Math

Twin primes. Find the first 15 pairs of twin primes.

(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139), (149,151), (179,181), (191,193), (197,199)

Rational or not. Is √2+√7 rational

No

Walk 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?

Yes It is possible to traverse the graph as requested. One such path is (AB)(BC)(CE)(EB)(BD)(DE)(EG)(GH)(HI)(IF)(FA)(AD)(DG)(GI)(IA)(AC)(CF)(FE)(EH)(HF)(FA).

Defininig gold. Explain what makes a rectangle a Golden Rectangle.

Φ = 1 + √5 / 2

Muchos 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?

21 x 4 = 84 mangos.

Approximating again. Which of the following objects in the drop down most closely resembles a Golden Rectangle?

3 x 05 inch index paper In decimal form, the four ratios are 5/3 = 1.66..., 11/8.5 = 1.29..., 14/11 = 1.2727..., and 1.5454.... Thus the first object, the 3x5 card, has proportions closest to those of a Golden Rectangle.

Floating in factors. What is the smallest natural number that has three distinct prime factors in its factorization?

30

One side to every story. What is a Möbius band?

A Möbius band is surface with one side and one edge.

Tangled up. Is the figure below a knot or a link?

A link. The given figure is a link because it contains two distinct closed curves.

Naturally 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?

Have the same cardinality by describing an explicit one-to-one correspondence between the two sets.

Odd couple. If n is an odd number greater than or equal to 3, can n + 01 ever be prime? What if n equals 01?

No. Then n + 01 is prime.

Stick number. What is the smallest number of sticks you need to make a trefoil knot? (Bending sticks here is not allowed.)

Six, see the figure.

Maybe moon. What features of the fractal forgeries of the cratered vista make it look realistic?

The craters and shadows appear real.

Hours 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?

10:45 6:45 3:30 7:30

Upping the ante. How many guards do you need for a gallery with 13 vertices?

4. For a gallery with 13 vertices, you need at most 13/3 ≈ 4 guards.

In 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?

8 x 05 Consecutive Fibonacci numbers (1,1,2,3,5,8,13...) are wonderful approximators of the Golden ratio. In a 10 × 10 grid, an 8 × 5 rectangle is your best bet. You can convince yourself by computing the 100 possible ratios and seeing which is closest to StartFraction 1 plus StartRoot 5 EndRoot Over 2 EndFraction period Surprisingly, it doesn't help to consider non-horizontal right triangles!

Numerical 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?

Ask your nephew what he gets if he adds one to his biggest number.

SpaghettiOs. Which of the following is the correct UPC for Campbell's SpaghettiOs? Show why the other numbers are not valid UPCs.

Check sum for 0 510000 2562 5 = 42 Check sum for 0 510000 2526 5 = 50 Check sum for 0 510000 2526 4 = 49 The correct one is UPC 2

Even 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.

Each even number is paired with the odd number to its left, and each odd number is paired with the even number to its right.

Map 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:

Figure 1

You 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?

In 12 hours it will read 7:15 In 17 hours it will read 12:15 In 26 hours it will read 9:15 In 336 hours it will read 7:15

How 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?

It should be easy to find your player because the number of mp3 players is 01.

Making 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?

Result remains unchanged. Total number of vertices goes up by 1, total number of edges goes up by 1, and the number of faces remains unchanged. The new edge cancels the new vertex in the computation so that the result remains unchanged. (V + 1) - (E + 1) + (F) = V - E + F.

Möbius 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.

The center strip has length L, and the additional two-sided loop has length 2L The center strip is a thinner version of the original, but the additional two-sided loop is twice as long.

Counting 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.

The number of red vertices is 20; The number of yellow vertices is 15; The number of blue vertices 15. The number of red vertices is 40% of 50 which equals 0.40 × 50 = 20. The number of yellow vertices is 30% of 50 which equals 0.30 × 50 = 15. The number of blue vertices is 50 - 20 - 15 = 15.

Twenty-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.

Two possible candidates: First, 29 is prime. Secondly, 29 happens to be the sum of three consecutive squares, 29 = 4 + 9 + 16. Lest the number 27 feel left out, it should be noted that 27 = 3×3×3, a perfect cube. Is the set of prime numbers sparser than the set of perfect cubes? Does this make it less interesting in your eyes?

What a character! What expression gives the Euler Characteristic?

V - E + F = 2

Undercover 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?

Yes Create a number with first digit not 10 and second digit not 6 and third digit not 4, and you have a number not on your friend's list. Some examples: 1155xx, 973xx. The last two digits can be anything.

Counting 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?

Yes. Given an arbitrary natural number, n, we find its mate by multiplying by n3.

You 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?

Yes; some shelf must have two CDs

Rubber 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.

(iii)-(iv)-(ii)-(i) It starts with the rectangular sheet. Glue the left and right sides to make a cylinder. Now, bend the cylinder into a U-shape, pass one end through the side of the cylinder (not possible in three dimensions), and then join the two ends together.

Approximating gold. Which of these numbers in the drop down is closest to the Golden Ratio?

1.62 The Golden Ratio is approximately 1.618, so the number 1.62 is closer than any of the other numbers.

Fattened fractions. Reduce these overweight fractions to lowest terms: 8/40, 25/15, -12/84, 38/4, -196/14

1/5 5/3 1/7 19/2 14

Lots 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.

286 Connected graphs on the plane satisfy the formula V - E + F = 2 which applies with F = 288. We can rewrite this as E = V + 286, showing that there are 286 more edges than vertices.

Super 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.

3√5, 6√5, 15 The Pinwheel Triangle has sides of length 3, 6, and 3√5. The super triangle has short leg equal to one hypotenuse of the Pinwheel Triangle, so it has length 3√5. The long leg of the super triangle has length equal to two hypotenuses of the Pinwheel Triangle and so has length 6√5. The hypotenuse of the super triangle has length equal to two long legs plus a short leg of the Pinwheel Triangle and so has length 02(6) + 3 = 15.

Decoding 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

67/100 598/100 123871/20000 -3363/10 -1/25000

Pile 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?

Ask each student to pick up his or her backpack.

Secret admirer. Use the scheme below to encode the message "I LOVE YOU."

B WXAU GXL

Same solution. Why does the equation Φ - 1 = 1/Φ have the same solution as the equation Φ/1 = 1/Φ-1

Because cross-multiplying both equations yields the same quadratic equation. There are several ways to see that these equations have the same solution. Inverting both sides of one equation yields the other. Also, cross-multiplying both equations yields the same quadratic: Φ to the 2nd power - Φ = 1

Celestial 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.

Check sum for 0 70743 00021 8 = 42 Check sum for 0 70734 00021 8 = 40 Check sum for 0 71734 00021 8 = 43 The correct one is UPC 2

Come on baby, do the twist. Which patterns have a rigid symmetry corresponding to a rotation?

First pattern (Fishscale) has no rotational symmetry Middle pattern (Hexagon) has rotational symmetry: through any multiple of 60 degrees Right-most (Parquet) pattern has only one rotational symmetry: through 180 degrees Do any of the patterns have more than one rotational symmetry? Yes

Underhanded 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?

No You're stuck here. Because you friend has shown you no digits of her last number, any number you pick might match the third on her list.

Keep 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?

Only one camera is needed. Place it at one of the four inside vertices.

Think 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?

Pair each positive real number with its additive inverse. Pair each positive real number with its additive inverse (i.e., its negative). The negative of any positive number is negative, and the negative of any negative number is positive. As a result, no numbers are left out. This one-to-one correspondence shows that the two sets have the same cardinality.

Symmetric 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?

Pattern on the left. Super tile: 4 tiles Super-super tile: 9 tiles Pattern on the right. Super tile: 4 tiles Super-super tile: 9 tiles

Out, 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?

Removing the red spot from the X leaves 4 piece(s), removing it from the Y leaves 3 piece(s), from the Z leaves 2 piece(s). The original letters are not equivalent by distortion.

A word you can count on. Define the cardinality of a set.

The cardinality of a set is the "number" of elements in the set.

Au natural. Describe the set of natural numbers.

The natural numbers are the whole numbers 1, 2, 03,...

Born φ. What is the precise number that the symbol φ represents? What sequence of numbers approaches φ? Choose all correct answers.

The symbol φ represents the infinitely long fractal expression / The symbol φ represents a solution to the equation / A sequence of numbers that approaches φ is the list of ratios of consecutive Fibonacci numbers.

Don'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.

There are the same number of the symbols

Singin' 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?

This Möbius band has one side with a blue portion and a white portion. Once you construct the band, you should see a white edge meeting a blue edge at the place where you taped the edges together. This Möbius band has one side with a blue portion and a white portion.

Standing 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?

Triangle 1. A gallery with three sides will have a floor plan in the shape of a triangle requiring only one guard.

Ones 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.

True

Describing distortion. What does it mean to say that two things are equivalent by distortion?

Two objects are equivalent by distortion if we can stretch, shrink, bend, or twist one, without cutting or gluing, and deform it into the other.

Detecting digits. Here's a list of three numbers between 0 and 01: 0.27272 0.13131 0.15151

What's the first digit of the first number? 2 What's the second digit of the second number? 3 What's the third digit of the third number? 1

Tricolor hue. For each triangulation, color the vertices red, blue, or green so that every triangle has all three colors.

Yes

Rubber 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?

Yes Circles and ellipses No The triangle, square, and pentagon are all equivalent by distortion because they can all be obtained by stretching or shrinking the same rubber band. All other polygons can also be obtained, as well as circles, ellipses, or shapes with curving boundaries. The rubber band cannot be stretched to make a rubber disk.

Getting 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?

Yes, it can be proved. Before moving the top triangle, side a represents the length of one side of the alleged square. After the triangle is moved, the same side represents the other side of the alleged square. This proves that the figure is indeed a perfect square. The second square is proved in a similar manner.

Putting guards in their place. For each floor plan, place guards at appropriate vertices so that every point in the museum is within view.

a) 3 b) 2 c) 2

Creating 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.

m = 8 (1,1), (3,3), (5,5), (7,7)

Knotty start. Which of the following knots are mathematical knots?

All knots with closed ends. All the figures represent mathematical knots except the one shown below, which is not a knot because it has loose ends and thus is not a closed curve.

On 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?

No Yes You cannot reduce the number of faces in a graph by adding edges. If the new edge is created by connecting two previous vertices, then one new region is added. If the new edge is created by adding one vertex and connecting it to an old vertex, then no new regions are created. In either case, the formula V - E + F remains unchanged. There is no other way to add an edge to a graph.

The 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?

One third. The picture of the Quacked Wheat box within the Quacked Wheat box is a copy, roughly one-third in size, of the original picture.

Dollar 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?

The paper clips are linked. The paper clips pass through each other, the dollar straightens out, and the paper clips fall to the floor. Surprisingly, the paper clips are linked!

The not knot. What is the unknot?

The unknot is a closed curve that can be untangled to resemble a circle.

Add 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?

V = 3 E = 3 F = 2 V = 4 E = 4 F = 2 Yes

What did you say? The message below was encoded using the following scheme: Decode the original message. VBV IXP VTV

TIT FOR TAT

Your 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.

(B), (P), (Q), (X) (A,R), (D,O), (H,K), (E,F,T,Y) (C,G,I,J,L,M,N,S,U,V,W,Z) Note that the category depends very much on the font. We have picked a simple font without serifs to simplify the categorization. There are nine groups (A,R), (B), (C,G,I,J,L,M,N,S,U,V,W,Z), (D,O), (E,F,T,Y), (H,K), (P), (Q), (X).

Discovering 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.

(Fn +1)2 + (Fn)2 = F2n + 1 for n ≥ 01

Real mayo (H). The following is the UPC for Hellmann's 08-oz. Real Mayonnaise. Find the missing digit. 0 48001 81 ▪ 18 6

5

Rational or not (ExH).For each of the following numbers, determine if the number is rational or irrational. Give brief reasons justifying your answers.

7/3 is rational because it can be written as the ratio of two integers 3.68 is rational because it can be written as the ratio of two integers √12/9√3 is rational because it can be written as the ratio of two integers √2/13 is irrational because it cannot be written as the ratio of two integers 5.99376 is rational because it can be written as the ratio of two integers

Different 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 ___________.

Reduced. The first copy is 0.85 as large as the original in height and in width. The second copy shrinks the width to 0.3 its original width and the height to 0.34 its original height. The third copy shrinks the width to 0.3 its original width and the height to 0.37 its original height. The fourth copy shrinks the width to zero and shrinks the height to 0.16 the original height. See the figures for the regions that delineate reduced copies. Note that some of the regions are tilted or turned over as well as reduced.

To tile or not to tile. Which of the following shapes can be used to tile the entire plane?

Shape 01: Yes Shape 02: Yes Shape 03: No Shape 04: Yes Shape 05: Yes The first, second, fourth, and fifth shapes can be used to tile the plane. The star cannot.

Expand 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?

The 16-unit super-tile has the same orientation as the original (center) equilateral triangle. Unlike the 4-unit super-tile, the 16-unit super-tile has the same orientation as the original (center) equilateral triangle.

Setting 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 =

Yes m = 96 No 77e = 385 4m = 384 77e − 4m = 1

Fifteen Fibonaccis. List the first 15 Fibonacci numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

Sum of Fibonacci. Express the natural number 31 as a sum of distinct, nonconsecutive Fibonacci numbers given in descending order (e.g. 55+21+3).

21+8+2

Multiplicity. 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.

1/2 There are a variety of answers. The largest such sub-figure is half the size of the original picture. Note that the original picture can be sub-divided into three of these sub-figures. There are also sub-figures that are 1/4, 1/8, 1/16, ... as large. (Any power of 1/2 will do.)

The 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 =

15 63 4 to the nth power - 1 minus 1 Looking at the figures you can easily see that at the first stage there is 1 segment with no bends, at the second stage there are 4 segments and 3 bends, and at the third stage there are 16 segments and 15 bends. You may want to gather your data in a table to make the pattern more evident: At each stage, the number of segments increases by a factor of 4. So. At the nth stage there are 4n-1 sides, with a bend between each adjacent pair, for a total of 4n-1 -1 bends.

Primal instincts. List the first 15 prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Delving 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.

Natural What's its 14th digit? 1 What's the 25th digit? 7 What's the 31st digit? 0

Who'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?

No. This construction does not yield a Möbius band. When the edges are taped together, the white edges will meet on one side and the blue edges will meet on the other, preserving the two-sided quality of the loop.

Count, then verify. What are the values of V, E, and F for the graph below? Compute V - E + F .

V = 7 E = 9 F = 4 7 - 9 + 4 = 2 V - E + F = 2

Always, 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.

Never, by definition. Never, by definition.

Human 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

Two. You only need two people who are willing to get very close. Jill stands with arms extended out to the side. Jack stands directly behind Jill, puts his left arm below her left arm, and his right arm above her right arm. Wrapping his right arm under his left arm, Jack grabs Jill's hands and completes the trefoil knot.

Still looking. Suppose a graph has n vertices and (1/2)n(n - 01) = 66 edges. How many vertices does the graph actually have?

12 From the hint, the equation becomes n2 - n - 132 = 0. Factoring yields (n - 12)(n + 11) = 00, so n = 12. (We can't have a negative number of vertices.)

Counting 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 =

16 64 4 to the Nth power

Dualing. 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

2 2 The regular solids and their duals have the same number of edges. The number of vertices in the dual equals the number of faces of the original solid, and vice versa. As a result, the computation V - E + F is the same for both objects. Because all regular solids are topologically equivalent to a sphere, their Euler characteristic is 02.

Catching 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.

54 The length of the second stage is (9 × 1/6 ) × 24 = 36 centimeters long. Because each Z in the third stage is again replaced by nine smaller Z's, each 1/6 as long, the length grows by another factor of 9/6. The third stage is (3/2)2 × 24 = 54 centimeters.

Too 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 =

9 27 81 3 to the Nth power At stage 2, the Sierpinski triangle has 9 filled-in triangles. At stage 3, there are 27 filled-in triangles. At each stage, the number of triangles increases by a factor of 3. At stage 4, there are 81 filled-in triangles. These values suggest that at stage n there will be 03n filled-in triangles.

Shake'em up. What did Georg Cantor do that "shook the foundations of infinity"?

He demonstrated that there are different "sizes" of infinity.

Scenic 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?

No It is not possible to add loop-d-loops to the given drive to obtain a trip that traverses each road exactly once and returns to the entrance. The given drive begins and ends at the entrance, but there is only one additional road that starts at the entrance. Any additional trip that traverses this road would not have an unused road on which to return to the entrance to complete a loop.

Snow 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?

No It is not possible to plow the streets as requested because there are several vertices (intersections) that are endpoints of an odd number of edges (streets).

The 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.

One side. Check how many sides a Möbius band has by doing this simple experiment. Take a marker and put the point on the middle of a side of the Möbius band and start drawing a line right down the middle all the way around the band. Keep drawing without lifting the marker. Keep going until you return to where you started. You have now drawn a circle on the Möbius band, and that circle, being one piece, all lies on the same side of the band. But wait. Look on the opposite side of the band. What do you see? That circle is present on the back side, too! The back side is the same as the front side. This illustrates that the Möbius band has just one side.

Count the crossings in each knot below. From left to right, the knots have

Three, zero, and five crossings, respectively. From left to right, the knots have three, zero, and five crossings, respectively.

Three 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.

Trefoil knot. Because the original strip has only one side and one edge, cutting along the center line will produce two sides and leave the two edges that are as long as the single edge in the original strip. The resulting strip is a knotted circle; it is a trefoil knot.

Who'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?

Yes Form an equilateral triangle by placing the three mirrors together. If you drew the path of a light beam inside the triangle, it would hit all three mirrors infinitely many times. So, in theory, if your eye is at that point, you would see each mirror infinitely many times.

New 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

Yes No The Euler paths in the graphs must start and end at a vertex of odd degree. The same argument that shows even degree vertices are required in order to have an Euler circuit applies here. Vertices other than the start and the end vertex must have edges that pair up as an "in" edge and an "out" edge as one traverses the Euler path. Thus, if a graph has more than two vertices of odd degree, it cannot have an Euler path.

Will 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?

Yes The requested walk is possible. Here's one way: AB, BC, CE, ED, DC, CD, DB, BA.

Linking 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?

Yes The walks can be linked as follows to create a single walk that begins and ends on landmass C, traversing each edge exactly once: CAAD(DCCBBEEBBAAD)DF(FCCBBCCEEF)FC.

A 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.

look A picture or object exhibits self-similarity if parts of it look identical to larger parts but at a different scale.

Not 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.

similar to The last picture looks strikingly similar to parts of the original Mandelbrot set though it is 10,000 times smaller.

Photo 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

to the right Represent the camera as a point light source and find all the paths that eventually come back to the camera. A line perpendicular to the parallel mirror will come back after one reflection. A light placed a little to the right will create a path that hits both mirrors before coming back to the camera. Thus, the pictures of the camera will be arranged going off to the right.

Musical 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?

A one-to-one correspondence exists between people and chairs in that short when the chairless person leaves and before the next chair is removed. A one-to-one correspondence doesn't exist between people and chairs at the begining of the game and also when the next chair is removed.

Bunch 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?

No

Social security. Is there a one-to-one correspondence between U.S. residents and their social security numbers?

No

Easy 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:

No 3-4-5 In any right triangle, the side opposite the right angle is the largest, so in our case, the length of the hypotenuse is 3. If this is indeed a right triangle, then the Pythagorean Theorem must hold, that is, 12 + 22 = 32, but 5 isn't 9, and so we don't have a right triangle. The only set of consecutive numbers that works here is 32 + 42 = 52.

Set 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).]

• The set of natural numbers less than 12: 1, 2,...11 • The set of all even natural numbers: 2, 4, 6,... • The set of solutions to the equation x2 - 25 = 0: -5, 5 • The set of all reciprocals of the natural numbers: 1, 1/2, 1/3

Living 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?

24 hours earlier it read 2:45 Three hours earlier it read 11:45 Thirty-one hours earlier it read 7:45 2400 hours earlier it read 2:45

This 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.

1, 1, 2, 3, 5, 8, 13, 21

Do 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

1.721 Not the same. The right triangle has base 1, and height 3, so its hypotenuse has length √10.

Fives 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

10n-5

Operating 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.

180 degrees When the angles are aligned, they will form a straight line or half of a complete rotation. Because a full rotation corresponds to 360 degrees, we have that the sum of the angles in a triangle is 180 degrees.

The 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.

2*2 to the 1/2 power (or 3√2?) No

Solve 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.

210 A rectangle is Golden if the ratio of the longer side to the shorter side is exactly Φ = 1 + √5 / 2 , the Golden Ratio. If the shorter side is 130 ft, the longer side is 130 times 1 + √5 / 2. Therefore the longer side is 210 ft. (130 x 1.618 = 210.34)

Assessing 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 We know the base of the rectangle; we need to find the height. The diagonal divides the rectangle into two right triangles, each with one leg of length 5 and hypotenuse of length 13. The length of the other leg will be the height of the rectangle. If x denotes the length of this leg, the Pythagorean Theorem tells us that 132 = x2 + 52. So x2 = 169 - 25 = 144, yielding x = 12. Thus the area of the rectangle is 5 × 12 = 60 square inches.

Sum 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

356= 233, 89, 34 948= 610, 233, 89, 13, 3 610= 610 1289= 987, 233, 55, 13, 1

Rooting 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.

√2 √3 (n + 1) to the second power

Two 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.

√5 √80 In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Hypotenuse hype. If a right triangle has legs of length 1 and x, what is the length of the hypotenuse? Enter the exact answer.

√x to the second power + 1

Fear factor. Express each of the following numbers as a product of primes: 77, 7889, 8, 39, 5643.

77 = 7 x 11 7889 = 7 x 7 x 7 x 23 8 = 2 x 2 x 2 39 = 3 x 13 5643 = 3 x 3 x 3 x 11 x 19

Still the one. What is a one-to-one correspondence?

A one-to-one correspondence is a pairing of objects from two sets in such a way that each object from one set is paired with exactly one object from the other set.

A rational being. What is the definition of a rational number?

A rational number is a number that can be expressed as a fraction - the ratio (or quotient) of two whole numbers.

Taking 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:

A symbol representing no company. A symbol representing two different companies. A company having no symbol.

Triangulating 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.

Each floor plan should be divided into triangles. a) 6 triangles b) 14 triangles c) 10 triangles

Art appreciation. State the Art Gallery Theorem.

If a polygonal closed curve in the plane has v vertices, then there are v/3 vertices from which it is possible to view every point on the interior of the curve. If v/3 is not an integer, then the number of vertices needed is the biggest integer less than v/3.

A 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?

No The set of all outcomes has a cardinality greater than that of the naturals. In fact, it's easy to construct a one-to-one correspondence between these two sets, so they are necessarily the same size.

Twisted 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?

No, she will not get a Möbius band. She will get a plain loop with no twists. The second half-twist in the same direction will simply undo the first one.

Flipping over symmetry. For each pattern below, describe a rigid symmetry corresponding to a flip. Which patterns have more than one flip symmetry?

Pattern 01: (Fishscale) Symmetry around horizontal axis Pattern 02: (Hexagon) Symmetry around the axes repeating with intervals of 30 degrees Pattern 03: (Parquet) Symmetry around horizontal and vertical axes The left-most and the right-most patterns have a flip symmetry obtained by flipping around a horizontal line drawn through the middle of each tile. The right-most pattern also has flip symmetry obtained by flipping through a vertical line. The center pattern has a symmetry around the axes repeating with intervals of 30 degrees.

Getting 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?

Sarah somehow found a 3 foot by 04 foot box and lay Adam's fishing pole along the diagonal. When we measure the length of a box, we don't measure the length of the diagonal, we measure the lengths of the sides—even though the diagonal is larger than either of the sides! So Sarah somehow found a 3 foot by 4 foot box and lay Adam's fishing pole along the diagonal. Once again, the 3-4-5 triplet comes to the rescue.

Suppose 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.)

To get started, let us estimate the thickness of an ordinary piece of paper by noting that packages of 200 sheets of paper are more than half an inch thick. So, a single piece of paper is at least 1/400 inches thick. Now, after one folding, the paper is twice the original thickness. After 2 foldings, the paper is 4 = 22 times as thick. The height would be more than 23000000000000 inches. [Round your answer to 02 significant digits.] After 53 foldings, the paper will be 0253 times as thick. The resulting paper is then 0253/400 inches thick. (That is about 2.3 × 1013 inches and about 355 million miles!)

I get around. Consider the following pairing: Honda . . . Deb Saab . . . Ed Lexus . . . Mike Trail-a-Bike . . . Julia

Who corresponds to the Saab? Ed What mode of transportation corresponds to Deb? Honda

Your 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?

Yes A toilet paper tube is equivalent by distortion to a CD, if we assume that the CD has thickness. Imagine the toilet paper tube is made of very squishy, malleable material. Just flatten it out until it's the diameter of a CD, then shrink the hole until it's the size of the hole in the CD. For the finishing touch, draw out the little tendrils from around the edges of the hole to match the "gripper teeth" that ring the edge of the center of the CD. Because the material of the original tube was only stretched or shrunk into a CD shape, the tube is equivalent by distortion to the CD.

Growing 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?

Yes Note that after we add the square, we have a rectangle with the following property: If we remove the largest square, we are left with a Golden Rectangle. Though not proved in the text, the Golden Rectangle is the only rectangle that has this feature, and so the original rectangle is indeed a Golden Rectangle. If the original sides had length 1 and j, then the new rectangle has sides of length 1 + j and j. Verify by calculator or by algebra that (1 + j)/j = j to prove that the new rectangle is also golden.

That 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.

Yes Removing the two intersection points leaves the theta curve in three pieces, while removing any two points on the circle always leaves exactly two pieces. Suppose there were a set of deformations that turned the theta curve into a circle. Mark the two intersection points red and follow the deformation process. At any stage, the removal of the two red points should leave the distorted theta curve in three pieces, yet when we get to the final stage, the circle, we find that the object falls into just two pieces. This contradiction shows that our assumption is wrong. The theta curve is not equivalent to the circle.

Starry-eyed. Consider the two stars below. Are they equivalent by distortion?

Yes Suppose the five- and six-pointed stars are made of flexible plastic. Imagine placing a metal cylinder with a small radius inside each star and expanding the radius until both stars are stretched into tightly fitted circles. This shows that both stars are equivalent to a circle and also shows how to distort one star into the other.

Fold 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?

Yes The dimensions of the new rectangle are exactly half of the original. The ratio between the longer and shorter sides remains unchanged and so still represents a Golden Rectangle.

Half 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?

Yes They are equivalent. It doesn't matter whether you view the objects as having thickness or not. If you view them as ideal two-dimensional objects, the straw is a cylinder, and the defaced coin is a disk with a hole in it. Deform the straw so that the bottom of the straw is the same radius as the drilled hole and stretch the top so that it is as wide as the coin. Now squash the straw vertically so that the entire straw lies in a single plane. Voila, you've deformed the straw into the coin. A similar process works for the thickened three-dimensional objects.