Episode 404 - “Thirteen” airdate: 10/19/07
Episode 404, entitled “Thirteen,” revolves around Don’s team (with Charlie’s help) trying to catch a killer who leaves clues in the form of number grids at the site of each violent murder. Teachers and parents, please note that this episode contains some fairly violent scenes. Below is a discussion of the math involved, but you may want to screen the episode before discussion with students.
Fibonacci sequence
At the scene of each murder, the killer has left a message for the FBI in the form of an elaborate grid of numbers on the wall. When a similar grid is found at another crime scene, Don asks Charlie for help. Upon viewing the grids, Charlie realizes that each grids display the Fibonacci sequence with certain numbers from the sequence omitted (the omitted numbers form the phone number of the killer’s next victim). The Fibonacci sequence is a series of numbers starting 1, 1, 2, 3, 5, 8, …, where each new number in the sequence is determined by adding the previous two. While these numbers may seem to be quite ordinary, they are far from it! The number of flower petals for many plants is a Fibonacci number; and the number of opposing spirals in a pineapple and pine cone are each a Fibonacci number. In fact, many things in nature can be represented by a Fibonacci number. What is very interesting about this number sequence is that it approaches true beauty – the golden ratio, to be more precise. Try evaluating ratios of consecutive numbers in the sequence (3/2, 5/3, 8/5…). These ratios approach the number 1.61803399…, which is known as the “golden ratio,” often represented by the Greek letter Phi (?). The ancient Greeks were very familiar with this ratio and created much of their art and architecture to conform to phi (I first learned this while a student in my 3rd grade class watching the Disney classic “Donald Duck in Mathmagic Land.”)
The activities and web sites below allow students to explore more about the Fibonacci sequence.
- It All Started With a Pair of Rabbits - This activity is an exploration of Fibonacci numbers.
- Class of Gold – This activity, from the TI Activities Exchange, has students investigate Fibonacci numbers and the ratios of successive Fibonacci numbers.
- Continued Fractions - This activity, from the TI Activities Exchange, investigates the ratio of successive terms of a Fibonacci sequence.
- Learn more about how much the golden ratio is abundant in our lives.
- Read about the use of the golden ratio in art.
- Read about the use of the golden ratio in music & architecture.
- See if your face contains a golden ratio.
Cryptography
The killer’s clues become more and more complex as the information is further embedded in the number grids left at the murder sites. Charlie needs to find a way to filter out the unneeded numbers to arrive at the message left behind. To visualize this, we need to look at an interesting coding system that utilizes a variation of this technique called visual cryptography. If you look at both objects below, you will notice that they do not look like a message at all.
However, when they are placed on top of each other, the message below appears.
The activity from a previous episode and web sites below allow students to explore more about visual cryptography.
- Episode #:208 - “In Plain Sight ” - Now You See It, Now You Don’t. This activity leads students through the process of creating images like the Numb3rs image above.
- View a different hidden message using visual cryptography.
- Create your very own visual cryptography.
- Better yet, download an applet to visually encrypt at a higher resolution.
Numerology
Charlie hits a wall when trying to decode the next few number grids left by the killer. Knowing the killer’s fondness for bible quotes, Charlie enlists CalSci’s Religion and Philosophy professor, Alex Trowbridge, who suggests that the numbers might be gematria. Gematria is a branch of numerology. Numerology is the belief in a relationship between numbers and objects (the Pythagoreans were big into this). An example of numerology is called “number summing,” where letters are replaced with their equivalent number values (the letter A is 1, B with 2, etc.) and summed. Gematria uses this idea, but with the Hebrew alphabet. Charlie does not believe in numerology, but he decides to give gematria a shot. This leads to a possible encryption key, Isaiah 53 or I-53 or 153, which Amita runs through her computer and gets another code, whose result turns out to be a web address which leads to the next murder victim. To illustrate how difficult it is to decode an encrypted matrix, try decoding the following number grid.
Any idea how to decode it? Here is a hint: Once the message is decoded, use A = 1, B = 2, C = 3, etc. to find the message. Since I encoded it, I‘ll share the encryption key. It is a little different from Charlie’s. It is:
The activities and web site below, which are from previous episodes, allow students to explore more about the visual cryptography.
- Episode #: 305 - “The Mole” - Coded Messages. Decoding matrices like the one outlined above.
- Episode #:106 - “Sabotage” - Breaking the Code. Students encrypt their own messages using a variety of techniques.
- Use numerology to find your life path number.
Enumerative combinatorics – coin sorting machine
The FBI has realized the killer is picking his victims from the phone book- focusing on men with the same names as apostles. There could be tens of thousands of possible names with some variation on an apostle’s name. Amita offers to sort through the phone book digitally, with search specifications she’ll program herself. “It’s classic enumerative combinatorics,” she explains. When met with blank stares, Amita explains through an audience vision of a coin-sorting machine. “Imagine dumping in a bucket full of coins. Not just U.S. coins, but currencies from around the world. We can set the machine to sort the coins based on certain criteria- size, mass, composition…to isolate one specific group. Just like our algorithm can search the online telephone database for the one specific group of entries that satisfy the gematria criteria defined by our killer, reducing a potential target list of tens of thousands to perhaps just a few hundred.” A classic example of this is the sieve of Eratosthenes. Eratosthenes was yet another Greek mathematician. (It seems like everyone was a Greek mathematician back then!) He wanted to create a process that would find every prime number, which he did (he also came upon the concept of infinity as he did this).
- Except for 2, remove all the numbers that are multiples of 2.
- Except for 3, remove all the numbers that are multiples of 3.
- Except for 4, remove all the numbers that are multiples of 4.
- And so on
This process can be carried out to infinity and will only keep those numbers that are prime (and the number one).
For example, starting with the integers from 1 to 24, let’s find all the prime numbers.
- After removing numbers greater than 2 that are multiples of 2, we have:1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23
- After removing numbers greater than 3 that are multiples of 3, we have: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23
We can now stop, since only prime numbers are remaining. The activity below, which is from a previous episode, allows students to explore more about this topic.
- Episode #:311 - “Killer Chat” The Perfect House . This activity has students decide which houses a killer might strike at next, based on the characteristics of the house.
Encryption solution:
As outlined in the listed WAUMED activity, multiply by the inverse of the encryption matrix:
Now just substitute letters for the numbers and find out something about my school.
Happy Numb3rs,
Pat Flynn, Olathe East High School, Olathe, Kansas
