Episode 406 - "In Security"

In this episode, Charlie continues to anxiously await the release of his new book as well as his first book signing. Of the many thoughts that occupy his mind are concerns over how to sign his name and how long it will take to sign his name. Should he use an initial for "Charles," or should he use "Charlie"? Will writing his full name take too long? In his own words "I could use an adaptive algorithm adjusting the variable of time per person. Three hours, less than 1,000, but greater than…I better bring a stopwatch." It sounds like Charlie is obsessing over this point just a bit too much.

However, this seemingly simple concept might present some interesting discussion opportunities with students. Or, perhaps you could run a little experiment, throw in a bit of proportional reasoning, and determine the estimate for the time it takes to sign 1,000 books. Arrange students in pairs, having one student play the role of the author and the other student playing the role of the consumer. Each pair of students should spend 5 minutes acting out the book signing (5 minutes should allow for some handwriting fatigue to set in). To be really particular, have the person getting their book signed (or notebook) walk up to the author, hand their notebook to him or her, receive the notebook, and repeat the process until 5 minutes have passed. For this first simulation, instruct the author to simply sign their name each time.

Next, have the two students switch roles and simulate the book signing experiment for another 5 minutes. This time, however, have the book signer include personal messages with each signature such as "All the Best!" or "Best of Luck".

At the conclusion of the activity, have each student calculate the time to sign 1000 books under each scenario using the proportion

where x represents that amount of time it will take to sign 1000 books. Ask each pair of students to report their findings to the class and note any similarities or differences. Incidentally, I tried this experiment myself and came up with a range of between 2 and 4 hours depending on whether or not a phrase is used to accompany each signing.

As an extension to this activity, it might be interesting to come up with a linear model where the number of characters represents the independent variable and the length of time to sign 1000 books (in minutes) represents the dependent variable.

While Charlie is obsessing over his book signing event, Don is out on a date with a federally protected witness. Soon after dropping her off at her house, an intruder enters her home and kills her, thus setting up the premise for which the episode is based. Don is distraught, thinking that he must be responsible for leading the killer to her home. He asks Charlie to perform "that escape-radius thing you do" to help track the killer. This technique is based on the idea that, if the origin of the crime is known (the center of a circle), as well as the mode of transportation and estimated speed of flight, then the perpetrator can be expected to be within a circle whose radius is given by the product of the speed of the killer and the time since the attack. Charlie is skeptical since this technique is much less effective over time.

In fact, if you study this situation using related rates, you’ll see that the rate of change in the area of the circle that contains the perpetrator is directly proportional to the radius of the circle. For example, consider that the assailant leaves the scene of the crime by foot, traveling at 5 miles per hour. Using related rates

Using this formula, we see that, after 1 hour (when the assailant is 5 miles from the scene of the crime), the search area covers

square miles with an instantaneous rate of change in area of

These results suggest that Charlie had reason to be skeptical.

Here are some other NUMB3RS activities that have activities related to circles.

• Episode 221, "Rampage" – Activity: Circling Around. This activity focuses on using Venn diagrams and the notion of using circles or spheres to help locate the position of a suspect.
• Episode 207, "Convergence" – Activity: Two-Dimensional Trilateration. Two-dimensional trilateration is a process that uses intersecting circles to find a location, much the way GPS receivers are used to pinpoint a person’s latitude and longitude.

Later in the episode, Charlie explains how he used Decision Analysis to help determine if Don was responsible for the murder of a federally protected witness. A Charlie Vision appears:

The Vision...	Charlie Says...
A small tree sprout grows from the ground and evolves into a tree.	"Decisions all start somewhere and grow. Choices made. Until at the end, you have complex branches of decision making…good or bad. My job is to work backwards, pruning the tree by assigning values to each decision, and then compare those values. Pitting what could have happened against what actually happened.

Charlie’s vision describes the notion of using the strategy of working backwards to solve a problem. As math teachers, we constantly refer to a variety of problem-solving strategies such as working backwards; finding a pattern; solving a simpler analogous problem; making a visual representation; guessing, checking, and revising; organizing data; and using logical reasoning… just to name a few.

The concept of working backwards is a problem-solving strategy that can and should be used by students to help them find a solution efficiently. In fact, some problems require that you work backwards. Consider the problem below, which was posed by American Mensa, the high IQ society:

Jim was buying barbecue supplies for the Fourth of July. He spent half of his money, plus $2, on meat; half of what was left, plus $2, on salad fixings; then half of what was left, plus $2, on soft drinks. He then had $4 left for paper plates and other necessities. How much did he start with?

You could work forwards, by assigning the variable x to represent the initial amount of money and then solve the following equation:

Actually, it might be fun having students try to find an equation such as the one above that will yield the correct solution, x = 60.

Or, you could work backwards …

You are left with $4. Add $2 to this amount and double this sum to get $12. This is the amount Jim had before he purchased the soft drinks. Now add $2 to $12 and double this sum to get $28. This is the amount Jim had before he purchased the salad fixings. Using the same process, add $2 to $28 and double this sum to get $60. This is the initial amount that Jim had before he purchased the meat. This result can be checked easily by starting working forward with the answer of $60 to see if the final result is in fact $4.

Here’s a similar problem that combines the strategy of working backwards with the strategy of drawing a diagram to represent the situation:

My favorite aunt gave me some money for my birthday. I spent one-third of it on a new CD. I spent half the remainder to take my friend to the movies. Then I bought a magazine with half of what was left. When I went home, I still had $6. How much did my aunt give me for my birthday?"

The un-shaded portion in the fourth diagram represents the unspent money, $6. Since this represents one-sixth of the entire diagram, then we can deduce that the aunt gave 6•6 = $36 for the birthday gift.

Towards the end of this episode, Charlie is watching surveillance film of a prisoner during a visitation. He notices a piece of paper with a graffiti-type image being passed from the visitor to the prisoner and suspects that there is a secret code contained within the image. Using a morphological Image Cleaning Algorithm, Charlie is able to decipher the image, revealing the address of a planned hit.

Here are two previously written NUMB3RS activities that use a similar process called steganography to reveal a hidden message.

• Episode 208-21, "In Plain Sight" – Activity: Now You See It, Now You Don’t
• Episode 305, "The Mole" – Activity: Coded Messages

That’s it for Episode 406. Enjoy!

Steve Ouellette, Westwood High School, Westwood, MA