The Numbers Behind NUMB3RS (28 page)

My student soon faced a dealer's 10 up-card with a total of 13 in his hand, requiring him to hit. Keeping his cool in the face of possible second-dealing, he signaled for a hit. What happened next was worthy of a scene in
NUMB3RS
. The dealer moved his hand sharply to deliver the requested card, but that same motion launched another card in a high arc above the table, causing it to fall to the floor. Fortunately, the card my student was dealt was an 8, giving him a total of 21, which not surprisingly beat the dealer's total. This dramatic scene taught three lessons: that when “dealing seconds” unskilled hands might unintentionally move the top card too much to conceal the cheating; that the second card (unseen by the cheat dealer) might turn out to be even better for the player than the top card; and finally, that it was clearly time for our Caltech heroes to cash in their winnings, leave that casino, and never go back.

A few weeks after I was let in on his secret life, my student told me that he and his classmate had ended their adventures in Las Vegas. They had earned a net profit of $17,000—pretty good in those days—and they knew it was time to quit. “What makes you think so?” I asked innocently. He proceeded to explain how the “eye in the sky” system works. Video cameras are positioned above the casino ceiling to enable the casino to watch the play at the tables. They detect not only cheating but also card counting. The casino personnel who monitor the play through the camera are taught to count cards too, and by observing a player's choices, when to bet larger and smaller amounts, they can detect pretty reliably whether or not card counting is in progress.

At one well-known casino, my student and his friend returned to play after a month's absence, using all their usual techniques to avoid being spotted as card counters. They sat down at a blackjack table, bought some quarters, and placed their bets for the first hand. Suddenly, a “pit boss” (dealer supervisor) appeared, pushed their stacks of chips back to them, and politely informed them that they were no longer welcome at that casino. (Nevada law allows casinos to bar players arbitrarily.)

When my student, feigning all the innocence he could muster, asked why on earth the casino would not want to let him and his friend play a simple game of blackjack, the pit boss said, “We figure you're into us for about $700, and we're not going to let you take anymore.” A full month after their last appearance, and for a mere $700. The casinos may depend on mathematics in order to make a healthy profit, but they cry foul when anyone else does the same.

IS THE MATH IN
NUMB3RS
REAL?

Both of us are asked this question a lot. The simplest answer is “yes.” The producers and writers go to considerable lengths to make sure that any math on the show is correct, running script ideas by one or more professional mathematicians from the hundreds across the country that are listed in their address book.

A more difficult question to answer is whether the mathematics shown really could be used to solve a crime in the way depicted. In some cases the answer is a definite “yes.” Some episodes are based on real cases where mathematics actually was used to solve crimes. A couple of episodes followed the course of real cases fairly closely; in others the writers exercised dramatic license with the real events in order to produce a watchable show. But even when an episode is not based on a real case, the use of mathematics depicted is generally, though not always,
believable
—it could happen. (And experience in the real world has shown that occasionally even “unbelievable” applications of mathematics do actually occur!) The skepticism critics express after viewing an episode is sometimes based on their lack of awareness of the power of mathematics and the extent to which it can be applied.

In many ways, the most accurate way to think of the series is to compare it to good science fiction: In many cases, the depiction in
NUMB3RS
of a particular use of mathematics to solve a crime is something that could, and maybe even may, happen someday in the future.

One thing that is completely unrealistic is the time frame. In a fast-paced, 41-minute episode, Charlie has to help his brother solve the case in one or two “television days.” In real life, the use of mathematics in crime detection is a long and slow process. (A similar observation is equally true for the use of laboratory-based criminal forensics as depicted in television series such as the hugely popular
CSI
franchise.)

Also unrealistic is that one mathematician would be familiar with so wide a range of mathematical and scientific techniques as Charlie. He is, of course, a television superhero—but that's what makes him watchable. Observing a real mathematician in action would be no more exciting than watching a real FBI agent at work! (All that sitting in cars waiting for someone to exit a building, all those hours sifting through records or staring at computer screens…boring.)

It's also true that Charlie seems able to gather masses of data in a remarkably short time. In real-life applications of mathematics, getting hold of the required data, and putting it into the right form for the computer to digest, can involve weeks or months of labor-intensive effort. And often the data one would need are simply not available.

Regardless of whether a particular mathematical technique really could be used in the manner we see Charlie employ it, however, the one accurate thing that we believe comes across in practically every episode is the
approach
Charlie brings to the problems Don presents him. He boils an issue down to its essential elements, strips away what is irrelevant, looks for recognizable patterns, sees whether there is a mathematical technique that can be applied, possibly with some adaptation, or—and this has happened in several episodes—failing the possibility of applying some mathematics, at least determines whether there is a piece of mathematics that, while not applicable to the case in hand, may suggest, by analogy, how Don should proceed.

But all of the above observations miss the real point.
NUMB3RS
does not set out to teach math, or even to explain it. It's entertainment, and spectacularly successful entertainment, at that. To their credit, the writers, researchers, and producers go to significant lengths to get the math right within the framework of producing one of the most popular fictional crime series on U.S. network television. From the point of view of good television, however, it is only incidental that one of the show's lead characters is a mathematician. After all, the series is aimed at an audience that will of necessity contain a very small percentage of viewers knowledgeable about mathematics. (There are nothing like 11 million people in the country—the average
NUMB3RS
audience at an episode's first broadcast—with advanced mathematical knowledge!) In fact, Nick Falacci and Cheryl Heuton, the series' original creators and now executive producers, have observed that what persuaded the network to make and market the program in the first place was the fascination of a human interaction of two different kinds of problem solving.

Don approaches a crime scene with the street-smart logic of a seasoned cop. Charlie brings to the problem his expertise at abstract logical thinking. Bound together by a family connection (overseen by their father, Alan, played as it happens by the only family member who actually understands quite a lot of the math—Judd Hirsch was a physics major in college), Don and Charlie work together to solve crimes, giving the viewer a glimpse of how their two different approaches intertwine and interact. And make no mistake about it, the interaction of mathematical thinking with other approaches to solve problems is
very much
a real-world phenomenon. It's what has given us, and continues to give us, all of our science, technology, medicine, modern agriculture, in fact, pretty well everything we depend upon every day of our lives.
NUMB3RS
gets that right in spades.

In what follows, we provide brief, episode-by-episode synopses of the first three seasons of
NUMB3RS
. In most episodes, we see Charlie use and refer to various parts of mathematics, but in our summaries we indicate only his primary mathematical contribution to solving the case.

FIRST SEASON

1.23.05
– “Pilot”

A serial rapist/killer is loose in Los Angeles. Don leaves a map showing the crime locations on the dining table at his father's home, and Charlie happens to see it. He says he might be able to help crack the case by developing a mathematical equation that can trace back from the crime locations to identify the killer's point of origin. He explains the idea in terms of a water sprinkler, where you cannot predict where any individual droplet will land but, if you know the pattern of all the drops, you can trace back to the location of the sprinkler head. Using his equation (which you see on a blackboard in his home at one point), he is able to identify a “hot zone” where the police can carry out a sweep of DNA samples to trace the killer.

1.28.05
– “Uncertainty Principle”

Don is investigating a series of bank robberies. Charlie uses predictive analysis to accurately predict where the robbers will strike next. He likens the method to predicting the movements of fish, describing his solution as a combination of probability modeling and statistical analysis. But when Don and his team confront the thieves, a massive shootout occurs leaving four people, including an officer, dead. Charlie is devastated, and retreats into the family garage to work out a famous unsolved math problem (the P versus NP problem) that he also plunged into after his mother became terminally ill a year earlier. But Don needs his brother's help and tries to get Charlie to return to the case. When Charlie does involve himself again, he notices that the pattern of the bank robberies resembles a game called Minesweeper. The gang uses information gathered from each robbery to choose the next target.

2.4.05
– “Vector”

Various people in the L.A. area start to become sick; some of them die on the same day. Don and Charlie are called in independently (to Don's surprise) to investigate a possible bioterrorist attack, in which someone has released a deadly virus into the environment. The CDC official who calls in Charlie says they need him to help run a “vector analysis.” Charlie sets out to locate the point of origin of the virus. Announcing that his approach involves “statistical analysis and graph theory,” he plots all the known cases on a map of L.A., looking for clusters, and tries to trace out the infection pattern. He later explains that he is developing a “SIR model” (so-called for susceptibility, infection, recovery) of the spread of the disease, in order to try to identify “patient zero.”

2.11.05
– “Structural Corruption”

Charlie believes that a college student who allegedly committed suicide by jumping from a bridge was instead murdered, and that his death is related to an engineering thesis he was working on about one of Los Angeles's newest and most important buildings, which may not be as structurally safe as the owner claims it to be. Charlie bases his suspicions on the location of the body relative to the bridge, which his calculations reveal is not consistent with the student throwing himself off the bridge. Starting with the student's data on the building, Charlie builds a computer model that demonstrates it to be structurally unsafe when subjected to certain unusual wind conditions. Suspicion falls on the foundations. By spotting numerical patterns in the company's records, Charlie determines that the records had been falsified to cover up the use of illegal immigrant workers.

2.18.05
– “Prime Suspect”

A five-year-old girl is kidnapped. Don asks for Charlie's help when he discovers that the girl's father, Ethan, is also a mathematician. When Charlie sees the mathematics Ethan has scribbled on the whiteboard in his home office, he recognizes that Ethan is working on Riemann's hypothesis, a famous math problem that has resisted attempts at solution for more than 150 years. A solution could not only earn the solver a $1 million prize, but could provide a method for breaking Internet security codes. When Don is able to determine the identity of one of the kidnappers, and learns that the plan is to “unlock the world's biggest financial secret,” it becomes clear why Ethan's daughter was kidnapped. But when Charlie finds a major error in Ethan's argument, they have to come up with a way to fool the kidnappers into believing that he really can provide the Internet encryption key they are demanding, and trace their location to rescue the daughter.

2.25.05
– “Sabotage”

A serial saboteur claims responsibility for a series of deadly train accidents. At each crash site the perpetrator leaves a numerical message, claiming in a telephone call to Don that the message tells him everything he needs to know about the series of crashes. The FBI team assumes the message is in a numeric code, which Charlie tries to crack. Charlie sees lots of numerical patterns in the message but is unable to crack the code. Charlie and the FBI team soon realize that each accident was a re-creation of a previous wreck, and eventually Charlie figures out that there is no code. The message is a compendium of data about a previous crash. Charlie says, “It's not a code, it's a story told in numbers.”

3.11.05
– “Counterfeit Reality”

A team of forgers has taken an artist hostage to draw the images to produce small-denomination counterfeit bills. The counterfeiters murder at least five people, leading Don to believe that if the missing artist isn't located soon she will be killed when she finishes her work on the phony money. Charlie is brought in to run an algorithm to enhance the image quality on some store-security videotapes relevant to the case. After studying the fake bills, he notices some flaws that appear to be deliberate, but do not seem to have any pattern. His student Amita suggests that if he looks at the image at an angle, he may be able to discern a pattern. In this way, he is able to read a secret clue, written by the kidnapped artist, that leads the FBI to the gang's location.

4.1.05
– “Identity Crisis”

A man wanted for stock fraud is found garroted in his apartment, and the crime is eerily similar to a murder committed a year earlier, a case which Don closed when an ex-con confessed. Now, Don must re-investigate the old case to determine whether he put an innocent man in jail. He asks Charlie to go over the evidence to see if he missed anything the first time around. Charlie questions the procedure used for identification of suspects from photographs and the method of using fingerprints for identification. He carries out a statistical analysis of eyewitness evidence reliability.

4.15.05
– “Sniper Zero”

Los Angeles is plagued by a spate of sniper killings. Charlie initially tries to determine the location of the sniper by calculating the trajectories of the bullets found in the victims, mentioning his use of “drag coefficient models.” By graphing the data and selecting axes appropriately, Charlie concludes that more than one shooter is at work. He suspects that the data is following an exponential curve, suggesting that there is an epidemic of sniper attacks, inspired by an original “sniper zero.” He compares the situation to the decisions of homeowners to paint their houses a certain color, mentioning the much discussed “tipping point” phenomenon. He analyzes the accuracy of the shooters in terms of “regression to the mean,” and concludes that the key pattern of sniper zero is not in the locations of the victims but in where the sniper fired the shots.

4.22.05
– “Dirty Bomb”

A truck carrying radioactive material is stolen, and the thieves threaten to set off a dirty bomb in L.A. in twelve hours if they aren't paid $20 million. While Don attempts to track down the truck, Charlie analyzes possible radiation dispersal patterns to come up with the most likely location where the bomb may be detonated to inflict the most damage to the population. However, the gang's real aim is for the FBI to evacuate an entire city square, in order to steal valuable art from a restoration facility. Eventually the FBI is able to identify and capture the three criminals, who use the threat of detonating a dirty bomb to try to negotiate their release. Observing that the isolation and individual interrogation of the three criminals is reminiscent of the so-called prisoner's dilemma, Charlie has the three brought together to present them with a risk-assessment calculation, which shows how much each has to lose. This causes the one with the greatest potential loss to come clean and say where the radioactive material is hidden.