Weekly Episode Blog

This episode has a quite prominent factorial in Charlie’s boardwork. The value 28! is part of a formula Charlie uses to evaluate the threat risks for a journalist who has disappeared and may have been kidnapped. With a long list of suspects, Charlie suggests creating an Asymmetric Threat Assessment to determine motive.

Colby says they know the motive is revenge, and Charlie replies, “True. But each one has been motivated to a different degree. Think of that cool carnival game where you shoot a squirt gun in the clown’s mouth in order to fill your balloon. An Asymmetric Threat Assessment allows us to measure the factors that feed motive in each suspect in the same way the mechanism of the game measures the flow of water into each clown’s mouth. It doesn’t matter whether the suspects have the same motive or not. We only need to identify the one case in which the motive has been fed to the point where the suspect eventually pops.”

The 28! in the episode seems like product placement for a math class, offering an opportunity to talk about factorials. What does 28! mean? How large is it? What is the largest factorial a typical calculator will evaluate? How can expressions with multiple factorials be reduced? Some students have developed a misconception that the answer to every counting question is n!, for some value of n (and they usually start guessing combinations of numbers in the question). Exploring questions for which the answer is not a simple factorial is an important endeavor when studying combinatorics. Students get a chance to see that not every question has a simple formula and that understanding when to apply a formula is just as important as–if not more important than–knowing the formula.

Here is a fun question involving factorials: (3!)(5!)(7!) = n!. Find the value of n. If students find the answer quickly, ask them to write a similar question. Is there a general rule that applies? (n = 10)

On the general topic of numbers, the 28 in the episode arises from the number of suspects. But 28 is an interesting number all by itself. The factors of 28, other than itself, are 1, 2, 4, 7 and 14; and the sum of those factors is 28. This fact makes 28 a “perfect number”. Another perfect number is 6 (1 + 2 + 3 = 6). Ask students to find more. Related categories of numbers are deficient (sum of the factors is less than the number) and abundant (sum of the factors is greater than the number). Looking for patterns in these numbers is a fun investigation. Also on the topic of 28, there is a movement by some scientists to switch from a 24-hour day to a 28-hour day, with a 6-day week instead of a 7-day week.

The episode includes much discussion of green energy, specifically solar panels for the Eppes house. Consider the following. The Earth’s axis of rotation is not straight up and down. It is tilted 23.5° from vertical. On the day of the June Solstice, the sun’s rays hit the Tropic of Cancer — which is the line at 23.5° N latitude — at a right angle. That is, if you looked to the sky at noon on that day while standing on the Tropic of Cancer, the Sun would be directly overhead. (See the image below.) On the same day, at what angle would the sun’s rays hit Charlie’s home in Los Angeles? The angle of the sun is the angle that the sun’s rays make with the tangent line at a point on the Earth’s surface.

The latitude of Los Angeles is roughly 34°. Since the sun is 11° south of overhead, then the angle of the sun is 11°. On the day of the summer solstice, Charlie would be best served to have his panels at the same angle to catch the maximum amount of solar energy.

Later in the episode, David and Colby find a pad in the missing reporter’s office with a block of numbers written on it. Charlie can find no discernible pattern. Several numbers repeat in each row, but there is no indication of their significance. When Charlie learns the reporter was investigating a real estate scam, the numbers start to make more sense. The reporter listed the property ID number, square footage, and assessed value of 49 parcels of land and obscured the real data with meaningless digits.

One method to analyze large amounts of data is to use Checksum calculations (a pre-production version of the script for this episode indicates this as Charlie’s approach, but that detail may have been lost in production). Checksums are a way to detect error in data, particularly large or sensitive sets sent electronically. One example of numbers that use checksums is UPC bar codes, which are comprised of 12 digits. The first 6 digits identify the manufacturer, and the next 5 identify the product. The final digit is the check value with the following rule:

for the UPC abcdefghijkm:

m = 10 – [3(a + c + e + g + i + k) + (b + d + f + h + j)] mod 10.

For example, the UPC for the TI-84 Plus Silver Edition Graphing Calculator is 033317192069. The first 6 digits, 033317, identify Texas Instruments as the manufacturer and 19206 identifies the specific calculator. The final 9 is the check digit, which is found as follows:

10 – [3(0 + 3 + 1 + 1 + 2 + 6) + (3 + 3 + 7 + 9 + 0)] mod 10
10 – [3(13) + (22)] mod 10
10 – [61] mod 10 = 10 – 1 = 9

So if a UPC is listed as 033317192065, it must be incorrect because the check digit is wrong. For practice, find the check digit for Nintendo Wii, with manufacturer code 045496 and product code 90008 (answer is at the end of the paragraph). Check sums are also used for ISBN numbers for books, credit card account numbers and airline ticket numbers. The NCTM Illuminations website includes a lesson, “Check That Digit”, on modular arithmetic, UPC and ISBN bar codes. (The check digit for the Wii UPC is 3.)

Related activities from previous episodes are listed below.

• Checksum, from season 1 episode 102, “Uncertainty Principle”, involves checksums for bank routing numbers.
• I Never Metadata I Didn’t Like, from season 3 episode 318, “Democracy”, explores the idea of making sense of data through additional information.

Go Pats!

Kathy Erikson

This entry was posted on Tuesday, January 15th, 2008 at 2:22 pm and is filed under Weekly Episode Blogs. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.