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- 1. Open EndedMuchos mangos. You inherit a large crate of mangos. The top layer has 21 mangos. Peering through the cracks in the side of the crate, you estimate there are four layers of mangos inside. About how many mangos did you inherit?
- 2. Open EndedYou have an empty CD rack consisting of 13 shelves and you just bought 13 totally kickin' CDs. Can each CD go on a different shelf? What if you had 14 new CDs?
- 3. Open EndedTwenty-nine is fine. Find the most interesting property you can, unrelated to size, that the number 29 has and that the number 27 does not have.
- 4. Open EndedSuppose you were able to take a large piece of paper of ordinary thickness and fold it in half 53 times. What would the height of the folded paper be? Would it be less than a foot? About one yard? As long as a street block? As tall as the Empire State Building? Taller than Mount Everest? (Assume that packages of 200 sheets of paper are more than half an inch thick.)
- 5. Open EndedFifteen Fibonaccis. List the first 15 Fibonacci numbers.
- 6. Open EndedBorn φ. What is the precise number that the symbol φ represents? What sequence of numbers approaches φ? Choose all correct answers.
- 7. Open EndedThis question gave the Fibonacci sequence its name. It was posed and answered by Leonardo of Pisa, better known as Fibonacci. Suppose we have a pair of baby rabbits: one male and one female. Let us assume that rabbits cannot reproduce until they are one month old and that they have a one - month gestation period. Once they start reproducing, they produce a pair of bunnies each month (one of each sex). Assuming that no pair ever dies, how many pairs of rabbits will exist in a particular month? During the first month, the bunnies grow into rabbits. After two months, they are the proud parents of a pair of bunnies. There will now be two pairs of rabbits: the original, mature pair and a new pair of bunnies. The next month, the original pair produces another pair of bunnies, but the new pair of bunnies is unable to reproduce until the following month. Thus we have: Continue to fill in this chart and search for a pattern. Here is a suggestion: Draw a family tree to keep track of the offspring.
- 8. Open EndedDiscovering Fibonacci relationships (S) We'll use the symbol F1 to stand for the first Fibonacci number, F2 for the second Fibonacci number, F3 for the third Fibonacci number, and so forth. So F1 = 1, and F2 = 1, and, therefore, F3 = F2 + F1 = 2, and F4 = F3 + F2 = 3, and so on. In other words, we write Fn for the nth Fibonacci number where n represents any natural number; for example, we denote the 10th Fibonacci number as F10 and hence we have F10 = 55. So, the rule for generating the next Fibonacci number by adding up the previous two can now be stated, in general, symbolically as: Fn = Fn − 1 + Fn − 2 By experimenting with numerous examples in search of a pattern, determine a simple formula for (Fn +1)2 + (Fn)2; that is, a formula for the sum of the squares of two consecutive Fibonacci numbers.
- 9. Open EndedSum of Fibonacci (H). Express each of the following natural numbers as a sum of distinct, nonconsecutive Fibonacci numbers: 356, 948, 610, 1289. In each case, enter the Fibonacci numbers in descending numerical order: 356, 948, 610, 1289
- 10. Open EndedSum of Fibonacci. Express the natural number 31 as a sum of distinct, nonconsecutive Fibonacci numbers given in descending order (e.g. 55+21+3).
- 11. Open EndedPrimal instincts. List the first 15 prime numbers.
- 12. Open EndedFear factor. Express each of the following numbers as a product of primes: 77, 7889, 8, 39, 5643.
- 13. Open EndedOdd couple. If n is an odd number greater than or equal to 3, can n + 01 ever be prime? What if n equals 01?
- 14. Open EndedAlways, sometimes, never. Does a prime multiplied by a prime ever result in a prime? Does a nonprime multiplied by a nonprime ever result in a prime? Always? Sometimes? Never? Explain your answers.
- 15. Open EndedTwin primes. Find the first 15 pairs of twin primes.
- 16. Open EndedFloating in factors. What is the smallest natural number that has three distinct prime factors in its factorization?
- 17. Open EndedYou own a very expensive watch that is currently flashing "7:15." What time will it read in 12 hours? In 17 hours? In 26 hours? In 336 hours? What time is it when an elephant sits on it?
- 18. Open EndedLiving in the past. Your watch currently reads "2:45." What time did it read in 24 hours earlier? Three hours earlier? Thirty-one hours earlier? What time did it read 2400 hours earlier?
- 19. Open EndedHours and hours. The clock now reads 10:45. What time will the clock read in 60 hours? What time will the clock read in 860 hours? Suppose the clock reads 07:30. What did the clock read 28 hours earlier? What did the clock read 96 hours earlier?
- 20. Open EndedCelestial seasonings (S). Which of the following is the correct UPC for Celestial Seasonings Ginseng Plus Herb Tea? Show why the other numbers are not valid UPCs.
- 21. Open EndedSpaghettiOs. Which of the following is the correct UPC for Campbell's SpaghettiOs? Show why the other numbers are not valid UPCs.
- 22. Open EndedReal mayo (H). The following is the UPC for Hellmann's 08-oz. Real Mayonnaise. Find the missing digit. 0 48001 81 ▪ 18 6
- 23. Open EndedWhat did you say? The message below was encoded using the following scheme: Decode the original message. VBV IXP VTV
- 24. Open EndedSecret admirer. Use the scheme below to encode the message "I LOVE YOU."
- 25. Open EndedSetting up secrets. Let p = 7 and q = 17. Are p and q both prime numbers? Find (p − 1)(q − 1), the number we call m. m = Now let e equal 5. Does e have any factors in common with m? Finally, verify that 77e − 4m = 1. 77e = 4m = 77e − 4m =
- 26. Open EndedCreating your code (S). Suppose you wish to devise an RSA coding scheme for yourself. You select p = 3 and q = 5. Compute m, and then find (by trial and error if necessary) possible values for e and d. m = Possible values for e and d are Hint: m is calculated as m = (p − 1)(q − 1). e must be relatively prime to m. For each possible value of e, find d and y that satisfy de − 8y = 1.
- 27. Open EndedA rational being. What is the definition of a rational number?
- 28. Open EndedFattened fractions. Reduce these overweight fractions to lowest terms: 8/40, 25/15, -12/84, 38/4, -196/14
- 29. Open EndedDecoding decimals. Show that each of the decimal numbers below is actually a rational number by expressing it as a ratio of two integers. 0.67 5.98 6.19355 -336.3 -0.0004
- 30. Open EndedRational or not (ExH).For each of the following numbers, determine if the number is rational or irrational. Give brief reasons justifying your answers.
- 31. Open EndedRational or not. Is √2+√7 rational
- 32. Open EndedStill the one. What is a one-to-one correspondence?
- 33. Open EndedI get around. Consider the following pairing: Honda . . . Deb Saab . . . Ed Lexus . . . Mike Trail-a-Bike . . . Julia
- 34. Open EndedNumerical nephew. At a family gathering, your four-year-old nephew approaches you and proudly proclaims he has found the biggest number. How would you gently refute his naive notion?
- 35. Open EndedPile of packs. You walk into class late and notice a bunch of backpacks lying against one wall. How could you check to see if there's a one-to-one correspondence between the backpacks and the students in the room? Is there a way to pair up each backpack with a student?
- 36. Open EndedBunch of balls. Your first job every morning at tennis camp is to get the ball machine ready for action. You open up some new cans of tennis balls and empty them into a large hopper. Is there a one-to-one correspondence between the balls and the cans?
- 37. Open EndedTaking stock. It turns out that there is a one-to-one correspondence between the New York Stock Exchange symbols for companies and the companies themselves (for example, PE is Philadelphia Electric Company). Explain why this correspondence must be one-to-one. What would happen if it were not? Describe potential problems. If the correspondence were not one-to-one, we would have one of the following situations:
- 38. Open EndedDon't count on it. The following are two collections of the symbols @ and ©: @@@@@@@@@@@@@@@@@@@@@@@ ©©©©©©©©©©©©©©©©©©©©©©© Are there more @'s than ©'s? Quickly answer the question without counting using the notion of a one-to-one correspondence.
- 39. Open EndedSocial security. Is there a one-to-one correspondence between U.S. residents and their social security numbers?
- 40. Open EndedMusical chairs. Musical chairs is a fun game in which a group of people march around a row of chairs while music is played. There is one more person than there are chairs. The moment the music stops, everyone scrambles for a chair. The person left chairless loses and moves to the sidelines. Then everyone in a chair gets up, one chair is removed, and the music and marching begin again. At what points in this game do we have a one-to-one correspondence between chairs and people, and at what points do we not have such a correspondence?
- 41. Open EndedAu natural. Describe the set of natural numbers.
- 42. Open EndedSet setup. We can denote the natural numbers symbolically as {1, 2, 3, 4, ...}. Use this notation to express each of the sets described below. • The set of natural numbers less than 12. • The set of all even natural numbers. • The set of solutions to the equation x2 - 25 = 0. [Write your answers in ascending order.] • The set of all reciprocals of the natural numbers. [Give exact answers (in the form of a fraction if needed).]
- 43. Open EndedA word you can count on. Define the cardinality of a set.
- 44. Open EndedEven odds. Let E stand for the set of all even natural numbers (so E = {2, 4, 6, 8, ...}) and O stand for the set of all odd natural numbers (so O = {1, 3, 5, 07, ...}). Show that the sets E and O have the same cardinality by describing an explicit one-to-one correspondence between the two sets.
- 45. Open EndedNaturally even. Let E stand for the set of all even natural numbers (so E= {2, 4, 6, 8, ...}). Which is true among the set E and the set of all natural numbers?
- 46. Open EndedFives take over. Let EIF be the set of all natural numbers ending in 05 (EIF stands for "ends in five"). That is, EIF = {5, 15, 25, 35, 45, 55, 65, 75,...} Describe a one-to-one correspondence between the set of natural numbers and the set EIF. For any natural number n there is the corresponding number in the set EIF, which can be written in terms of n as
- 47. Open EndedCounting cubes (formerly Crows). Let C stand for the set of all natural numbers that are perfect cubes, so C = {1, 8, 27, 64, 125, 216, 343, 512, ...}. Do the set C and the set of all natural numbers have the same cardinality?
- 48. Open EndedHow many mp3s? In the Lost and Found Office of your school, there are two boxes. One box contains a bunch of mp3 players and the other box contains a bunch of earbud headphones. You are told that these collections have the same cardinality. A deranged algebra instructor noticed that if x2 - x - 89 represents the cardinality of the mp3 players then 3x - 29 represents the cardinality of the headphones. Untangle this cryptic observation to determine how many mp3 players there are. Suppose your lost mp3 player ended up in that first box. How easy would it be to locate yours?
- 49. Open EndedShake'em up. What did Georg Cantor do that "shook the foundations of infinity"?
- 50. Open EndedDetecting digits. Here's a list of three numbers between 0 and 01: 0.27272 0.13131 0.15151
- 51. Fill in the BlankDelving into digits. Consider the real number 0.12345678910111213141516... Describe in words how this number is constructed. This number is constructed by writing down the _______________ numbers in sequence.
- 52. Open EndedUndercover friend. Your friend gives you a list of three, five-digit numbers, but she only reveals one digit in each: 10???? ?6??? ??4?? Can you describe a five-digit number you know for certain will not be on her list?
- 53. Open EndedUnderhanded friend. Now your friend shows you a new list of three, five-digit numbers, again with only a few digits revealed: 4???? ?5??? ????? Can you describe a five-digit number you know for certain will not be on her list?
- 54. Open EndedThink positive. Which of the following is useful to prove that the cardinality of the positive real numbers is the same as the cardinality of the negative real numbers?
- 55. Open EndedA penny for their thoughts. Suppose you had infinitely many people, each one wearing a uniquely numbered button: 1, 2, 3, 4, 05,... (you can use all the people in the Hotel Cardinality if you don't know enough people yourself). You also have lots of pennies (infinitely many, so you're really rich; but don't try to carry them all around at once). Now you give each person a penny; then ask everyone to flip his or her penny at the same time. Then ask them to shout out in order what they flipped (H for heads and T for tails). So you might hear: HHTHHTTTHTTHTHTHTHHHTH... or you might hear THTTTHTHHTTHTHTTTTTHHHTHTHTH... and so forth. Consider the set of all possible outcomes of their flipping (all possible sequences of H's and T's). Does the set of possible outcomes have the same cardinality as the natural numbers?
- 56. Open EndedOnes and twos. The set of all real numbers between 0 and 1 just having 1s and 02s after the decimal point in their decimal expansions has a greater cardinality than the set of natural numbers.
- 57. Open EndedTwo out of three. If a right triangle has legs of length 1 and 2, what is the length of the hypotenuse? Enter the exact answer. If it has one leg of length 1 and a hypotenuse of length 9, what is the length of the other leg? Enter the exact answer.
- 58. Open EndedHypotenuse hype. If a right triangle has legs of length 1 and x, what is the length of the hypotenuse? Enter the exact answer.
- 59. Open EndedAssessing area. Suppose you know the base of a rectangle has a length of 5 inches and a diagonal has a length of 13 inches. Find the area of the rectangle.
- 60. Open EndedOperating on the triangle. Using a straightedge, draw a random triangle. Now, carefully cut it out. Next, amputate the angles by snipping through adjacent sides. Now, move the angles together so the vertices all touch and the edges meet.
- 61. Open EndedEasy as 1, 2, 3? Can there be a right triangle with sides of length 1, 2, and 03? Can you find a right triangle whose side lengths are consecutive natural numbers? The triangle with sides:
- 62. Open EndedGetting a pole on a bus. For his 13th birthday, Adam was allowed to travel down to Sarah's Sporting Goods store to purchase a brand new fishing pole. With great excitement and anticipation, Adam boarded the bus on his own and arrived at Sarah's store. Although the collection of fishing poles was tremendous, there was only one pole for Adam and he bought it: a five-foot, one-piece fiberglass "Trout Troller 570" fishing pole. When Adam's bus arrived, the driver reported that Adam could not board the bus with the fishing pole. Objects longer than four feet were not allowed on the bus. In tears, Adam remained at the bus stop holding his beautiful five-foot Trout Troller. Sarah, seeing the whole ordeal, rushed out and said, "Don't cry, Adam! We'll get your fishing pole on the bus!" Sure enough, when the same bus and the same driver returned, Adam boarded the bus with his fishing pole and the driver welcomed him aboard with a smile. How was Sarah able to have Adam board the bus with his five-foot fishing pole without breaking the bus line rules and without cutting or bending the pole?
- 63. Open EndedThe scarecrow. In the 1939 movie The Wizard of Oz, when the brainless scarecrow is given the confidence to think by the Wizard (by merely handing him a diploma, by the way), the first words the scarecrow utters are, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." An isosceles right triangle is just a right triangle having both legs the same length. Suppose that an isosceles right triangle has legs each of length 2. What is the length of the hypotenuse? Do not round. Is the scarecrow's assertion valid? This question illustrates the true value of a diploma without studying.
- 64. Open EndedRooting through a spiral. Start with a right triangle with both legs having length 01. What is the length of the hypotenuse? Do not round. Suppose we draw a line of length 01 perpendicular to the hypotenuse and then make a new triangle as illustrated (see figure below). What is the length of this new hypotenuse? Suppose we continue in this manner. Describe a formula for the lengths of all the hypotenuses, i.e. find the length of the hypotenuse of the Nth triangle.
- 65. Open EndedGetting squared away. In our proof of the Pythagorean Theorem, we stated that the second figure is actually two perfect squares touching along an edge. Can you prove that they are indeed both perfect squares?
- 66. Open EndedStanding guard. Draw the floor plan of a gallery with three vertices. What shape do you get? What is the smallest number of guards you need?
- 67. Open EndedArt appreciation. State the Art Gallery Theorem.
- 68. Open EndedUpping the ante. How many guards do you need for a gallery with 13 vertices?
- 69. Open EndedKeep it safe. At what vertices would you place cameras so that you use as few cameras as possible and so that each point inside the curve is visible from a camera?
- 70. Open EndedPutting guards in their place. For each floor plan, place guards at appropriate vertices so that every point in the museum is within view.
- 71. Open EndedTriangulating the Louvre. Triangulate the floor plans by adding straight segments that do not cross each other yet span the insides and extend from one vertex to another.
- 72. Open EndedTricolor hue. For each triangulation, color the vertices red, blue, or green so that every triangle has all three colors.
- 73. Open EndedCounting the colors. Your polygon has 50 vertices. fourty percent have been colored red, 30% yellow, and the remainder blue. Determine the number of vertices of each of the three colors.
- 74. Open EndedDefininig gold. Explain what makes a rectangle a Golden Rectangle.
- 75. Open EndedApproximating gold. Which of these numbers in the drop down is closest to the Golden Ratio?
- 76. Open EndedApproximating again. Which of the following objects in the drop down most closely resembles a Golden Rectangle?
- 77. Open EndedSame solution. Why does the equation Φ - 1 = 1/Φ have the same solution as the equation Φ/1 = 1/Φ-1
- 78. Open EndedSolve this Golden Rectangle. In the 20th century, artists were still fascinated with the beautiful proportions of the Golden Rectangle. An architect designed a villa based on the concept of a Golden Rectangle. The Golden Rectangle has a shorter side of 130 ft. How long is the longer side? Round the answer to the nearest integer.
- 79. Open EndedFold the gold. Suppose you have a Golden Rectangle cut out of a piece of paper. Now suppose you fold it in half along its base and then in half along its width. You have just created a new, smaller rectangle. Is that rectangle a Golden Rectangle?
- 80. Open EndedGrowing gold. Take a Golden Rectangle and attach a square to the longer side so that you create a new larger rectangle. Is this new rectangle a Golden Rectangle?
- 81. Open EndedIn the grid. Consider the 10 × 10 grid below. Find the four points that, when joined to make a horizontal rectangle, make a rectangle that is the closest approximation to a Golden Rectangle. (Challenge: Suppose the rectangle can be tilted.) What is its size?
- 82. Open EndedDo we get gold? Let's make a rectangle somewhat like the Golden Rectangle. As before, start with a square; however, instead of cutting the base in half, cut it into thirds and draw the line from the upper right vertex of the square to the point on the base that is one-third of the way from the right bottom vertex. Now use this new line segment as the radius of the circle, and continue as we did in the construction of the Golden Rectangle. This produces a new, longer rectangle, as shown in the diagram. What is the ratio of the base to the height of this rectangle (that is, what is base/height for this new rectangle)? Let the sides of the square be 03 units long. Now remove the largest square possible from this new rectangle and notice that we are left with another rectangle. Are the proportions of the base/height of this smaller rectangle the same as the proportions of the big rectangle? The proportions are
- 83. Open EndedTo tile or not to tile. Which of the following shapes can be used to tile the entire plane?
- 84. Open EndedFlipping over symmetry. For each pattern below, describe a rigid symmetry corresponding to a flip. Which patterns have more than one flip symmetry?
- 85. Open EndedCome on baby, do the twist. Which patterns have a rigid symmetry corresponding to a rotation?
- 86. Open EndedSymmetric scaling. Each of the two patterns below has a symmetry of scale. For each pattern, determine how many small tiles are needed to create a super-tile. How many are required to build a super-super-tile?
- 87. Open EndedExpand again. Take your 4-unit equilateral triangle and surround it with 12 equilateral triangles to create a 16-unit super-triangle. Which way is it oriented?
- 88. Open EndedSuper total. Recall that the Pinwheel Triangle has sides of length 3, 6, and 3√5. The figure below shows a super triangle made of five Pinwheel Triangles. Find the lengths of the sides of the super triangle.
- 89. Open EndedDescribing distortion. What does it mean to say that two things are equivalent by distortion?
- 90. Open EndedYour last sheet. You're in your bathroom reading the liner notes for a newly purchased CD. Then you discover that you've just run out of toilet paper. Is a toilet paper tube equivalent by distortion to a CD?
- 91. Open EndedRubber polygons. Find a large rubber band and stretch it with your fingers to make a triangle, then a square, and then a pentagon. Are these shapes equivalent by distortion? What other equivalent shapes can you make with the rubber band? Can you stretch it to make a rubber disk?
- 92. Open EndedOut, out red spot. Remove the red spot from the letters below. For each letter, how many pieces result? Are the original letters equivalent by distortion?
- 93. Open EndedThat theta. Does there exist a pair of points on the theta curve whose removal breaks the curve into three pieces? If so, the existence of those two points would provide another proof that the circle is not equivalent by distortion to the theta curve.
- 94. Open EndedYour ABCs. Consider the following letters made of 01-dimensional line segments: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Which letters are equivalent to one another by distortion? Group equivalent letters together.
- 95. Open EndedHalf dollar and a straw. Suppose we drill a hole in the center of a silver dollar. Would that coin with a hole be equivalent by distortion to a straw?
- 96. Open EndedStarry-eyed. Consider the two stars below. Are they equivalent by distortion?
- 97. Open EndedOne side to every story. What is a Möbius band?
- 98. Open EndedSingin' the blues. Take an ordinary strip of white paper. It has two sides. Color one side blue and leave the other side white. Now use the strip to make a Möbius band. What happens to the blue side and the white side?
- 99. Open EndedWho's blue now? Take an ordinary strip of white paper. It has two sides. Color one side blue and leave the other side white. Now give one end of the strip two half twists (also known as a full twist). Tape the ends together. Do you get a Möbius band?
- 100. Open EndedTwisted sister. Your sister holds a strip of paper. She gives one end a half twist, then she gives the other end a half twist in the same direction, then she tapes the ends together. Does she get a Möbius band?
- 101. Open EndedThe unending proof. Take a strip of paper and write on one side: "Möbius bands have only one side; in fact while." Next, turn it over on its long edge and write "reading The Heart of Mathematics, I learned that." Now, tape the strip to make a Möbius band. Read the band. This activity illustrates how many sides a Möbius band has.
- 102. Open EndedThree twists. Take a strip of paper, put three half twists in it, and glue the ends together. Cut it lengthwise along the center core line. Find an interesting object hidden in all that tangle.
- 103. Open EndedMöbius lengths. Use the edge identification diagram of a Möbius band to find the lengths of the two bands we get when we cut the Möbius band by hugging the right edge. Give the lengths in terms of the length of the original Möbius band, L.
- 104. Open EndedRubber Klein. Suppose you have a rectangular sheet of rubber. Carefully illustrate how you would associate and then glue the edges of the sheet together to build a Klein bottle. Arrange the sequence illustrating the construction of the rubber Klein bottle.
- 105. Open EndedKnotty start. Which of the following knots are mathematical knots?
- 106. Open EndedThe not knot. What is the unknot?
- 107. Open EndedCount the crossings in each knot below. From left to right, the knots have
- 108. Open EndedTangled up. Is the figure below a knot or a link?
- 109. Open EndedHuman trefoil. What is the minimum number of people you need to make a human trefoil knot? The minimum number of people you need to make a human trefoil knot is
- 110. Open EndedStick number. What is the smallest number of sticks you need to make a trefoil knot? (Bending sticks here is not allowed.)
- 111. Open EndedDollar link. Take two paper clips and a dollar and fasten them as illustrated below. Now, pull the ends of the dollar so as to straighten it out. What happens to the paper clips?
- 112. Open EndedMap maker, map maker make me a graph. Represent the map below using a graph, with a vertex (dot) for each landmass and an edge (line or arc) for each bridge. Input the correct figure number Entry field with correct answer:
- 113. Open EndedWill the walk work? Can you take a walk around the town shown in the map below, cross each bridge exactly once, and return to where you started?
- 114. Open EndedWalk the line. Is it possible to traverse the graph below with a path that uses each edge exactly once and returns to the vertex at which you started?
- 115. Open EndedLinking the loops. In the map below, the following walks can be taken from various starting points: CAADDFFC, FCCBBCCEEF, DCCBBEEBBAAD Can these walks be linked together to create one walk that starts on landmass C, crosses each bridge exactly once, and then returns to C?
- 116. Open EndedScenic drive. Below is a map of Rockystone National Park. One scenic drive is Entrance to Moose Mountain to Rockystone Lake to Lookout Below to Entrance. Can you add loop-d-loops to this drive to obtain a trip that traverses each road in the park exactly once and returns to the entrance?
- 117. Open EndedUnder-edged. Is it possible to traverse the graph below with a path that uses each edge exactly once and returns to the vertex at which you started? If so, find such a path. If not, add the fewest number of new edges until such a path is possible. (Remember that edges can be curved.)
- 118. Open EndedSnow job. Below is a map of the tiny town of Eulerville (the streets are white; the blocks are orange). After a winter storm, the village snowplow can clear a street with just one pass. Is it possible for the plow to start and end at the Town Hall (shown in black), clearing all the streets without traversing any street more than once?
- 119. Open EndedNew Euler. You were presented with graphs (shown below) that had no Euler circuit because they had vertices with odd degree (an odd number of incident edges). But in three of the four graphs, you could find a path that traversed each edge exactly once. Such a path is called an Euler path. Each of your Euler paths started and ended at a vertex of odd degree. Did this have to happen for these graphs? If you had more than two vertices of odd degree, could an Euler path exist? Entry field with correct answer
- 120. Open EndedStill looking. Suppose a graph has n vertices and (1/2)n(n - 01) = 66 edges. How many vertices does the graph actually have?
- 121. Open EndedStill looking. Suppose a graph has n vertices and (1/2)n(n - 01) = 66 edges. How many vertices does the graph actually have?
- 122. Open EndedWhat a character! What expression gives the Euler Characteristic?
- 123. Open EndedCount, then verify. What are the values of V, E, and F for the graph below? Compute V - E + F .
- 124. Open EndedAdd one. Find the values V, E, and F for the graph below. Now add a vertex in the middle of one of the edges. How do the values of V, E, and F change? V' = Do the new values still satisfy the Euler Characteristic formula?
- 125. Open EndedMaking a point. Take a connected graph and add a vertex in the middle of an edge, making two edges out of the one. What happens to V - E + F?
- 126. Open EndedOn the edge. Is it possible to add an edge to a graph and reduce the number of regions? Is it possible to add an edge and keep the same number of regions?
- 127. Open EndedDualing. What is the relationship between the Euler Characteristic for a regular solid and its dual? The Euler characteristic for a regular solid is The Euler characteristic for its dual is
- 128. Fill in the BlankLots of separation. Suppose we are told that a connected graph cuts the plane into 288 regions. How many more edges than vertices are there? There are _____________ more edges than vertices.
- 129. Open EndedThe incredible shrinking duck. On the Quacked Wheat box, outline the sub-picture that is an identical, but reduced, copy of the whole picture. Roughly what fraction of the height is the reduced picture compared to the whole?
- 130. Fill in the BlankMultiplicity. In the Sierpinski Triangle, outline three sub-figures that are identical but reduced copies of the whole figure. For each sub-figure you outlined, compare its width, as a fraction, to that of the whole Sierpinski Triangle. The largest such sub-figure is ________ of the size of the original picture.
- 131. Fill in the BlankDifferent sizes. Shown below are four reduced copies of the whole picture of a fern—one is about 85% as large as the whole figure, the other three are much smaller—that make up the whole picture except for the stem. Some of the regions of the above figures are tilted over as well as ___________.
- 132. Fill in the BlankNot quite cloned. In the Mandelbrot set shown, the whole is not identical to any subpart. Find some subparts that nevertheless look similar to some yet smaller subparts. The last picture looks strikingly_________________parts of the original Mandelbrot set.
- 133. Open EndedMaybe moon. What features of the fractal forgeries of the cratered vista make it look realistic?
- 134. Fill in the BlankA search for self. What does self-similarity mean? A picture or object exhibits self-similarity if parts of it ________________ identical to larger parts but at a different scale.
- 135. Fill in the BlankToo many triangles? At stage 0, the Sierpinski triangle consists of a single, filled-in triangle. (See the figure below.) At stage 1, there are three smaller, filled-in triangles. How many filled-in triangles are there at stage 2? How many at stage 3? What's the pattern? How many triangles are there at stage 04? How many will there be at stage n? At stage 02, the Sierpinski triangle has _____ filled-in triangles. At stage 03, there are _____ filled-in triangles. At stage 04, there are _____ filled-in triangles. At stage n, there will be N filled-in triangles, where N =
- 136. Fill in the BlankCounting Koch. Look at the early stages of the Koch curve in the figure below. The top figure is stage 1; it has four line segments. The next figure is stage 2. How many line segments does it have? How many line segments do you think there are at stage 3? (Count them to check your answer!) What's the general pattern? At stage 02, the Koch curve has _____ line segments. At stage 03, it has _____ line segments. At stage n, there will be N line segments, where N =
- 137. Open EndedWho's the fairest? Can you position three mirrors in such a way that in theory you could see infinitely many copies of all three mirrors?
- 138. Open EndedPhoto op. Suppose you arrange two mirrors facing each other at a slight angle, as shown below. Place a camera parallel to one of the mirrors. Snap the picture. The picture will contain many increasingly smaller pictures of the camera. Will they be arranged going off to the right, the left, or up? The pictures of the camera will be arranged going off
- 139. Fill in the BlankThe Kinks. Koch's kinky curve is created by starting with a straight segment and replacing it with four segments, each 1/3 as long as the original segment. So, at the second stage the curve has three bends. At the next stage, each segment is replaced by four segments, and so on. How many bends does this curve have at the third stage? The fourth stage? The nth stage? At the third stage there are _______ bends. At the fourth stage there are ______ bends. At the nth stage there are N bends, where N =
- 140. Fill in the BlankCatching Z's. Take a Z. Put in nine smaller Z's, as shown, to create the second stage. If the smaller Z's are 1/6 as long as the large one, roughly how long is the line through the Z's at the third stage if the line through the original, big Z is 24 centimeters long? The third stage is ______ centimeters.
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