1/15 2/19 3/23 4/28 5/12 6/23 Total/120 % Please do not write in the spaces above.

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1 1/15 2/19 3/23 4/28 5/12 6/23 Total/120 % Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before attempting any problems. You must show all work, or risk losing credit. Be sure to answer all questions asked. NOTE: This exam is out of 120 points. To receive full credit on problems, they must not only be mathematically correct, but they must also be solved using the correct notation and terminology. The following list of all primes below 100 may or may not be helpful: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Good luck! MATH Version I Fall 2015 Dr. Morton Name: Exam I

2 1. (15 points) True/False: True or False? (Circle the correct answer.) If false, tell me why it is false. Be as specific as possible. a) True or False: G.H. Hardy was careful to spend almost every Sunday morning in Church. b) True or False: G.H. Hardy s protégé Srinivasa Ramanujan lived a happy and long life in England after G.H. Hardy brought him there from India. c) True or False: The number 2 is irrational. d) True or False: Our friend the square lived out the remainder of his life heralded a hero in Flatland, one who had brought the truth of the 3 rd dimension to the world. e) True or False: The method of proof used to prove that there are infinitely many primes is called a proof by contradiction. f) True or False: G.H. Hardy loved the game of cricket and always followed the cricket scores. g) True or False: Mobius strips have no real world applications.

3 2. (19 points) Short answer: a) Give the definition of the word Apology that was used by G.H. Hardy (one sentence maximum). b) Give two different examples of twin primes. (Note there is a list of primes under 100 on the front of this exam.) a. Example 1: b. Example 2: c) Does 9 63? Use the definition from class to clearly answer yes or no (i.e. show all work). d) Find all divisors of 100. You do not have to show your work. e) Using your answer above, find all proper divisors of 100. Again, you do not have to show your work. f) Draw one figure that has all of the following properties: it has genus three, four 3-crossings, and infinitely many cut points. (Points will be awarded for how well your figure has these three properties, but it may have as many 4-crossings and you like, and the maximum number of pieces is also up to you.) g) Color in the following map, using the correct theorem. You may number the areas rather than color them in.

4 3. (23 points) To help remind you of the patterns of figurate numbers, I am giving you the following information about adding up sequences of numbers (odds on the left, evens in the middle, and natural numbers on the right). Note: This is just to remind you of the pattern you are not supposed to do anything until part a below. 1=1 1+3= = = =25 2=2 2+4= = = =30 1=1 1+2= = = =15 a. Illustrate the sum of numbers =10 using x s and o s as a figurate number. b. Suppose we want to add up the first 645 odd numbers. What is the 645 th odd number? Show your work. What is the sum when we add up (645 th odd number)? Show your work. Illustrate your answer as a figurate number (without using x s and o s just give the shape and the dimensions.) c. Suppose we want to add up the first 645 even numbers. What is the 645 th even number? Show your work. What is the sum when we add up (645 th even number)? Show your work. Illustrate your answer as a figurate number (without using x s and o s just give the shape and the dimensions.) d. Suppose we want to add up the first 645 whole numbers. What is the 645 th number? (No work is needed.) What is the sum when we add up (645 th number)? Show your work. Illustrate your answer as a figurate number (without using x s and o s just give the shape and the dimensions.)

5 4. (28 points) a. We wish to figure out which of the following figures are homeomorphic. First, fill in the information for each figure, making sure to put 0 if the answer is zero. The last column is in case there is a symbol with a 5-crossing or 6-crossing (etc). NOTE: the first column here is for symbol number, so that when I ask for which symbols are homeomorphic, you can just give the symbol number rather than rewriting the symbol. Use the symbols themselves to fill in the table not the symbol numbers. Symbol number Symbol # of cut points Genus Max # of Pieces # of 3- crossings # of 4- crossings Any other crossings? If so, put the type here (otherwise leave blank) b. Now classify all the 6 symbols as to which are homeomorphic (i.e. tell me everything that is homeomorphic to the first symbol, to the second symbol etc). You may use the symbol numbers from the first column to tell me which are homeomorphic, rather than rewriting the symbols themselves. Also let me know which symbols are homeomorphic only to themselves.

6 5. (12 points) Suppose you and I are playing a game of nim. a) If there is one pile of 100 stones in front of us, do you want to go first or second: What is your best strategy in this case? Be specific. b) If there are two piles in front of us, both with 100 stones, do you want to go first or second: What is your best strategy in this case? Be specific. c) If there are two piles in front of us, one with 100 stones and one with 113 stones, do you want to go first or second: What is your best strategy in this case? Be specific. d) If there are 516 piles in front of us, each with 1 stone, do you want to go first or second: This space intentionally left blank

7 6. (23 points) There are seven figures that can be built using continuous motion from a rectangle: A disc, tube, sphere, torus, Mobius strip, Klein bottle, real projective plane. Answer the following (about one sentence each). a. Fill in the following information about each of the figures in the open spaces below: Figure Genus Number of sides Can all sides be touched (yes/no) Disc / Square Number of dimensions needed to build Tube/ band Sphere Mobius strip Torus Klein bottle Real projective plane b. How do we know that the Klein bottle is not homeomorphic to the tube? (Give a convincing reason.) c. How do we know that the Mobius strip is not homeomorphic to the tube? (Give a convincing reason.) d. How does continuous motion differ from what is allowed in homeomorphisms? e. What is the arrow (gluing) diagram for the Mobius strip? For the real projective plane? Carefully fill in the squares below (under the correct description) to answer: Mobius strip: Real projective plane: f. What happens when we cut a Mobius strip in ½? (Be specific.) g. What happens when we cut a Mobius strip in 1/4 s? (Be specific)