Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.1 Car Industry A3.1 Because competition is heavy, car manufacturers are cooperating with each other more and more. By doing so they lower the costs of re s e a rc h, development, and sales. In Figure 1 you can see which major car manufacturers are working together. When two car manufacturers are directly connected in the figure, it means that they are cooperating. Figure 1. Car manufacturers. Which of the graphs in Figure 2 fit the diagram above? Copy all graphs that fit and add names of manufacturers to the vertices. Figure 2. Five graphs. 4 6 9
4 7 0
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.2 A rc h a e o l o g y A3.2 page 1 of 2 A rchaeologists study old civilizations. In the past there were transitions as one civilization evolved into another. Some civilizations had more than one s u c c e s s o r. In Figure 1 you can see, for example, that civilization Matri-Hawaiian evolved into the civilizations: Bi-Eskimo, Patri-Eskimo, and Normal Hawaiian. From these three, only Patri-Eskimo had a successor that is shown: Normal Eskimo. The figure can be read like a matrix: a black box means that there was a direct succession; a white box means that direct succession was impossible. Figure 1. Several civilizations and their successors. This information can also be represented by a graph. The graph in Figure 2 represents just a part of the information in Figure 1. Copy and complete the graph by using all the information in Figure 1. Figure 2. The beginning of a civilization graph. 4 7 1
4 7 2
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.3 Joints or Links A3.3 Mechanical engineers sometimes use graphs to describe how parts of machinery are connected. Figure 1 is a locking pliers, a simple example of such machinery. Figure 1. A simple mechanical linkage. The locking pliers shown above have links 1, 2, 3, 4, 5, and 6 and joints a, b, c, d, e, f, and g. As you see in the figure, every joint belongs to two links. There are two different ways to make a graph of this situation: (1) use the joints as vertices, and (2) use the links as vertices. 1. Draw the graph with the joints as vertices. An edge indicates that the joints belong to the same link. 2. Draw the graph with the links as vertices. The joints are now edges. 3. The figure has six links and seven joints. So you may expect to find the same numbers (6 and 7) in the graphs, because vertices and edges represent links and joints. Check your graphs from Items 1 and 2 for the numbers of edges and vertices. Explain your findings. 4 7 3
4 7 4
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.4 F rom Town to To w n A3.4 page 1 of 4 Building new houses in a town can attract people from other towns (this is called migration). Local and regional governments need to predict, as p recisely as possible, how many people will move from one town to another. A software company offers a program to help governments make these predictions. They guarantee that this program, which uses a simulation, gives good predictions. You are assigned to look at the results of the program to see whether it does what is promised. The simulation is based on some rules. Two of these rules are: Rule one. The migration from town X to town Y depends on the size of the population of X: a larger population leads to more migration (if there are enough houses available in Y). Rule two. The migration also depends on the distance between X and Y: a larger distance leads to less migration. 1. Without knowing the exact numbers that go with the two rules, it is not possible to draw an exact graph. Nevertheless, you can draw a qualitative graph describing the behavior in rules one and two. Draw the qualitative graphs for rules one and two in Figure 1. Figure 1. 4 7 5
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.4 page 2 of 4 Your first test on the program involves three towns, A, B, and C (Figure 2). Figure 2. You feed the program with the necessary numbers. The exact numbers are not important, but they should show: a) that A and C are big towns with about the same population. b) that B is a small town, with many new houses. c) that the distances AB and CB are about the same. The program gives you the result presented in a graph in Figure 3. Figure 3. 2. Describe the meaning of this graph. 3. Remember: it s your job to test the program, which should be operating according to rules one and two (stated above). Discuss whether the rules for the program can be used to explain the output described below: a) There is no edge between A and C. b) 256 units (individuals or families) move from A to B, and eight move from B to A. c) 260 units move from C to B, and 256 move from A to B. 4 7 6
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T For a more challenging test of the program, a fourth town (D) is added (Figure 4). A3.4 page 3 of 4 Figure 4. The graph that is presented by the program is shown in Figure 5. Figure 5. 4. The following questions may have some overlap. a) Use rules one and two to explain the migration to and from D. b) The migrations among A, B, and C have changed considerably from the previous simulation. Use rules one and two to explain these changes. (Of course you should consider the influence of D.) 4 7 7
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.4 page 4 of 4 5. For a final test of the program, you ask a colleague to add a fifth town (E) to the four that are already loaded in the program. Your colleague feeds the program with the numbers (population size and distances to the other towns). Now you must interpret the graph that is generated by the program (Figure 6). Identify the location of town E and estimate its population. If your interpretations match the numbers that were used by your colleague, you will accept the program as a useful tool. Figure 6. 4 7 8
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.5 M i k a d o A3.5 page 1 of 3 Mikado (also known as Pick Up Sticks) is a game played with many colore d sticks. The sticks are spread criss-cross on the table. A player has to try to pick up sticks out of this mess without moving any of the other sticks. The color of the stick indicates its value. The player who gains the most points (the sum of the values of all the sticks gathered) wins the game. Figure 1 shows a messy heap of sticks. In Figure 1 it is unclear whether a stick lies on top of or under another stick. Figure 2 shows the convention used for drawing two sticks in which stick b lies on top of stick a. The positions of two sticks can also be represented in a graph. The vertices represent the sticks. The edges indicate that a stick is on top of the stick being pointed to by the arrow. Thus the graph is a directed graph. So the situation in Figure 2 is represented by the graph in Figure 3. For two sticks there are only two possibilities: either the two sticks are lying loose, or one stick is lying on the other. For three sticks there are more possibilities. In Figure 4, you see a graph that represents the position of three sticks and a real picture of the three sticks, lying on the table. 1. In Figure 4, the graph and the real position don t tell the same story. Explain. Figure 1. Figure 2. Figure 3. 2. Draw the graph that fits the real position in Figure 4. Also draw a real position that fits the graph in Figure 4. Figure 4. Real position. 4 7 9
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.5 page 2 of 3 If you don t use the letters a, b, and c in Figure 4, the graph and the re a l position on the table do match. Without the letters, the graph simply gives the message: one stick is on top of another, and a third stick is on top of both of the first two. This message also describes the real position on the table. So, without using letters, the two graphs in Figure 5 describe the same situation. Figure 5. 3. If you don t use letters for the sticks, there are seven different ways to arrange three sticks on the table. Find all seven different possibilities for three sticks. For every possibility, draw both the graph and the real position on the table. Suppose you are one of the players and it is your turn to gather sticks f rom the table. Remember that you are only allowed to pick up a stick without moving another one. So you cannot pick up a stick that is lying under another stick. The sticks are lying on the table as shown by the graph in Figure 6. Figure 6. 4. Which of the six sticks can you pick up first? 5. Explain why it is impossible to pick up all six sticks from the table. 6. By changing the direction of only one arrow, it is possible to pick up all six sticks. Which arrows can be considered for this? 4 8 0
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T Nine sticks are lying on the table. The graph in Figure 7 describes the s i t u a t i o n. A3.5 page 3 of 3 7. In which order can these nine sticks can be picked up from the table? Some hints: a) Look carefully at the direction of the arrows at the vertices. b) When a stick is picked up, draw the graph again without the vertex that belongs to that stick. 8. Describe an algorithm for the order in which you can pick up the sticks in any given situation. Make sure that somebody who wants to use your algorithm clearly understands what to do. (Use the back of this page for your answer.) Figure 7. 9. Are you sure that somebody who uses your algorithm will pick up the sticks in Figure 7 in exactly the same order as you did? If so, explain why you are sure. If not, can you improve your algorithm in such a way that you can be sure? 4 8 1
4 8 2
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.6 The Tower of Hanoi A3.6 page 1 of 2 The Tower of Hanoi is a puzzle that was invented by a French mathematician. (See Figure 1.) It consists of a number of discs of various diameters on a peg, called peg A, with the largest disc on the bottom and the smallest disc on the top. The challenge is to transfer the whole tower, with the largest disc on the bottom and the smallest disc on top, to peg B or peg C, according to the following ru l e s : Rule one. You can only move one disc at a time. Rule two. You can never place a larger disc on top of a smaller one. The goal is to transfer the tower to another peg using the least possible number of moves. Figure 1. The Tower of Hanoi Normally, the tower of Hanoi has at least eight discs. However, you will first play with only three discs to get an understanding of how this game works. 1. Play the game. Use the table in Figure 2 to keep track of the situation after each move. You should be able to complete the transfer using no more than seven moves. Also note the location of disc 1 after every move. Later on, this will help you to design an algorithm for the best way of playing this game. 2. You played the game with three discs. Do you see anything noteworthy in your table? If yes, write down what caught your attention. Move # A B C On which peg is disc 1? Start 1 2 3 A 1 2 3 1 B 2 3 1 2 B 3 4 5 6 7 Figure 2. 4 8 3
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.6 page 2 of 2 3. Now play the game with a tower consisting of four discs. Use the table in Figure 3. You will definitely need more than six moves, so extend the table or use a blank sheet of paper and make a new table. If you need more than 15 moves, check the moves you made for a situation that appeared more than once. Move # A B C On which peg is disc 1? 0 1 2 3 4 - - A 1 2 3 4 1 - B 2 3 4 1 2 B 3 4 5 Figure 3. 6 4. You played the game again with four discs. Do you see anything noteworthy in your table? If yes, write down what caught your attention. Is it the same pattern as in Item 2? 5. Write an algorithm for playing this game. (If you don t see the pattern in Items 2 and 4, you can play the game with five discs or more until you do.) 4 8 4
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.7 Hit the Road, Jack A3.7 page 1 of 2 The weighted graph in Figure 1 re p resents a map of roads connecting towns (distances are in miles). 1. Delete as many roads as possible without isolating any town so that the total combined length of the remaining roads is as small as possible. 2. How many roads can be deleted? 3. Look at the original graph. Somebody living in town A has to visit clients in towns B, C, D, E, and F and return home by the end of the day. What route should this person follow in order to drive as few miles as possible during this trip? Figure 1. Another map of towns and roads is given by the matrix in Figure 2. In this matrix you can see the distances between towns. For example, between P a n d Q there is a road seven miles long. You can also read from this matrix that t h e re is no road connecting S and T. 4. Draw a weighted graph of this map. Figure 2. 5. Again, delete as many roads as possible without isolating any town and so that the total length of the remaining roads is as small as possible. 4 8 5
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.7 page 2 of 2 6. a) How many roads can be deleted? b) Find the minimum spanning tree. Explain your answer. c) What is the total length of the roads in the minimum spanning tree? 7. Look at your graph from Item 4. Someone living in town P has to visit clients in towns Q, R, S, and T and wants to be home again by the end of the day. Determine which route to travel that day in order to drive as few miles as possible. 8. Someone living in town S has to visit clients in towns P, Q, R, and T in one day and wants to be home again by the end of the day. Tell this person how to travel in order to drive as few miles as possible. 9. Compare and comment on your answers to Items 7 and 8. 4 8 6
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.8 Pass or Fail A3.8 page 1 of 2 In a small country in Europe there is a four-year vocational school. Every year the same number of students enter grade 1 of this school. At the end of grades 1, 2, and 3 students go up to the next grade or (if their pro g ress was i n s u fficient) stay in their present grade. At the end of grade 4, students can graduate from school. During their career at that school, students who cannot go up to the next grade or do not pass their final exam can repeat the grade they failed. However, repeating a grade is permitted only once during their school career. Not passing a second time means that the student leaves school without a diploma. In Figure 1 you see a tree diagram that shows the school career of a student who repeated grade 3 and graduated school after five years: PPFPP. P means the student rises to the next grade or graduates. F means the student does not rise or fails at the final exam. The last letter of the school career is put in a box to show that this is the end of the student s school career. Figure 1. 4 8 7
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.8 page 2 of 2 1. The tree diagram also shows another student s school career. Describe what happened to this student. 2. Draw all possible school careers on the figure. Indicate the end of the school career by drawing a box around the last letter. (Every branch of the tree has been drawn on the figure for your convenience, but you probably won t need them all.) Assume for every student every year that the probability of rising to the next grade or of passing the final exam is 75%. So the probability of remaining in the lower grade is 25% every year. 3. For the students who are starting in grade 1, calculate the percentage of students who will graduate after four years. 4. What percentage of students beginning grade 1 will graduate in exactly five years? 4 8 8
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.9 Playing Dominoes A3.9 page 1 of 4 I n t ro d u c t i o n The game of dominoes is played with many pieces. On every piece there are two numbers, represented by dots. The numbers vary from 0 (no dots at all) to 6. Three examples of pieces that you can find in the game are shown in Figure 1. Figure 1. Each piece can be described by a pair of numbers: (1, 5) is the piece with one dot on the left side and five dots on the right side. Two pieces can be placed next to each other if they have a number in common. In Figure 1, the two pieces on the right match and can be placed on a table, connected as shown in Figure 2. Figure 3 shows another example of a correct connection and an example of an incorrect connection. Figure 2. Two questions are posed in this problem: (A)How many pieces are in the game? (B) Is it possible to connect all the pieces so that they form a closed chain? Before these questions are posed for the full game, you will study the same questions for a simpler case: the game of dominoes with a small number of pieces. Incorrect connection of two pieces. Correct connection of two pieces. Figure 3. DOMINOES 0 2 The dots on the pieces only vary from 0 to 2. There are six different pieces in this game, as shown in Figure 4. 1. If piece (1, 2) is placed on the table, you have a choice of four pieces, each of which can be connected to (1, 2). What are your choices? Figure 4. 4 8 9
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.9 page 2 of 4 2. Try to complete the chain shown in Figure 5. Explain why this is not a very good start for a chain of all six pieces. Figure 5. 3. Find a chain that uses all six pieces. A chain can be made into a closed chain if the first and last piece of the chain can be connected because they match each other. A closed chain with six pieces looks like Figure 6. 4. Can you make the chain that you found in Item 3 into a closed chain? If so, how? If not, find a chain that can be closed. Figure 6. For the questions posed so far, you were able to use a trial-and-error method because there were only six pieces in the game. The complete game of dominoes has too many pieces for a trial-and-error method. There f o re, here is another way to look at this problem that is helpful when answering questions about the complete game. The pieces in the game can be described by a graph. In Figure 7 you see a graph with vertices 0, 1, and 2. The vertices represent the number of dots on one side of a piece. The edges in the graph represent the pieces themselves. So, the edge between 1 and 2 stands for piece (1, 2). Figure 7. 5. What is the meaning of the edge that goes from 1 to 1? (An edge that connects a vertex to itself is called a loop.) 4 9 0
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T 6. To find the number of pieces used in the game, you could count the number of edges (there are six edges!). But there is also another way to find the number of pieces by reasoning instead of counting. How can you be sure (using reasoning) that this graph shows all the pieces in the game? A3.9 page 3 of 4 Building a chain of pieces as you did in Items 3 and 4 is re p resented on the graph as a walk along the edges. The small chain in Figure 3 is re p resented on the graph as a path, shown in Figure 8. 7. Copy the graph and draw the closed chain that you found in Item 4 as a path on the graph. Figure 8. 8. Use the graph to prove that the chain in Figure 5 can never be a closed chain. DOMINOES 0 6 Now it is time to investigate the original problem: (A)How many pieces are in the full game? (B) Is it possible to build a closed chain that uses all the pieces in the full game? Remember that each piece of the full game has two numbers from 0 to 6 on it. Investigate questions (A) and (B) using the graph of the full game. The beginning of the graph is drawn in Figure 9. 9. In this beginning part of the complete graph, all of the pieces of the full game that have a 0 (no dots at all) on one side are represented. How can you be sure about that? Figure 9. 4 9 1
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.9 page 4 of 4 10. Complete the graph for the remaining pieces. Now back to the original questions (A) and (B). You can use the complete graph that you drew in Item 10, representing the full game of dominoes, to find the answers. 11. How many pieces are in the full game? Explain your answer. 12. Is it possible to arrange all of the pieces of the full game into a closed chain? 4 9 2
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.10 I n t e r s e c t i o n s A3.1 0 page 1 of 2 The drawing of an intersection in Figure 1 shows six s t reams of traffic. The traffic should be controlled by lights to avoid a jam during rush hour. A good system of traff i c lights regulates which lights are red or green with the p roper timing. A possible start for the design of the system is the translation of the situation into a graph of the streams of traffic that can go together. For example, 5 and 6 can go together; 4 and 6 cannot go together; 1 and 4 also cannot go together, because there are not enough lanes to merge the streams. 1. Complete the compatibility graph in Figure 2. Figure 1. Figure 2. 2. One suggestion for designing a traffic light system is to take one stream, for example stream 1. Then identify all the streams that are compatible with stream 1. Let these streams have the same light as stream 1. Comment on this suggestion. 4 9 3
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.1 0 page 2 of 2 3. It is also possible to draw a conflict graph for the intersection. For example, 2 and 6 are in conflict; 2 and 5 are in conflict. Complete the conflict graph in Figure 3. Figure 3. 4. Both the compatibility graph and the conflict graph can be derived from the traffic situation at the intersection (Figure 4). Is it possible to draw the conflict graph from the compatibility graph without looking at the intersection? Explain. Figure 4. 4 9 4
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.11 Tr a ffic Lights A3.1 1 page 1 of 3 Figure 1 shows an intersection at which traffic lights will be placed to reduce traffic jams. Streams are compatible when they can go at the same time without causing accidents. For example, streams a and b a re compatible because they can be given green lights at the same time. The municipal works department of the local government is looking for an optimal way to give green lights to the d i ff e rent traffic streams in a cycle of 60 seconds. 1. One way to adjust the lights is by giving each stream ten seconds of green ( Figure 2). This kind of picture is called a clock diagram. You can read from Figure 2 that stream a first has ten seconds of green light, then stream b has ten seconds of green light, and so on. After a minute the cycle starts again. How long does every stream in Figure 2 have to wait at a red light? Why is this not a very wise choice? Figure 1. 2. Another way to adjust the lights is shown in Figure 3. You can see that streams a, b, and c get a green light during the first 15 seconds. a) How can you tell from Figure 3 that streams a, b, and c are compatible? b) For how long does each stream have a green light and for how long does each stream have a red light during the 60-second cycle? c) Compare this light schedule with the first one. Do you think this one is better? Why? a b c d e f a b c d e f 0 45 15 f e b a 30 d c Figure 2. f e d c b a Figure 3. 4 9 5
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.1 1 page 2 of 3 3. Design a clock diagram with a change from red to green every 20 seconds. 4. Is it possible to design a cycle with fewer than three periods? Explain your answer. How do you design these clock diagrams? And how do you design a model with the shortest total waiting time? One way to do this is by using a compatibility graph: a graph that connects two points if the s t reams are compatible (Figure 4). This compatibility graph can be divided into subgraphs that cover all the points of the graph. For example: aef, de, and bc. By choosing three subgraphs you can make a cycle of three periods. In each period, the streams of one subgraph have a green light. Figure 4. 5. a)design a clock diagram for the example in Figure 4. b) Is this an efficient model? What is the waiting time for each stream? 6. Why is it not possible to have a subgraph abf among the subgraphs you choose? 4 9 6
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T 7. You made a clock diagram as an answer to Item 3. What are the subgraph divisions used in the clock diagram in Item 3? Why is this a better use of subgraph divisions than the example in Item 5? A3.1 1 page 3 of 3 8. What are the subgraph divisions for the clock diagram in Figure 3? 9. Compare the models used in Item 2 and Item 3. Which one is better and why? 10. Another traffic situation is represented by the graph in Figure 5. As you can see, only b, d, and f are incompatible. Find an optimal way of giving green lights to the different streams. Explain your solution. Figure 5. 4 9 7
4 9 8
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.12 P re d a t o r- P rey 1 A3.1 2 In ecological systems, animals eat other animals. These are called pre d a t o r- p rey relationships, and they are part of the food chain. In nature there are many species that eat other species, so there exists a large, very complicated food chain. In this problem, however, you will study a small food chain. A, B, C, D, E, F, G, and H are the names of the eight species involved. B H means B eats H, so B is the predator and H is the prey (Figure 1). 1. Create a matrix to represent this food chain. Explain what the numbers in your matrix mean. Figure 1. 2. Ecologists reduce the size of the matrix by deleting all rows and columns having numbers that indicate that no predator-prey relationship exists. Reduce the matrix you made in Item 1. 3. The competition between predators can be represented in a different kind of graph. The vertices of this graph are the predators. When two predators have a prey in common, there is an edge between their vertices. Draw this graph. 4 9 9
5 0 0
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.13 P re d a t o r- P rey 2 A3.1 3 page 1 of 2 Some animals eat other animals to survive. If species A eats species B, A is the predator and B is the pre y. The food chain goes from A to B: A B. Consider the eight species in the matrix representation in Figure 1. A 1 in the matrix indicates that the predator eats the prey. For example, predator A eats prey B (row 1, column 2). 1. Column A and row G consist of only zeroes; there is not a single 1. Explain what this means for the species A and G. Figure 1. 2. A food chain goes from A to B and from B to G. A B G. This food chain is called A-B-G. What is the longest food chain in the food web represented by Figure 1? 3. Two animals are competitors if they have at least one prey in common. Identify all of the pairs of competitors that exist in this food web. 5 0 1
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.1 3 page 2 of 2 4. A toxic material is dumped in this environment. Some species store this poison in their bodies. If they are eaten, they pass the poison to their predator. No matter which species comes into contact with the toxic material first, in the end the poison will reach species A. Explain why this is true. 5. It is possible to draw a graph to represent the food web in Figure 1. Add the letters A through H to the vertices of the graph in Figure 2, and place arrows on the edges to represent the direction of the food chain. Figure 2. 5 0 2
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.14 S n o w b o u n d A3.1 4 page 1 of 3 A small town has only one snow plow. The maintenance department wants to find a route for the plow to remove snow from all the streets without having to retrace any street it has already plowed. But no one knows if there is such a route. So they decide to experiment with a few layouts of simpler towns. 1. Help them solve that problem for each of the small towns shown in Figures 1 (a) (l). Is there a route passing through each street exactly once and returning to the starting point? (Such a route is called an Euler circuit.) Keep a record of which of the towns in Figure 1 do have an Euler circuit. If there is an Euler circuit, what point (or points) could be used as the starting point? a) b) c) Figure 1. d) e) f) g) h) i) j) k) l) 5 0 3
A S S E S S M E N T Unit 3: HIDDEN CONNECTIONS Mathematics: Modeling Our World A3.1 4 page 2 of 3 2. Look back at those towns in Item 1 that do have Euler circuits. Try to find a property that all the vertices of all the graphs have in common. Use that property to predict whether each of the graphs in Figures 2 (a) (d) has an Euler circuit. Then check whether your prediction is right. a) b) Figure 2. c) d) 3. How do you know that none of the graphs in Figures 3 (a) (d) has an Euler circuit? Show how to add exactly one edge to each of the graphs so that the new graph does have an Euler circuit. (It is permissible to add a second edge between a pair of vertices that are already connected.) a) b) Figure 3. c) d) 5 0 4
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T 4. Euler circuits are named in honor of the great Swiss mathematician Leonhard Euler, who solved a famous problem about the town of Königsberg, which is built upon the two banks of a river and the two islands in between. (Figure 4) A3.1 4 page 3 of 3 Figure 4. The seven bridges of Königsberg. The problem was to find a way to stroll though the city crossing each bridge exactly once and return to the starting point. Euler solved the problem for them. Use a graph to explain his answer. 5. The graph in Figure 5 contains an Euler path (not circuit). An Euler path passes through each edge, but does not have to start and end at the same vertex. How can you tell just by looking at the graph where the Euler path starts and ends? Trace the Euler path. Figure 5. 5 0 5
5 0 6
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.15 E u l e r s House, Version A A3.1 5 Figure 1 is the floor plan of a house with five rooms (I to V). Between ro o m s you can see openings through which you can walk from one room into the o t h e r. From every room you can also step outside through an opening. Figure 1. The floor plan of a house. 1. Is it possible to design a path through the house so that you go through each opening (between two rooms and between a room and outside) exactly once? Explain. 2. By closing one of the 12 openings, you can design such a path. It is not necessary to start and finish in the same room. Choose an appropriate opening to close and describe the path. 5 0 7
5 0 8
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.15 E u l e r s House, Version B A3.1 5 Figure 1 is the floor plan of a house with five rooms (I to V). Between ro o m s you can see openings through which you can walk from one room into the o t h e r. From every room you can also step outside through an opening. Figure 1. The floor plan of a house. 1. Make a graph of this situation, in which outside is also treated as a room. 2. Use the graph to explain why it is not possible to design a path through the house so that you go through each opening (between two rooms and between a room and outside) exactly once. 3. By deleting an edge in the graph, it is possible to design such a path. Choose an appropriate edge to delete and design the path. Draw this path on the floor plan in Figure 1. What does deleting an edge in the graph mean on the floor plan? 5 0 9
5 1 0
Mathematics: Modeling Our World Unit 3: HIDDEN CONNECTIONS A S S E S S M E N T PROBLEM A3.16 Is it Legal? A3.1 6 The matching problems in Lesson 6 involved situations in which individuals had pre f e rences. Some matching problems involve situations in which individuals have qualifications. For example, a law firm assigns a lawyer to a case only if the case involves an area of the law in which the lawyer has expertise. The incidence matrix in Figure 1 shows which lawyers are qualified to work on cases A G. A 1 means that the lawyer is qualified A B C D E F G Beth 1 0 0 0 0 1 0 Carl 0 0 0 0 1 1 0 Jack 0 0 1 1 0 0 0 Annette 1 0 0 0 1 0 0 Amalia 1 0 0 0 0 0 1 Paulette 0 0 1 1 0 0 0 Figure 1. Terri 0 1 0 0 0 0 1 1. Use a bipartite graph to represent this situation. 2. A complete matching is one in which every person and every case is part of a pair. Is there a complete matching for this situation? 5 1 1
5 1 2