Book summary “The Numbers Behind NUMB3RS”: summary

Introduction

You often see television detectives solve crimes through gut instinct, interrogation room drama, and dramatic shootouts. Real-world investigations demand rigorous quantitative methods to process massive arrays of evidence, spatial data, and behavioral clues. You might wonder how complex mathematics translates into practical police work without slowing down the fast-paced nature of criminal investigations.

Keith J. Devlin bridges this gap by examining the hit television series NUMB3RS, showing how real-world mathematical tools solve actual crimes. The central idea is that mathematics offers a powerful lens for uncovering hidden patterns, predicting criminal behavior, and securing justice.

The book is structured into four core areas of applied mathematics. You will explore geographic profiling, link analysis, Bayesian inference, and game theory.

• How do investigators find a serial criminal's home using spatial data?
• Can algorithms predict when a criminal's behavior changes?
• What makes public key cryptography secure?

1. Geographic Profiling and Statistics in Investigations

Spatial distributions of crimes can be mathematically analyzed to backtrack to a perpetrator's base of operations, while statistical hypothesis testing evaluates anomalies.

The mathematical applications depicted in the television series NUMB3RS are rooted in real-world mathematics. The timeline and scope of the fictional mathematics superhero Charlie Eppes are vastly exaggerated. An average NUMB3RS audience of 11 million viewers watched an episode's first broadcast during the 41 minutes that represent the duration of a fast-paced television episode.

The show's producers consult professional mathematicians to ensure the underlying math is correct. Real-world crime solving using mathematics is a slow process that requires weeks or months of labor-intensive data gathering.

In the pilot episode, Charlie uses a water sprinkler analogy and an equation to trace back crime locations to a killer's point of origin.

Geographic Profiling

Criminal geographic profiling works by analyzing the spatial distribution of crime locations to backtrack to the perpetrator's base of operations. This functions much like determining the position of a hidden lawn sprinkler from the pattern of water drops.

Serial criminals tend to commit crimes close to their home but maintain a buffer zone where they do not strike immediately around their residence. By calculating distances from crime sites using mathematical models, investigators generate concentric hot zones where the criminal is most likely to reside or work.

In the pilot episode of NUMB3RS and in real life with the South Side Rapist case in Lafayette, Louisiana, mathematician Kim Rossmo used his Rigel software to pinpoint a hot zone that ultimately led police to the culprit. Rossmo started developing the technique in the 1980s decade and scribbled the formula on a napkin on a Japanese bullet train in 1991. Geographic profiling failed in the 2002 Beltway Sniper case where 10 people were killed and 3 were critically injured.

Rossmo's formula balances two opposing distance-based phenomena to calculate the probability that a criminal lives in a specific grid square. The formula uses two fractions combined by summation. The first fraction represents the decreasing probability of crimes as distance from the base increases, using an exponent resembling an inverse square law. The second fraction represents the buffer zone effect near the criminal's residence. Tuning parameters and adjustments are required to fit past crime patterns.

You can apply the mathematical model by following these steps:

• Superimpose a grid of little squares with row and column coordinates i and j over the crime map.
• Calculate the traveling distance from the center point of each grid square to each crime location.
• Place the distance in the denominator raised to power f for the first part of the formula.
• Subtract the distance from 2B (where B defines the buffer zone size) and raise to power g for the second part.
• Sum the contributions from each crime using the Greek letter sigma to find the probability p_ij for each grid square.
• Color the map squares based on probabilities, assigning bright yellow to the highest probability hot zone.

Fighting Crime with Statistics

Hypothesis testing uses probability theory to evaluate whether an observed outcome, such as an elevated number of deaths on a specific hospital shift, is too unusual to be mere coincidence.

Statisticians establish a baseline probability of an event occurring randomly. If the likelihood of the observed anomaly happening by chance is infinitesimally small, it provides compelling circumstantial evidence to suspect foul play and justify an indictment.

In United States v. Kristen Gilbert, statistician Stephen Gehlbach calculated that the probability of having 40 out of 74 deaths occur on nurse Gilbert's shifts randomly was less than 1 in 100 million. This occurred at the VA hospital where Gilbert stopped working in the 1996 year. The analysis examined 1,641 total shifts and 74 total deaths recorded during the study period.

2. Link Analysis, Neural Networks, and Data Mining

Advanced computational tools, data-mining algorithms, and neural networks uncover hidden connections and patterns in massive law enforcement datasets.

Link analysis connects dots across disparate data points to reveal associations among criminals, terrorists, or transactions. Analysts represent linked data as a network on a computer screen. Nodes stand for individuals, organizations, or locations, while links represent transactions or relationships. You can track call logs, cash transactions, and travel records to identify key players or find previously unknown collaborators.

In the NUMB3RS episode "Brutus," Charlie Eppes analyzes gun sales data from a Nevada dealer to track gun distribution using link analysis.

Link Analysis and Neural Networks

Neural networks simulate brain circuitry through layers of connected nodes to recognize complex patterns and classify transactions or behaviors. Data flows from input layers through hidden layers to output nodes. Connection weights adjust automatically based on signal strengths and feedback during training. You can use signal strengths ranging from 1 to 100 to govern transmission values.

Trained networks often achieve a 95 percent accuracy rate. Visa Corporation used neural networks beginning in the 1993 year to reduce credit card fraud, achieving near record-low fraud rates by June 12, 2006. On that date, Visa published a full-page Washington Post advertisement noting that fraud occurred in only 1 in every 50,000 credit card transactions.

To build an effective classifier, you follow these steps:

• Present the network with a large set of historical records with known outcomes.
• Have the network make predictions on authenticity or classification.
• Reward the network with positive feedback for correct predictions and punish it for wrong ones to adjust connection weights over thousands of cycles.

Facial recognition systems translate facial features into a numerical face print to match targets against a database using neural network comparisons. Software measures distances at and between 80 nodal points on the face, allowing rapid comparison despite surface changes like beards or hats. You must photograph the subject full-face against a neutral background.

On December 25, 2004, the Los Angeles Times reported a police stop where officers used a portable facial recognition system to identify a suspect with a 94 percent probability of accuracy.

Crime Data Mining

Successful data-mining investigations require extensive human-driven preparation and manual data cleaning before advanced algorithms can be applied. Data miners spend significant time re-keying records, removing extraneous text headers, formatting structured fields, and filtering out duplicate or erroneous entries.

Dr. Colleen McCue analyzed an unpaid 37-page conference call invoice containing 2,017 initial call entries. Fewer than 5 percent of the entries contained a customer name. The total dropped to 1,047 calls after removing duplicates before applying a Kohonen neural net.

To process messy investigative logs, you execute these actions:

• Obtain an electronic copy of records in a text format and double-check against hard copies.
• Remove information not directly pertinent to the analysis, such as headers and payment details.
• Format the document into a structured form, separating area codes, phone numbers, and enhancing dates with days of the week.
• Perform a visual check to remove duplicate entries caused by repeated connection attempts.
• Submit the cleaned data to a Kohonen-style neural network for cluster analysis.

Geometric clustering uses multidimensional vector space and distance metrics to group descriptions of minor offenses, identifying crimes committed by the same individual or gang. Descriptions of perpetrators are coded into numerical variables and represented as vectors in a high-dimensional space. Euclidean distance measures similarity between points.

A study in England in 2000 and 2001 by researchers at the University of Wolverhampton and West Midlands Police analyzed 800 bogus official burglaries involving 1,292 offenders. When restricted to female distracters, the researchers analyzed 89 crimes and 105 offender descriptions.

To isolate distraction burglary patterns, you apply these methods:

• Code each of the selected variables numerically.
• Normalize the numerical variables so that each one varies between 0 and 1, or use binary encoding.
• Represent each perpetrator description as a vector in multidimensional geometric space.
• Use a Kohonen self-organizing map to reduce the multi-dimensional array of points into a two-dimensional grid.
• Examine grid squares containing several entries to identify potential gang activity.

The normalization process yielded 46 binary variables from the study data. Researchers chose a 5-by-7 grid for the Kohonen self-organizing map to group the variables.

3. Bayesian Inference and Changepoint Detection

Bayesian probability models and changepoint detection procedures process shifting data streams to track fugitives and detect behavioral changes.

Changepoint Detection

Changepoint detection uses mathematical formulas to identify sudden shifts in individual performance or patterns over time. By studying performance metrics such as long-ball hitting, aggressive play, and temper tantrums, analysts can spot when an athlete suddenly changes behavior, which historically served as an indicator of steroid use.

In the NUMB3RS episode Hardball, a high school dropout named Oswald Kittner used the Shiryayev-Roberts changepoint detection procedure to successfully track steroid use among baseball players. The steroid dosage found in the dead baseball player's locker compared to normal dosages reached thirty times more powerful.

Page's procedure provides an efficient numerical index method to detect frequency increases of specific events while balancing false alarms. By tracking an index S that updates daily using probability calculations and resets to 1 if it drops below 1, systems can detect changes rapidly without triggering excessive false positives.

Before a change, the baseline probability of an event happening on any given day is 1 out of 30. After a change, the increased probability of the event happening on any given day is 1 out of 7. You can implement this detection technique through specific calculations:

• Start with an index S equal to 1.
• On days when the event occurs, multiply S by the probability of the event assuming a change occurred, and divide by the probability assuming no change.
• On days when the event does not occur, multiply S by the no-change probability factor and divide by the baseline probability factor.
• If the new value of S is less than 1, reset S to 1.
• Declare that a change has occurred when S reaches or exceeds a pre-assigned threshold such as 50.

Using a pre-assigned threshold of 50 yields an approximate interval between false indicators of change of 1,250 days. The average maximum quickness of detection after a change occurs is thirty-three days.

Bayesian Manhunt Tracking

Bayesian statistics combines prior beliefs with new observational data using likelihood ratios to recalculate accurate probabilities.

Thomas Bayes, who lived from 1702 to 1761, developed a formula that multiplies prior odds by a likelihood ratio representing the probability of new observations under competing hypotheses. This prevents human intuition from ignoring baseline population ratios.

Consider a town with 75 black taxis and 15 blue taxis, making 90 total taxis in the hit-and-run thought experiment town. An eyewitness is right 4 out of 5 times, or 80 percent of the time. Bayes' calculation shows a hit-and-run taxi seen as blue was actually more likely to be black due to the 5-to-1 baseline population ratio, resulting in an actual probability that the taxi involved in the hit-and-run was blue of 44 percent.

You can execute this update method systematically:

• Estimate the relative prior probabilities P(A) / P(B) of two possibilities.
• Calculate the likelihood of new information observation X under both possibility A and possibility B.
• Multiply the prior odds by the Likelihood Ratio of observing X under A divided by X under B to get updated posterior odds.

Bayesian probability distribution maps can tease out hidden movement patterns from massive volumes of contradictory citizen sighting reports. By assigning prior probabilities to map grid squares and continuously updating them based on the timing, location quality, and behavior reported in each new tip, analysts can isolate the true trajectory of a fugitive.

In the NUMB3RS episode Manhunt, which broadcasted on May 13, 2005, Charlie used Bayesian statistical analysis to process hundreds of scattered public sighting reports and locate an escaped prison inmate. You can apply the same spatial tracking approach through structured steps:

• Assign initial prior probabilities to each grid square on a map based on known separation points or assume equal distribution.
• Establish time increments such as five-minute intervals for tracking.
• Receive new sighting reports containing time, location, and witness quality details.
• Update the probability of every grid square using likelihood calculations based on whether a fugitive could travel that distance in the given timeframe.
• Repeat the application of Bayes' theorem across all incoming tips until high probability concentrates on specific plausible locations.

4. DNA Profiling, Cryptography, and Game Theory

Forensic DNA matching, public key cryptography, and game theory provide rigorous frameworks for identification, secure communication, and counterterrorism.

DNA Profiling and Cryptography

The reliability of DNA matching is evaluated using probability theory to calculate the random match probability across multiple loci. The FBI's CODIS system uses a product rule of probabilities, multiplying the odds of a random match at individual sites. When thirteen independent loci match, the random match probability becomes infinitesimally small, making accidental matches virtually impossible under ideal conditions.

In the Raymond Jenkins murder case, the FBI estimated that the probability of a random person sharing Jenkins' DNA profile was 1 in 26 quintillion. The convicted offender CODIS database contains 3 million records. The random match probability for a thirteen-locus match is 1 in 10 trillion.

Public key cryptography separates encryption and decryption keys, relying on the extreme computational difficulty of factoring large composite numbers. Invented by Rivest, Shamir, and Adleman, the RSA system uses a public enciphering key made by multiplying two large prime numbers and a secret deciphering key consisting of the original primes. While multiplying large primes is computationally simple, reversing the process by factoring their product is practically impossible for computers.

In 1994, 600 volunteers distributed a quadratic sieve factorization problem across the Internet to crack the 129-digit RSA challenge number, revealing the hidden message 'squeamish ossifrage'. The RSA system uses primes of approximately 150 digits. Secure keys use composite numbers of approximately 300 digits.

Fermat's mathematical observation enables efficient primality testing without exhaustively checking every possible factor. Fermat observed that if p is prime and a is less than p, then a raised to the power of p minus 1, minus 1, is divisible by p. Using modular arithmetic and remainders, computers can apply this property to test large numbers for primality, though composite pseudoprimes can occasionally satisfy the test.

Testing the number 61 using modular arithmetic verifies that it satisfies the primality condition with zero remainder. The smallest composite number possessing the Fermat property is 341. There are 245 pseudoprimes below one million.

To test primality:

• Compute the number a raised to the power of n minus 1, minus 1.
• Check whether n divides the resulting number using modular arithmetic remainders to ignore multiples of n.

Game Theory and Operations Research

The prisoner's dilemma demonstrates how rational self-interest can lead to a suboptimal outcome for all parties, whereas cooperation yields a better collective result. Formulated as a payoff matrix, game theory shows that individual players acting rationally and selfishly will choose to defect or talk because it is a dominated strategy to trust. However, if both trust each other, the collective outcome is superior to mutual defection.

In the NUMB3RS episode 'Dirty Bomb,' Don and Charlie use the prisoner's dilemma to break the solidarity of three captured terrorists who refuse to disclose the location of a radioactive cesium 137 bomb. The payoff matrix yields a one-year sentence if neither prisoner talks, zero time for a prisoner who talks while the other stays quiet, five years for the prisoner who stays quiet while the other talks, and two years if both prisoners talk.

In repeated prisoner's dilemma games without prior communication, simple programs based on reciprocity outperform aggressive strategies. Robert Axelrod organized a computer tournament where programs played sequences of prisoner's dilemma games. The winning program, Tit for Tat, starts by cooperating and subsequently mimics the opponent's previous move, successfully punishing defection while rewarding cooperation.

Anatol Rapoport wrote the winning computer program called Tit for Tat for a 1980 tournament organized by Robert Axelrod at the University of Michigan.

In repeated interactions, you follow these rules:

• Choose trust in the first game.
• In later games choose whatever strategy the other player chose in the game before.

Operations research and queueing theory can optimize complex real-world physical systems like port container scanning without causing operational bottlenecks. Lawrence M. Wein, Yifan Liu, Zheng Cao, and Stephen E. Flynn analyzed radiation portal monitor placements at overseas ports. By adjusting detector locations and testing times based on queueing models, they improved neutron detection limits while staying within budget constraints.

At the world's second-busiest port in Hong Kong, a demonstration project deployed radiation portal monitors and automated targeting systems to scan cargo trucks. The system operates with a portal monitor placed 75 meters from the front gate. Trucks pass the monitor at a regulated speed of 10 miles per hour. A 40-foot shipping container undergoes a scan time of 3 seconds. The neutron detector has an area of 0.3 square meters and an efficiency of 0.14. The distance from the RPM to the center of the container is 2 meters. The targeting system flags 5 percent of containers for VACIS inspection. High-energy X-rays cost $250 each, while manual container inspections cost $1,500. Each RPM machine has an annualized cost of $100,000. Security requires a minimum probability of 95 percent of detecting the source level. The queue branching structure after the front gate features four lanes. Trucks experience an average processing time of 60 seconds at lane heads. Design changes achieved an overall factor of reduction in the source level of neutron radiation detected from Design 1 to Design 4 of 13.

The Kelly gambling system provides a money management rule that eliminates the risk of gambler's ruin by limiting bets to a fixed percentage of current capital. Invented by a Bell Laboratories physicist, the system dictates that you should never bet more than a percentage of your current capital roughly equal to your average percentage advantage over the casino, protecting you from losing your entire stake during temporary downswings.

Caltech graduate student Lorden studied Edward Thorp's explanations of the gambler's ruin problem and the Kelly system while exploring blackjack strategies.

When you manage funds, apply these steps:

• Determine your average percentage advantage over the casino.
• Calculate the equivalent percentage of your current capital.
• Never bet more than that calculated percentage on any single play.

10 Key Ideas

1. Criminal geographic profiling works by analyzing the spatial distribution of crime locations to backtrack to the perpetrator's base of operations.
2. Hypothesis testing uses probability theory to evaluate whether an observed outcome, such as an elevated number of deaths on a specific hospital shift, is too unusual to be mere coincidence.
3. Neural networks simulate brain circuitry through layers of connected nodes to recognize complex patterns and classify transactions or behaviors.
4. Changepoint detection uses mathematical formulas to identify sudden shifts in individual performance or patterns over time.
5. Bayesian statistics combines prior beliefs with new observational data using likelihood ratios to recalculate accurate probabilities.
6. The reliability of DNA matching is evaluated using probability theory to calculate the random match probability across multiple loci.
7. Public key cryptography separates encryption and decryption keys, relying on the extreme computational difficulty of factoring large composite numbers.
8. Link analysis connects dots across disparate data points to reveal associations among criminals, terrorists, or transactions.
9. The prisoner's dilemma demonstrates how rational self-interest can lead to a suboptimal outcome for all parties, whereas cooperation yields a better collective result.
10. The Kelly gambling system provides a money management rule that eliminates the risk of gambler's ruin by limiting bets to a fixed percentage of current capital.