The natural next question after my initial lukewarm response to the show and my later diatribes is, "Okay, Adam, if you're so smart, let's hear some good math ideas from you."

     I could parry twice.

     First, the guys who wrote the show are professional writers with professional consultants and they're supposed to come up with good ideas. Even though I'm a professional mathematician and a reasonably well-renowned teacher (not known for my modesty, alas), writing television drama isn't my forté and these are the people who are supposed to come up with good shows.

     Second, the show is walking a very thin line. The math in the show has to satisfy three criteria:

(A) The math has to be useful for solving crime.
(B) The math has to be non-obvious to the FBI.
(C) The math has to be accessible to a television audience.

     Okay, there's some wiggle room here for the audience to suspend disbelief. Do I really believe the difference between pseudo-random and truly random numbers is a big element in detecting serial killers? Do I really believe the FBI didn't think of that? Do I really believe the typical television viewer really got Charlie's explanation? No, no, and not really, but enough to make a good forty-three-minute drama. (That's how long the shows are when you filter out the commercials.)

     If the pilot could come up with some good examples, then maybe I can, too. So far I only have two I like, but maybe somebody else can make good plots of the ideas below where I couldn't.

     Somebody is smuggling diamonds from Holland to Saudi Arabia and the FBI suspects it's an art shipper bringing them in with paintings into New York City and then out to Saudi Arabia. The only catch in the analysis is that none of the paintings coming from Holland are going to Saudi Arabia. So how could they be involved with smuggling if there's no connection between the paintings coming in from one place and going out to the other?

     In my proposed episode of "NUMB3RS," Charlie points out that movement of paintings is a sequence of events in a "state space" where the state of a painting is its location. If there were no connection between inbound paintings and outbound paintings, then the movement of paintings would be a "Markov chain," a state-space sequence where each move is dependent only upon the current state with no knowledge of prior states, a "memoryless" process.

     18% of the paintings leaving New York are going to Saudi Arabia. If painting movement were truly independent, truly a Markov chain, then 18% of the paintings coming from Holland that went anywhere else would go to Saudi Arabia. The fact that all of the inbound Holland paintings went somewhere other than Saudi Arabia, and all the outbound Saudi paintings came from somewhere other than Holland, tells us that the shipping sequence is not a Markov chain. The two shipping routes must be connected. Charlie could draw a diagram with circles and arrows to show the state space and its transitions.

     Whoever is managing the shipments from Holland must be connected to whoever manages the shipments to Saudi Arabia, provable by the absence of common paintings, and Charlie helps the FBI find the smugglers.

     The FBI is solving some kind of serious crime, maybe a string of robberies (not always a serial killer) and they have a profile of the criminals with a set of low-likelihood alternatives. Initially they go for the high-probability profile, say Asian gangs, because it has the highest likelihood of success.

     Over the course of the investigation, facts come to light that make the initial profile less and less likely. So far this is what you expect to happen in a television drama where the crime fighters chose the most probable criminal type. But the FBI now has several originally-rejected profiles and no obvious way to select their new profile.

     But here's the cool part. As the initial probabilities are cast into doubt, Charlie is able to build a Bayesian probability model which has joint probabilities of the various observations with the profile assumptions. The same lack of neighborhood specificity of the crimes that rules out Asian gangs also reduces the chance of Mexican gangs but leaves Black gangs or a white fencing ring as options, but the choice of consumer electronics being stolen points toward gangs rather than fences. So Charlie can determine the most likely alternative as each piece of factual evidence comes in. He could display some combinations with Venn diagrams to explain the Bayesian probability calculations.

     My primary field is optimization, finding ways to things better. Most of those fall flat on their faces when I try to find crime-fighting solutions simply because the FBI isn't limited the same way my employers have been. Maybe a clever script writer could find a way to use these ideas, but the constraints that make the problems mathematically interesting seem like hokey plot twists when I try to apply them to fighting crime. I use these methods and similar techniques frequently in my work as an industrial mathematician making business run better.

     Trying to adapt any of these for the information highway in these times of high-speed, low-cost data transfer seems even more contrived. A computer geek can send an entire episode of "NUMB3RS" to a friend using a file-copy over the Internet, who is going to notice the extra cost of an unnecessary link in a spreadsheet transfer that is part of a crime?

     Shortest Path: Given a network of nodes (think airports, warehouses, waypoints) and edges between them (think flights, routes, connections) with some kind of cost (think time or money), we can find the lowest cost path from one node to another. The criminals have to get from Los Angeles to the Seychelles in a hurry, minimum elapsed time, and their only option is airline flights. Then Charlie might calculate the clear-cut minimum going through Miami and have the FBI waiting there to apprehend the bad guys.

     Hamiltonian Path: Again, consider a network where, this time, somebody has to visit each node exactly once, the so-called "Traveling Salesman Problem." (I have been a traveling salesman for a summer and I can tell you we didn't use any fancy math to figure out where to go next.) The crooks have to visit a set of places, warehouses by truck, cities by air, and, for some plot-twist reason, can't go to the same one twice, so Charlie figures out the route they would have to take. There are also "Eulerian Paths" where each edge has to be traversed exactly once, mathematically interesting but not well suited to a crime story that I can see.

     Work Flow Optimization: A whole bunch of things have to get done but there is a complicated set of rules telling which have to be done before others can start. For example, you can't start the assembly of a product until all the parts are finished. The crime involves such a complex sequence of events that the FBI thinks it will take a month to get it all done and Charlie sees a way to optimize all the work into an elapsed time of two weeks and uses the specific time and place of one of the steps to figure out how to find the bad guys.

     Decision Choice Analysis: Maybe there's a better name for this, but it comes up in user-interface design where somebody has to make choices to run some software and my associates have shown designers that three yes/no choices can't possibly articulate nine options.

23 = 8

This kind of logic comes up in those logic puzzles where you have, for example, twelve coins one of which is fake, not the same weight as the others, and you have three weighings on a balance to determine which is fake and which way, heavy or light. Maybe there are nine possible crime scenarios and somehow the FBI only gets to ask three informants yes/no questions about it, but that sure sounds like a goofy, contrived plot line from here. Maybe Charlie could deduce that the facts the FBI knows the criminals have couldn't be enough to determine what they knew, so they had to know something more and it had to be an inside job.

     There are some plots that have been terribly overused in this and other crime shows. If asked for recommendations about plots for the show (fat chance of that), then I would exhort the producers not to use these story lines. (They would be beating the proverbial dead horse.)

     Geek-to-geek challenge: The criminal is another mathematician seeking to challenge Charlie by leaving clues only Charlie would figure out, perhaps related to his published academic research papers. Tension is increased as the perpetrator gives specific time limits after which victims held hostage will die horrible deaths.

π = 3.14159265358979323...

e = 2.71828182845904523...

Having the smart protagonist face a personal challenge from a criminal was tiresome when they did it in "CSI" or having Hannibal Lechter torment Clarisse Starling so much that she changed from Jodie Foster into Julianne Moore. When they finally catch the evil bastard, maybe it turns out to be Charlie's mentor, Professor Larry Fleinhardt, angry at losing his protogé to FBI work and excusing his crimes by pointing out that all the victims were failing his course and wouldn't contribute much to theoretical physics anyway.

     Super statistical detection: The idea that mathematical or statistical analysis can see things invisible to mere mortals is cute and sometimes even true, but it's a tiresome plot.

μ = Σ xj / n

σ2 = Σ (xj−μ)2 / n

If Charlie had used seven radar returns to see an airplane none of them could resolve individually, then I wouldn't have minded, but getting photographic resolution is just plain silly. Sure, I have seen forecasters predict an election from one vote (they were right and their reasons were sound), but those were teams of subject-matter experts with years of experience. Just being a math whiz doesn't give you that kind of insight.

     Expert at everything: This is a nice segue to the show's failure that frustrates me the most. Charlie changed from a math genius to an expert on everything. He recognizes an artist from her fake ten-dollar bills, knows all about radar noise reduction, and figures out how to catch a fugitive, more than seasoned experts in each case. Math isn't magic, just a different and more comprehensive way to look at solving problems. It's more about asking the right questions than coming up with snotty, know-it-all answers.

     I did my Ph.D. work in this area of topology and used it to solve multi-dimensional systems of polynomial equations. In spite of my familiarity with it, I can't yet think of a way to solve crime with homotopy theory or any other topology I can recall. Sorry.

     I don't have a lot of great plots for this situation drama, but that could be that I'm just not creative enough or it could be that they picked too hard a theme. In any case, they should have ended the series before going to the pompous-ass know-it-all in the middle episodes and the simple-minded, preachy plots they used in the last three of the season.

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