Math 1001: Excursions in Mathematics Final Exam: 9 May :30-4:30 p.m.

Transcription

1 Math 1001: Excursions in Mathematics Final Exam: 9 May :30-4:30 p.m. Name: Section Number: You have three hours to complete this exam. There are ten problems on twelve pages, worth a total of 100 points. No books, notes, calculators, or cell phones are allowed. The appearance of any of these items during the exam will result in immediate forfeiture of the exam paper. Show your work and explain your reasoning whenever it is reasonable to do so. 1

2 Problem 1 (10 points) Indicate whether each statement is TRUE or FALSE. No justification necessary. (a) (2 points) In an election with two candidates, the plurality, Borda count, instant-runoff and pairwise-comparison methods all return the same winner. (b) (2 points) A fair share to you, if the booty is being divided among you and three other people, is never a fair share to you if the booty is being divided among you and two other people. (c) (2 points) There is a complex number whose square is 4. (d) (2 points) If a connected graph has an Euler path, then it cannot have a Hamilton path. (e) (2 points) When an honest coin is flipped five times, it is equally probable that an even number of heads will appear or that an odd number of heads will appear. 2

3 Problem 2 (10 points) Show that the sum of any 10 consecutive Fibonacci numbers is a multiple of 11. More specifically, if F N is any Fibonacci number, show that F N + F N F N+8 + F N+9 = 11(5F N + 8F N+1 ) 3

4 Problem 3 (10 points) Decide, in the following four cases, whether the given complex number is or is not in the Mandelbrot set: s = 1, s = 1, s = i, s = i. In each case, you should compute the first four terms of the number s Mandelbrot sequence, and decide whether that sequence is attracted, periodic, or escaping. 4

5 Problem 4 (10 points) Consider an election with ten candidates (A, B, C, D, E, F, G, H, I, J) and 100 voters. Suppose the results of this election are as follows: 40 voters ranked the candidates A, B, C, D, E, F, G, H, I, J; 30 voters ranked the candidates J, I, H, G, F, E, D, C, B, A; 20 voters ranked the candidates I, J, H, G, F, E, D, C, B, A; 10 voters ranked the candidates J, A, I, B, H, C, G, D, F, E. Find the winner of this election according to the Borda count method. 5

6 Problem 5 (10 points) (a) (3 points) Explain the majority, Condorcet, and monotonicity criteria for fairness. Each fairness criterion can be stated in the form if (such-and-such involving candidate X), then candidate X should win the election ; it is your task here to fill in the blanks. (b) (2 points) Give an example, involving three candidates and ten voters, showing that the plurality counting method violates the Condorcet criterion for fairness. (In this part, as well as parts (c) and (d), you need to explain why your example works.) 6

7 (c) (2 points) Give an example, involving three candidates and ten voters, showing that the Borda counting method violates the majority criterion for fairness. (d) (3 points) Give an example, involving three candidates and fifteen voters, showing that the instant-runoff method violates the monotonicity criterion for fairness. 7

8 Problem 6 (10 points) Suppose some booty has been divided into six shares, numbered 1 through 6. The six people among whom the shares are being divided have the following preferences: Only odd-numbered shares are fair to Hannah; only even-numbered shares are fair to Taylor; only shares whose number is divisible by 3 are fair to Mindy; only shares that are not fair to Hannah are fair to Marcus (that is, his list of fair shares is exactly opposite to hers); only shares that are not fair to Taylor are fair to Justin; only shares that are not fair to Mindy are fair to David. Find three different fair divisions of the booty among these six people, using the shares

9 Problem 7 (10 points) Recall that the complete graph K N has N vertices, and exactly one edge between each pair of vertices, with no multiple edges or loops. For which values of N does the graph K N have an Euler circuit? (This problem is asking for a complete description of such values of N, not merely a list of a few examples. This description can be given in one sentence, although you should then explain why it s right.) 9

10 Problem 8 (10 points) (a) (5 points) Give an example of a connected graph with six vertices such that each vertex has degree 3, and such that a Hamilton circuit exists in this graph. (b) (5 points) Assign the weights 1, 2, 3, 4, 5, 6, 7, 8, and 9 to the edges of this graph (each edge should have a different weight) in such a way that there exists a Hamilton circuit of weight 21; then find this Hamilton circuit. 10

11 Problem 9 (10 points) (a) (2 points) A pizza parlor offers four toppings pepperoni, sausage, mushroom, and Canadian bacon that can be put on a basic cheese pizza. How many different pizzas can be made? (A pizza can have anywhere from no toppings to all four.) (b) (2 points) Assume the pizza parlor now offers a fifth topping, olives. Now how many different pizzas can be made? 11

12 (c) (6 points) Five Guys Burgers and Fries offers fifteen toppings for their burgers. How many different burgers can be made? (Hint: There is a pattern to these answers, and the previous two answers should help you to guess the answer for this part. To get credit for this part, however, you must understand and explain why your answer works, and not just correctly guess it.) 12

13 Problem 10 (10 points) A prime number is a number divisible only by 1 and itself. For example, 5 and 11 are prime numbers; 4 and 9 are not prime numbers. If a pair of honest ten-sided dice (whose faces read 1 through 10) is rolled, what is the probability that the sum of the numbers rolled is prime? 13