Episode 405 - “Robin Hood” airdate: 10/26/07
A bank for the rich and powerful is robbed but the stolen contents of the safety deposit boxes are being liquidated and donated to charities, and it’s up to Don and Charlie to uncover the true motivation behind the humanitarian thief.
| Mathematical topics | Activity concepts | Appropriate course |
| Hand-path analysis | Linear and quadratic modeling | Algebra 1, Algebra 2, Pre-calculus |
| Robin Hood Math | Substitution | Algebra 1 |
| “405 is divisible by 9″ | Divisibility rules, modular arithmetic | Algebra 1, Algebra 2 |
| Machine Learning & Recommender Systems |
Two examples are given | General |
| Fields Medal | What is the Fields Medal? | General |
Hand-path Analysis
At least three of the safety deposit boxes have been burgled and then neatly closed so the FBI can’t tell which ones have been broken into. The only clues are several smears of acetone on the boxes–a chemical by-product of the method the thief used to bypass the biometric security on the boxes. Several of the smears are clear enough to indicate what position the robber’s hand was in when he touched the wall. From this evidence, Charlie offers to do a hand-path analysis and attempt to reconstruct the robber’s path around the vault, including which boxes he stopped at. It’s a big job, even by Charlie’s standards. Charlie explains his plan to use hand-path analysis to simulate the position of the robber by knowing where his hand was by comparing the situation with that of a golfer. Charlie says, “By watching and analyzing the swing, you can tell from the point of impact exactly where the golf ball will end up without ever looking at the fairway.” The trajectory of the golf ball is shown as Charlie is talking. It ball follows a parabolic path. Students explore parabolic paths in the following two NUMB3RS activities:
- The NUMB3RS activity Fresh Air and Parabolas from episode 319 - “Pandora’s Box” is an activity involving quadratic functions, trajectories, vectors, and parametric equations. In the activity students investigate linear and quadratic systems to solve problems both symbolically and graphically.
- The NUMB3RS activity Parabolic Food Fight from episode 224 - “Hot Shot” involves the use of simultaneous equations and matrices to fit curves to data.
The NUMB3RS activity Clearing All Obstacles is also associated with that episode.
In this activity students use the TI-Navigator system to explore changes of projectile motion due to the initial angle of flight. Suppose that there are two acetone smears and that the coordinates, in cm, of the center of each of these smears on the wall are A(45, 154) and B(89, 108). Where might the box be? Ask your students to find the location of the third point, using the assumption that the three points are collinear and that they are equally spaced. Some students will attempt this problem by first finding the equation of the line through A and B. Others will realize that B will be the midpoint of the segment from A to a third point, C (or that A is the midpoint of the segment from B to a point, D). Using the midpoint formula gives the coordinates of C as (133, 62). Another way to calculate the coordinates of C is to note that the change in the x-coordinate from A to B is 44, which also will be the change in the x-coordinate from B to C. Therefore the x-coordinate of C is 89 + 44 = 133. A similar argument works for C’s y-coordinate. Suppose, now, that there are three acetone smears and that the coordinates, in cm, of the center of each of these smears on the wall are A(45, 154), B(89, 108), and C(111, 94). Where might the box be then?
Where might the box be? (89 – 45) = 44, (111 – 89) = 22.So the next step might be 11, giving an x-coordinate of 122 and a y-coordinate of 89.25.
These two explorations are examples of mathematical modeling. In mathematical modeling there is no single, correct answer. Instead, the objective is to come up with a reasonable model of a situation in order to make predictions. COMAP, the Consortium for Mathematics and Its Applications, is conducting its 2007 10th Annual High School Mathematical Contest in Modeling (HiMCM) on November 2-19, 2007.
I recall that my students, at another school, participated in HiMCM in the late ’90s and were asked to come up with a better alternative to the presidential electoral system. COMAP’s problem was prescient because the 2000 presidential election occurred soon after the contest. In that election one candidate won the national popular vote and another won the electoral vote. This became the catalyst for much discussion about the American electoral process.
Robin Hood Math
This episode is titled “Robin Hood.” Ask your students “What do you think Robin Hood mathematics might be?” Ann Valera, from Eastern New Mexico University, states that “I also do Robin Hood math – take a quantity from one equation and substitute it into another equation – when we use substitution to solve a linear system, but that’s a whole other story.” An example: Consider the linear system -
Solve the first equation for y in terms of x and use “Robin Hood math” to substitute for y in the second equation in order to solve for x.
“405 is divisible by 9″
This was episode #405 of NUMB3RS. The 4 signifies that it is a season 4 episode and the 5 signifies that it was the 5th episode that was broadcast. In the episode, the bank robber divides the proceeds from the safe deposit boxes evenly among 9 charities. That led me to observe that the number 405 is evenly divisible by 9 because the sum of its digits is a multiple of 9. 405 can be written as 4(100) + 5 = 4(99 + 1) + 5 = 4×99 + 9. Since the term 4×99 is obviously divisible by 9, then it is sufficient to check whether the other terms are also divisible by 9. Have your students write down the last four digits of their phone number and see whether they can extend this method to determine whether it is divisible by 9. Example: The last four digits of my school’s phone number are: 7322.
In the last expression, the first three terms are divisible by 9, because each term has a factor that is a multiple of 9. But because 7 + 3 + 2 + 2 = 14 is not a multiple of 9, neither is 7322. Challenge your students to also find divisibility rules for dividing by 2, 3, 4, 5, 6, 7, 8, and 10. One of the methods for determining divisibility by 7 is:
- Take the last digit in a number.
- Double and subtract the last digit in the number from the rest of the digits.
- Repeat, if necessary, until the result is a two-digit number. If that number is divisible by 7, so is the original one.
Example: With 7322, the last digit is 2. Double it (to get 4) and then subtract 4 from the first three digits (732) to get 728. Now, double 8 (to get 16) and subtract 16 from the first two digits (72) to get 56 – which is clearly divisible by 7. Therefore, 7322 is divisible by 7.
More information on divisibility rules can be found at About.com, MathIsFun, and Wikipedia. The proofs for divisibility rules often involve modular arithmetic. In elementary school, modular arithmetic is often called “clock arithmetic.” An example: If it is 5 o’clock now, what time will it be in 100 hours? Since 100 = 8×12 + 4, it will be (5 + 4) or 9 o’clock in 100 hours.
The NCTM Illuminations activities Arithme-Tic-Toc and Check That Digit are secondary school level activities which that engage students in exploring modular arithmetic.
Machine Learning & Recommender Systems
The FBI has learned the stolen money from the bank has been donated to charity – 9 charities, in fact. But what do they have in common? Why these 9? Charlie comes to the rescue with Machine Learning and Recommender systems…
Machine learning is a branch of artificial intelligence. It involves algorithms and computer programs that permit computers to learn. My credit card company also uses machine learning to monitor my purchases. A large transaction, particularly if is from out of state, can often trigger a call from the bank’s security department.
Recommender systems organize information based on the experience of the user. The DVD rental company, Netflix, can recommend movies to its clients based on their previous choices. The supermarket I shop at uses recommender systems to collect data on the purchases made by its customers. It might determine that 70% of the customers who buy soda also buy chips. Therefore it might conclude that the 30% of soda buyers who don’t buy chips are potential customers for chips. The store might move a rack of chips closer to the soda aisle or send coupons to those 30% for discounts on chips in order to sell more chips. Through the use of computers, these conclusions can be made automatically.
The NUMB3RS activity The Perfect House from Episode 311 involves conjoint analysis which is similar to recommender systems.
Fields Medal
In the episode, Charlie wants to meet Amita’s father, but is willing to wait until she is ready for him to do so. Amita tells him that the meeting may happen “after you win your Fields Medal.”
The Fields Medal is the top honor a mathematician can receive, as there is no Nobel Prize in mathematics. Every four years, the International Mathematical Union awards 2 – 4 Fields medals to mathematicians who are not over 40 years of age. The award is named for the Canadian mathematician John Charles Fields who set the age limit because of his desire that “… while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others.” It was my honor to hear Andrei Okounkov, a 2006 Fields medalist, give both a formal lecture and an informal talk at the Park City (Utah) Mathematics Institute last summer. Dr. Okounkov, originally from Russia, is a professor of mathematics at Princeton University. Charlie asks Amita “Do your really think I’ll win the Fields?” Amita replies “I do. Why else do you think I go with you?” Do your students think that Charlie will win the Fields Medal? The youngest winner was Jean-Pierre Serre (France) who won it when he was 28 years old. Only 13 Americans have won it since its inception in 1936. Earlier in the episode, Charlie tells Larry “Love is one puzzle after another.” Mathematicians love puzzles.
Cheers,
John F. Mahoney, Benjamin Banneker Academic HS, Washington, DC
