salingeresque seclusion somewhere in the Pyrenees. No one really knows what math he’s working on, if any. Some say he herds sheep.

  These stories resonate with a popular myth about mathematics—that it drives you crazy, or is itself a species of craziness. David Foster Wallace, the most mathematical of modern novelists (he once took a break from fiction to write a whole book about transfinite set theory!) described the myth as the “Math Melodrama,” and described its protagonist as “a kind of Prometheus-Icarus figure whose high-altitude genius is also hubris and Fatal Flaw.” Movies like A Beautiful Mind, Proof, and π use math as a shorthand for obsession and flight from reality. And a best-selling murder mystery, Scott Turow’s Presumed Innocent, turned on the twist that the hero’s own wife, a mathematician, was
actually the demented killer. (In this case, the myth comes with a chaser of off-kilter sexual politics, the book’s strong implication being that the difficulty of stretching a woman’s brain onto a mathematical frame is what sent the murderess over the edge.) You can find a more recent version of the myth in The Curious Incident of the Dog in the Night-Time, in which mathematical ability presents itself as just another color in the autism spectrum.

  Wallace rejects this melodramatic picture of the mental life of mathematicians, and so do I. In real life, mathematicians are a pretty ordinary bunch, no madder than the average, and it’s not actually very common for us to slink off into isolation to wage lonely battles in unforgiving abstract realms. Mathematics tends to strengthen the mind rather than strain it to its breaking point. If anything, I’ve found that in moments of emotional extremity there is nothing like a math problem to quiet the complaints the rest of the psyche serves up. Math, like meditation, puts you in direct contact with the universe, which is bigger than you, was here before you, and will be here after you. It might drive me crazy not to do it.

  “TRYING TO MAKE IT ROLL”

  Meanwhile, in Massachusetts:

  The more people played Cash WinFall, the less profitable it was. Each big purchaser who entered the game split the prizes into more pieces. At one point, Gerald Selbee told me, Yuran Lu from Random Strategies suggested that they and the Selbee group agree to take turns playing the roll-downs, guaranteeing each group a higher profit margin. Selbee paraphrased Yuran’s proposal as “You’re a big player, I’m a big player, we can’t control these other players who are fleas in our hair.” By cooperating, Selbee and Lu could at least control each other. The plan made sense, but Selbee didn’t bite. He was comfortable with exploiting a quirk in the game, since the rules of the game were public, just as available to any other player as they were to him. But colluding with other players—though it’s not clear this would have violated any lottery rules—felt too much like cheating. So the cartels settled into an equilibrium, all three pouring money into every roll-down drawing. With the high-volume bettors buying 1.2 to 1.4 million tickets a drawing, Selbee estimated that lottery tickets on roll-down days had an expected value of just 15% more than their cost.

  That’s still a pretty nice profit. But Harvey and his confederates weren’t satisfied. The life of a professional lottery winner isn’t the cartoon of leisure you might imagine. For Harvey, running Random Strategies was a full-time job, and not a particularly fulfilling one. Before roll-down day, tens of thousands of lottery tickets had to be purchased and bubbled in by hand; on the day itself, Harvey had to manage the logistics of multiple team members scanning all those slips at the convenience stores that agreed to handle the team’s megapurchases. And after the winning numbers were announced, there was still the long slog of sifting the winning tickets from the worthless losers. Not that you could throw the losing tickets in the trash; Harvey saved those in storage boxes, because when you win the lottery a lot, the IRS audits you a lot, and Harvey needed to be able to document his gambling activities. (Gerald Selbee still has twenty-some plastic laundry tubs full of losing lottery tickets, about $18 million worth, occupying the back of a pole barn on his property.) The winning tickets required some effort, too. Each member of the group had to fill out an individual W-2G tax form for each drawing, no matter how small their share. Does it sound like fun yet?

  The inspector general estimated that Random Strategies made $3.5 million, before taxes, over the seven-year life of Cash WinFall. We don’t know how much of that money went to James Harvey, but we do know he bought a new car.

  It was a used 1999 Nissan Altima.

  The good times, the early days of Cash WinFall, when you could double your money with ease, weren’t so far in the past; surely Harvey and his team wanted to get back there. But how could they, with the Selbee family and the Doctor Zhang Lottery Club buying up hundreds of thousands of tickets for every roll-down drawing?

  The only time the other high-volume bettors took a break was when the jackpot wasn’t large enough to trigger the roll-down. But Harvey, too, sat those drawings out, for a good reason: without the roll-down money, the lottery was a crappy bet.

  On Friday, August 13, 2010, the lottery projected the jackpot for the next Monday’s drawing at $1.675 million, well short of the roll-down threshold. The Zhang and Selbee cartels were quiet, waiting for the jackpot to creep up over the roll-down level. But Random Strategies made a different play. Over the previous months, they’d quietly prepared hundreds of thousands of extra tickets, waiting for a day when the projected jackpot was close to $2 million, but not quite there. This was the day. And over the weekend, their members fanned across Greater Boston, buying up more tickets than anyone had before; around 700,000 in all. With the unexpected infusion of cash from Random Strategies, the jackpot on Monday, August 16, stood at $2.1 million. It was a roll-down, payday for lottery players, and nobody except the MIT students knew it was coming. Almost 90% of the tickets for the drawing were held by Harvey’s team. They were standing in front of the money spigot, all alone. And when the drawing was over, Random Strategies had made $700,000 on their $1.4 million investment, a cool 50% profit.

  This trick wasn’t going to work twice. Once the lottery realized what had happened, they put an early-warning system in place to notify top management if it looked like one of the teams was trying to push the jackpot over the roll-down line single-handedly. When Random Strategies tried again in late December, the lottery was ready. On the morning of December 24, three days before the drawing, the chief of staff of the lottery got an e-mail from his team saying “Cash WinFall guys are trying to make it roll again.” If Harvey was betting on the lottery being off-duty for the holiday, he wagered wrong; early Christmas morning, the lottery updated its estimated jackpot to announce to the world that a roll-down was coming. The other cartels, still smarting from their August snookering, canceled their Christmas vacations and bought hundreds of thousands of tickets, bringing profits back down to normal levels.

  At any rate, the game was almost up. Sometime shortly afterward, a friend of Boston Globe reporter Andrea Estes noticed something funny in the “20-20 list” of winners that the lottery makes public: there were a lot of people in Michigan winning prizes, and they were all winning one particular game, Cash WinFall. Did Estes think there was anything to it? Once the Globe started asking questions, the whole picture quickly came clear. On July 31, 2011, the Globe ran a front-page story by Estes and Scott Allen explaining how the three betting clubs had been monopolizing the Cash WinFall prizes. In August, the lottery changed the rules of WinFall, capping at $5,000 the total ticket sales any individual retailer could disburse in a day, effectively blocking the cartels from making their high-volume purchases. But the damage was done. If the point of Cash WinFall was to seem like a better deal for ordinary players, the game was now pointless. The last WinFall drawing—fittingly, a roll-down—was held on January 23, 2012.

  IF GAMBLING IS EXCITING, YOU’RE DOING IT WRONG

  James Harvey wasn’t the first person to take advantage of a poorly designed state lottery. Gerald Selbee’s group made millions on Michigan’s original WinFall game before the state got wise and shut it down in 2005. And the practice goes back much further. In the early eighteenth century, France financed government spending by selling bonds, but the interest rate they offered wasn’t enticing enough to drive sales. To spice the pot, the government attached a lottery to the bond sales. Every bond gave its holder the right to buy a ticket for a lottery with a 500,000-livre prize, enough money to live on comfortably for decades. But Michel Le Peletier des Forts, the deputy finance minister who conceived the lottery plan, had botched the computations; the prizes to be disbursed substantially exceeded the money to be gained in ticket receipts. In other words, the lottery, like Cash WinFall on roll-down days, had a positive expected value for the players, and anyone who bought enough tickets was due for a big score. />
  One person who figured this out was the mathematician and explorer Charles-Marie de La Condamine; just as Harvey would do almost three centuries later, he gathered his friends into a ticket-buying cartel. One of these was the young writer François-Marie Arouet, better known as Voltaire. While he may not have contributed to the mathematics of the scheme, Voltaire placed his stamp on it. Lottery players were to write a motto on their ticket, to be read aloud when a ticket won the jackpot; Voltaire, characteristically, saw this as a perfect opportunity to epigrammatize, writing cheeky slogans like “All men are equal!” and “Long live M. Peletier des Forts!” on his tickets for public consumption when the cartel won the prize.

  Eventually, the state caught on and canceled the program, but not before La Condamine and Voltaire had taken the government for enough money to be rich men for the rest of their lives. What—you thought Voltaire made a living writing perfectly realized essays and sketches? Then, as now, that’s no way to get rich.

  Eighteenth-century France had no computers, no phones, no rapid means of coordinating information about who was buying lottery tickets and where: you can see why it took the government some months to catch on to Voltaire and Le Condarmine’s scheme. What was Massachusetts’s excuse? The Globe story came out six years after the lottery first noticed college students making bizarrely large purchases in supermarkets near MIT. How could they not have known what was going on?

  That’s simple: they did know what was going on.

  They didn’t even have to sleuth it out, because James Harvey had come to the lottery offices in Braintree in January 2005, before his betting cartel placed its first bet, before it even had a name. His plan seemed too good to be true, such a sure thing that there must be some regulatory barrier to carrying it out. He went to the lottery to see whether his high-volume betting scheme fell within the rules. We don’t know exactly what conversation took place, but it seems to have amounted to “Sure, kid, knock yourself out.” Harvey and company placed their first big bet just a few weeks later.