Episode 402 - "Hollywood Homicide"

In last week's episode, a series of secrets and blackmail involving a young, up-and-coming actor are exposed when Don and his team investigate the murder of a young woman in his Hollywood Hills mansion. And now for the math...

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Game Theory

Charlie and Amita compare the people in a blackmail scheme to players in a stalking game, playing with risk and response. They say "game theory gives us a model…the idea is that each player works to maximize their own return." The idea of game theory and strategy is a common theme in Numb3rs. The activities below, which are from previous episodes, allow students to explore those ideas.

Episode 201, "Assassin"

• The Escape Game explores zero sum games, including the Prisoner’s Dilemma and discusses the work of John Nash.
• Tic-Tac-Toe looks at simple strategies in slightly altered versions of this easy to play game

Episode 321, "The Art of Reckoning"

• To C or Not to C introduces different strategies in the famous Prisoner’s Dilemma
• Spies Like Us analyzes a game in which a spy is attempting to break into a fort

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Working a question backward and Snell’s Law

Charlie talks about reconstructing the details of a face from a distorted image. At one point, he compares this to breaking a gumball machine (which seems to be a favorite reference for Charlie, is there some kind of obsession there?). He talks about analyzing the forces on the gumballs and working backward to find the original placement of each gumball. "Imagine striking it with a sledgehammer. The resulting chaos might seem random, it's not. The path and position of each and every gum ball is the result of the specific forces acting on it. Gravity, velocity, momentum, mass...in fact, it's conceivable that were we able to measure and quantify each of these forces, we could use the end positions of the gum balls to completely reconstruct the original gum ball machine, with each gum ball placed exactly where it started. It's the same with this image. It's being scattered by the forces of refraction...But using Snell's law, a dash of Goos-Haenchen shifts and a motion-tracking algorithm..." Charlie hits a button, and the image becomes clear.

This idea reminds me of the sprinkler analogy in the pilot episode. Charlie traced back the path of the water droplets to determine the placement of the nozzle (with the twist of TWO nozzles, but you probably know that already). At any rate, it is a nifty little idea, and models nicely a basic mathematical setup: here are the results, what happened? The theme appeared again in a season 3 episode when Charlie needed to 'unsmear' a smudged fingerprint. Finding an algorithm to determine a clear image of a fingerprint is an example of solving an inverse problem. The activity below was written to accompany that episode. Another connection to working a question backward is approaching the solving of an equation by peeling away layers, or reversing the order of operations.

Episode 319, "Pandora’s Box"

• Thinking Backwards explores inverse functions and geometric reflections

Charlie also mentions Snell’s Law, which refers to angles of reflection and refraction. An applet showing the phenomenon can be found at http://www.walter-fendt.de/ph11e/refraction.htm

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Beauty in Mathematics

Charlie and Larry have a nice discussion in the current episode about transcendence and finding the humanity in numbers. "You can’t always find transcendence and human connections… it’s similar to the bonds between atoms…some are eternal…others are fleeting, insubstantial… potentially catastrophic."

This is a great discussion point for a class. What is mathematics? Why do we study numbers? Is mathematics discovered or developed? Do we study math for its own sake or for its applications? Greater minds than my own have debated these questions and I do not presume to have answers, but I do enjoy asking them and hearing my students’ thoughts. I usually try to give examples of applications found for mathematics many years after the math itself was recorded. The proof of Fermat’s Last Theorem is the most commonly known example, but there are others. The discussion also brings to mind the idea of beauty in math and Charlie’s monologue in a season 1 episode on math and nature. Naturally, Charlie discusses Fibonacci numbers, and an accompanying activity is below.

Episode 106, "Sabotage"

• It All Started With a Pair of Rabbits

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Archimedes’ Principle

In a fit of procrastination, Charlie is cleaning his fish tank. While removing a rock, he is reminded of the principle stating that the water level drops "by an amount exactly equal to the volume of this rock." He will apply this principal to Don’s current case, involving a bathtub drowning. By determining how much the water level in the tub dropped, Charlie can estimate the size of the second person involved. A simple calculation check can be performed using the data from the episode. Charlie reports that 89.6 liters of water were displaced. You can have students research the density of both water and humans (roughly 1 kilogram per liter for both) and the conversion from kilograms to pounds (1 kilogram = 2.2 pounds). Students can then calculate the weight of the second person (197 pounds).

Several realistic difficulties with the application are discussed here.

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Random Apollonian Networks

Charlie mentions this topic as part of his research. Apollonius was an astronomer credited for coining the vocabulary of conic sections, particularly ellipse, parabola and hyperbola. A particularly interesting visual of an Apollonian fractal is at http://numb3rs.wolfram.com/403, and an overview of conic sections with a nice visual can be found at Wikipedia.

Go Sox!

Kathy Erickson, Monument Mountain Regional High School, Great Barrington, MA