Weekly Episode Blog

Summary: After being arrested for treason and espionage in last season’s final episode “The Janus List”, Colby Granger escapes from custody. Don and his team must track down their old friend before he can escape to the Chinese. But things are not always how they appear, and Charlie must come to the FBI’s aid once again.

Mathematical topics	Activity concepts	Appropriate math course
Nash Equilibria	Discrete mathematics, matrices	Algebra 2
Set Covering Deployment	Tessellations, area of parts of a circle	Geometry
Trust Metric	Directed graphs	Algebra 2
Tokarsky’s 26-Sided Floor plan	Reflections	Geometry

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Nash Equilibria

In this episode, Charlie teaches a small Introduction to Game Theory class. The lesson is on Nash Equilibria. “A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy.” (Source)

Relevant high school level mathematics: Nash Equilibria involve the use of matrices which are usually studied in Algebra 2. The 15 – 20 minute NUMB3RS activity (see link below) gives students an introduction to the use of matrices in game theory. In it, the “payoffs” for both player A and player B are combined into one matrix.

Activities:

1. To C or Not to C, written to accompany NUMB3RS Episode 321 “The Art of Reasoning,” features the Prisoner’s dilemma an example from Game Theory which relates to Nash Equilibria.
2. The Prisoner’s dilemma can also be played online.

Additional information:

• The life of John Nash is the subject of both the book and movie, A Beautiful Mind.
• Examples of Nash equilibria

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Set Covering Deployment

In the episode, Colby escapes. Charlie offers to optimally place officers around town to catch Colby using a “Set Covering Deployment.” Charlie says, “Picture a coastline at night. To keep us from crashing on the rocks, we build lighthouses. But lighthouses are a limited resource; they cost time, money, materials. Using set covering deployment we determine the best placement of our limited number of lighthouses to illuminate the ocean.”

Relevant high school level mathematics:
The set covering deployment problem is a classical problem in computer science. It involves determining the fewest number of subsets needed to cover a set. This concept is illustrated in Activity 1 below. In Geometry, a tessellation of a plane is a collection of tiles that cover the plane with no gaps or overlaps. A regular tessellation is made of tiles that are regular polygons. Regular hexagons, squares, and equilateral triangles can each tessellate the plane. The beam from a lighthouse sweeps out a circular path. But, circles can’t tessellate the plane as there will either be gaps or overlaps. Activities 2 and 3 focus on calculating the overlap of intersecting circles.

Activities:

1. Ask your students to imagine a standard 8 × 8 chessboard. Can it be covered with dominos that measure 2 × 1? If so, how many dominos would it take? Consider removing both the upper left and lower right squares on the chessboard. How many squares remain? How many 2 × 1 dominos would it take to now cover the board?
2. One way to cover a set of points is to use overlapping circles. Consider two congruent circles which each contain the other’s center.

   

   Find the area of the overlapping region in terms of the common radius.

3. Circling Around was written to accompany NUMB3RS Episode 221 “Rampage.” Part of the activity involves calculating the area the overlapping region of two intersecting circles.

Additional information:

• Many high school geometry books contain tessellation activities.
• An interactive tessellation activity
• A lesson on tessellations

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Trust Metric

The FBI team has to decide if they trust Colby, and if so, how much. The problem is that the man on the run is not the same man who’s been a part of the team for many years. Charlie offers to clear up the problem by creating a Trust Metric. A Trust Metric, sometimes using a trust network, is a measure of how a member of a group is trusted by the other members. The trust metric can be modeled as a directed graph, with the nodes representing people and each edge representing the level of trust one member has for another.

Relevant high school level mathematics: Directed graphs are commonly studied in Algebra 2 as part of the study of networks and finite graphs. The level of trust can be thought of in terms of the probability of telling the truth. Probability is another common Algebra 2 topic. Trust metrics are used by eBay and other Web sites to provide reliable feedback on sellers and buyers of items.

In this network, Janet tells Chyna the truth 80% of the time and Shana tells Chyna the truth 90% of the time. Taylor tells Janet the truth 60% of the time and Taylor tells Shana the truth 40% of the time. Give this information to your students and ask them to come up with an estimate of the extent Chyna should trust what Taylor, indirectly, tells her. One way of estimating this is to assume that these levels of truth are independent of each other. If so, based on the path through Janet, Chyna should trust what Taylor tells her 48% of the time since (0.8)x(0.6) = 0.48. Based on the path through Shana, Chyna should trust what Taylor tells her 36% of the time since (0.9)x(0.4) = 0.36. Averaging these two percentages gives the estimation that Chyna should trust what she hears, second hand, from Taylor 42% of the time.

Activity: Students should simulate this trust network in groups of four.

• One student (Taylor in the above diagram) generates two random four digit numbers on her calculator. Consider the first of the four digit numbers to be “true”, meaning the truth, and the other to be “false”, or a lie.

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• Next, she generates a random single digit number from 0 to 9. Since Taylor tells Janet the truth 60% of the time, Taylor checks this random digit. If it is less than 6 (from 0 to 5), she writes down the true four digit number and passes it to Janet. If the random digit is 6 or higher, she instead writes down the false four digit number and gives it to Janet.
• Now, Taylor generates another random single digit number from 0 to 9. Since Taylor tells Shana the truth 40% of the time, Taylor checks this random digit. If it is less than 4, she writes down the true four digit number and passes it to Shana. Otherwise, Taylor gives the false number to Shana.
• Janet and Shana don’t know whether they have received the true or false number from Taylor. They each generate a random four digit number and a random one digit number on their calculators.

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• Since Chyna believes Janet 80% of the time, Janet examines the one digit number. If it is less than 8, she writes down the four digit number she got from Taylor. Otherwise, she writes down the random four digit number that she generated. Likewise, Shana makes a decision about which number to give Chyna based on whether her random single digit is less than 9 or not.
• Janet and Shana give their numbers to Chyna anonymously. Chyna now has two four-digit numbers, but doesn’t know which number came from which student. She generates a random one-digit number. If it is even, she believes the first number is correct. Other wise, she believes the second is correct. She then compares her choice with Taylor’s original true number.

Conduct this simulation ten times and keep track of the results. Compare them with those of your classmates as well as with the estimate of the theoretical probability.

Additional information:

• Networks and directed graphs are common topics in Algebra 2 textbooks.
• http://www.trustlet.org/wiki
• http://en.wikipedia.org/wiki/Trust_metric

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Tokarsky’s 26-Sided Floor plan

The Plus Maths web site contains an excellent introduction to the Tokarsky’s 26-Sided Floor plan. “In the 1950’s, Ernst Straus asked a seemingly simple problem. Imagine a dark room with lots of turns and side-passages, where all the walls are covered in mirrors - just like the Hall of Mirrors in an old-fashioned fun-fair. Is it true that if someone lights a match somewhere in the room, then wherever you stand in the rest of the room (even down a side-passage) you can see a reflection of the match? For example, consider an L-shaped room, with the person holding the match standing near one corner. If I stand round the corner, I can see a reflection of the match because a light ray can bounce off the two opposite walls. This is true wherever I stand in the room, so the whole room is illuminated by just one match!

An illuminable room

The question Ernst Straus asked was whether there is any room which is so complicated that there’s somewhere you can hold the match which leaves part of the room shrouded in darkness because the reflections can’t reach it. Nobody knew the answer until 1995, when George Tokarsky of the University of Alberta in Canada showed that the answer was yes - there is such a room! He published this floor plan of a room with 26 sides - the smallest such room currently known. If the match is held in just the right place, then at least one other point in the room is dark.

A 26-sided unilluminable room

But a mystery still remains, lurking in a dim corner of our imaginations. With the room Tokarsky found, there is certainly one particular place where the match can be held which leaves part of the room dark; but if you move the match slightly, the whole room is lit up again.”

Relevant high school level mathematics: Reflection is one of the basic geometric transformations which are studied in high school geometry. In the first activity below, students are asked to use the fact that the angle of incidence equals the angle of reflection to construct a series of reflections such that a light source from point A illuminates point B. The second activity relates to a chessboard problem in which queens are placed so as to “illuminate” the other squares on the board.

Activities:

1. Give your students a handout with the following diagram on it. Ask them to try to construct a “light path” from point A to point B such that the path reflects properly from the sides of the room. One such path is shown below.

   

   

2. Imagine a standard 8×8 chess board. What is the least number of queens that could be placed on the board so that any square is reachable by at least one queen? Assume that each queen can move in the standard way.

Additional information:

• Jim Parks created a Geometer’s Sketchpad^® activity to find the inscribed triangle of minimum perimeter for a given triangle ABC. He states, “It is interesting to note that this same technique has been used recently to solve a long standing problem.” (See G. W. Tokarsky, “Polygonal Rooms Not Illuminable from Every Point,” Amer. Math. Monthly, vol.102, 1995, pp. 867-879.)

Cheers,

John F. Mahoney, Benjamin Banneker Academic High School, Washington, DC

This entry was posted on Monday, October 1st, 2007 at 11:28 am 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.