Math 104

lets make a deal

"Let's make a deal!" Monty Hall enthuses to the gentleman dressed as a giant singing raisin. The gleeful raisin, whose name is Warren Piece, is ready to wheel and deal as Monty Hall explains the game. "Behind one of these three doors is the Cadillac of your dreams. It is as long as a train and comes complete with a Jacuzzi. Of course, if you spend too much time in the Jacuzzi, your skin will wrinkle, but hey, you're a raisin, your skin's already wrinkled." Monty Hall continues by warning that, "Behind the other doors, howe- ver, are two other modes of transportation: two old pack mules. They don't come with Jacuzzis, although given their exotic odor, you may want to give them a bath." Of course, the crowd is laughing and applauding, just as the studio sign instructs. Monty sums it up: "So, there are three closed doors. Behind one is a lux- urious car, and behind the other two are mules. Now comes the moment of truth.What door do you pick?"The audience erupts,"Take Door Num- ber 1, take Door Number 1!!" "Door Number 2, Door Number 2!!" "Door Number 3's the one. Choose 3." Poor Warren Piece looks around at the crowd, confused and nervous. He considers Door Number 1, then 2, then 3. Finally Monty prompts, "Okay, Warren, which do you want?" The raisin-clad Warren shouts, "Okay, okay, I'll take Door Number 3, Door Number 3." As Monty Hall quiets the overly excited audience, he tells Warren, "I'll tell you what I'm going to do. I'm going to show you what's behind one of the doors you didn't pick. Let's take a look at what's behind Door Number 2." With that, Monty Hall turns to the Vanna White of the 1960s and says, "Please show us what is behind Door Number 2." The door dramatically swings open, the audience erupts, and Warren breathes once more—behind Door Number 2 is a mule! Monty, knowing where the mules are, always opens one of the mule doors first. Monty continues, "We now see that the Cadillac is not behind Door Number 2. You guessed Door Number 3. I'll tell you what I'm going to do. If you want, I'll let you change your mind and choose Door Number 1 instead. It's up to you. Do you want to stick to your original choice, or do you want to switch?" The audience goes nuts. "Stick, stick," yell half. "Switch, switch," advise the others. What to do, what to do? We now invite you to add your voice to the cacophony—although you need not shout. What should Warren Piece do? Should he switch choices, stick to his original guess, or does it not matter? Here a classic TV game show raises the question: How can we accurately measure the uncertain? he should switch-the probability is better, 2/3

cardinality

1 to 1 correspondence, the number of things in the set with the understanding that the set may contain infinitely many things. If the set contains finitely many things then the cardinatliy is the number if things in the set

Maine Voting article

1) ranked choice voting: a voter ranks all candidates in order of preference. If no one is ranked first by more than 50 percent of voters, the candidate least often ranked first is dropped. The process then repeats until a candidate does achieve 50 percent of the top ranking. In that sense, that candidate has majority support and wins. 2) is it a good idea, what are the drawbacks? vote splitting becomes a big issue

sum of fibonacci numbers

1) write down a natural number 2)find the largest fib number that doesn't exceed your number 3) subtract the fib number from yours 4) find the largest fib number that doesn't exceed the new number

Fibonacci sequence

A sequence of numbers in which each number is the sum of the previous two.

Thats a meanie genie

Alley asked to find the Rama Nujan and 9 identical stones appeared-she can only take one of the stones with her -8 weigh the same and the 9th weighs more containing the stone is this possible to fine using both the scales given just once? Answer: yes Divide the 9 gems into three groups of three. Weigh two of the groups against one another. If one is heavier, that group has the heavier gemstone. If both balance, the third group, unweighed, has the heavier gem. Take the heavier group of three and select two of them, and weigh them. If they balance, the remaining gem must be the heaviest one. If they do not, whichever is heavier is the chosen gem.

Dodgeball

Dodgeball is a game for two players—Player One and Player Two (although any two people can play it, even if they are not named "Player One" and "Player Two"). Each player has a special game board (shown below) and is given six turns. Player One begins by filling in the first horizontal row of his game board with a run of X's and O's. That is, on the first line of his board, he will write either an X or an O in each box. Then Player Two places either an X or an O in the first box of her board. So at this point, Player One has filled in the first row of his board with six letters, and Player Two has filled in the first box of her board with one letter. The game continues with Player One writing down either an X or an O in each box of the second horizontal row of his board. Then Player Two writes one letter (an X or an O) in the second box of her board. The game proceeds in this fashion until all of Player One's boxes are filled with X's and O's; thus, Player One has produced six rows of six marks each, and Player Two has produced one row of six marks. All marks are visible to both players at all times. Player One wins if any of his row exactly matches Player Two's row (Player One matches Player Two). Pla- yer Two wins if her row does not match any of Player One's rows (Player Two dodges Player One). Would you rather be Player One or Player Two? Who has the advan- tage? Can you devise a strategy for either side that will always result in victory? This little game holds within it the key to understanding the sizes of infinity. answer: the second person, the dodger should always win (has to do with infinity)

Division Algorithm

Let a and b be integers with b>0. Then there exists unique integers q and r with the property that a=bq+r, where 0<=r<b

factors

Numbers that are multiplied together to get a product

dot of fortune

One day three college students were selected at random from the studio audience to play the ever-popular TV game show, "Dot of Fortune." One of the students had already discovered the power and beauty of math- ematical thinking, while the other two were not nearly so fortunate. The stage contained no mirrors, reflective surfaces, or television monitors. The three students were seated around a small round table and blind- folded. As Pat, the host, explained the rules of the game, Vanna affixed a conspicuous but small colored dot to each student's forehead. "So, contestants," Pat explained, "at the sound of the bell you will remove your blindfolds. You will see your two companions sitting quietly at the table, each with a dot on his or her forehead. Each dot is either red or white. You cannot, of course, see the dot on your own forehead. After you have observed the dots on your companions' foreheads, you will raise your hand if you see at least one red dot. If you do not see a red dot, you will keep your hands on the table. The object of the game is to deduce the color of your own dot. As soon as you know the color of your dot, hit the buzzer in front of you. Do you understand the rules of the game?" All the students understood the rules, although the math fan understood them better. "Are you ready?" asked Vanna after affixing a red dot to each stu- dent's forehead. After the contestants nodded, Vanna rang the bell and they removed their blindfolds. The studio audience quivered with antici- pation. The students looked at one another's dots, and all raised their hands. After some time, the math fan hit her buzzer, knowing what color dot she had. Explain how she knew this. Why did the other students not know? This game requires creative logical reasoning—a powerful means to make discoveries whether they are in math, in life, or even (although rarely) on prime-time TV.

rolling around in vegas

Recently the swaggering burly billionaire, Mr. Bones, introduced an exciting new dice game at his glitzy High-Rollin' Bones' Hotel and Casino. An oversized gold bowl containing four dice is presented to the player. The player inspects each die, removes whichever die seems the luckiest, and throws a $100 chip in the bowl. Then Mr. Bones chooses one of the three remaining dice, takes a $100 chip (picturing his likeness) from his personal collection, and modestly places it into the bowl. Next the player and Mr. Bones roll their respective dice. Whoever rolls the higher number wins the two chips. Simple. To make the game interesting, the four dice are not the run- of-the-mill dice we remember from the gambling-free days of our youth. While each die does have six sides as usual, their faces are marked in unusual ways. The kit that accompanies each new copy of The Heart of Mathematics: An invitation to effective thinking contains these four special dice. Roll 'em on out of your kit. One die has two 6's and four 2's. Another has three 5's and three 1's. The third has four 4's and two blank faces. The last die has 3's on each face. The dice are not weighted— that is, any face is just as likely to land face-up as any other. Deep Pockets Drew strides up to the bowl to choose the winning die. Which die should Drew draw? Drew considers the die that has all 3's. Which die could Mr. Bones select that will beat the all-3's die two-thirds of the time? After finding that die, we know that the all-3's die would not be a particularly wise choice. Next Deep Pockets Drew considers the die with four 4's and two blank faces. Why will the die with three 5's and three 1's beat it two-thirds of the time? After verifying this dicey dominance, we know that selecting the die with four 4's and two 0's would not be a smart move. Drew next considers the die with three 5's and three 1's. Why will the die with two 6's and four 2's beat it two-thirds of the time? After con- firming this superiority, we know that the die with three 5's and three 1's would not be the best die. Only one possibility remains: the die with two 6's and four 2's. Is there a die that will beat it two-thirds of the time? Your surprising discovery will show that none of the four dice is the "best" one to select, because each one can be beaten by one of the other three dice two-thirds of the time. Amazing. So now Drew can put the dice in a circular order where each one beats its clockwise neighbor two-thirds of the time. What is that order? After doing the math, Deep Pockets Drew chooses not to play, and as a result his pockets become deeper. This intriguing dice game surprisingly leads to the seemingly unr- elated insight that the idea of a fair and democratic voting system is impossible—so much for "a government of the people, by the people, and for the people." answer:1/6 = 16.667% probability.

a tight weave

Sir Pinsky, a famous name in carpets, has a worldwide reputation for push- ing the limits of the art of floor covering. The fashion world stands agog at the clean lines and uncanny coherence of his purple and gold creations. Some call him square because his designs so richly employ that quaint quadrilateral. But squares in the hands of a master can create textures beyond the weavers' world, although not beyond human imagination. One day Sir Pinsky began a creation with, as always, a perfect, purple square. However, one square seemed too plain, so in the exact center of it he added a gold square. He saw that the central square implicitly defined eight purple squares surrounding it. As he pondered, he realized that those eight purple squares were identical to his original large square except for two things: (1) Each was one-third the size of the whole square; and (2) none of them had a gold square in its center. He wondered whether he could further modify his design so that each of the eight small squares would replicate the entire design except for being one-third its size. After much thought, he solved this puzzle and created a design with which his name is associated. Can you sketch and describe his design? Create this design in stages, adding more gold squares at each stage. Suppose the original square rug is 1 yard by 1 yard. How much gold material would be needed for the second stage? How much for the third stage? Continue computing the area of the gold squares at various stages of the process, and then guess how much gold material will be needed to create the final floor covering. The answer is surprising. answer: For part i, in 4 of 9 equal squares are colored gold. So it becomes 4/9 gold. (Which rounds to 44%, correct) For part ii, the remaining 5/9 is made 4/9 gold. So the amount of gold is: 4/9 (from step 1) + 5/9 x 4/9 (from step 2) = 36/81 + 20/81 = 56/81 which rounds to 56/81 x 100% = 69% gold

dropping trou

The highlight of Professor Burger's April 1993 talk to more than 300 Williams College students and their parents occurred when, after remov- ing his shoes, he tied his feet together with a stout rope, leaped onto the table, dramatically removed his belt, unzipped his zipper, and dropped his pants. The purple cows (Williams mascots) mooing about on his baggy boxer shorts completed an image not soon forgotten in the annals of mathematical talks. The more conservative parents in the audience were contemplating transferring their sons and daughters to a less "progres- sive" school. But then, at the moment of maximum shock and bewilderment, Profes- sor Burger performed the seemingly impossible feat of rehabilitating his fast-sinking reputation. Without removing the rope attached to his feet, he turned his pants inside out and pulled his trousers back to their accus- tomed position (though now inside out). Thus he simultaneously restored his modesty and his credibility by demonstrating the mathematical tri- umph of reversing his pants without removing the rope that was tying his feet together. Please attempt to duplicate Professor Burger's amazing feat—in the privacy of your room, of course. You will need a rope or cord about 5 feet long. One end of the rope should be tied snugly around one ankle and the other end tied equally snugly about the other ankle. Now, without removing the rope, try to take your pants off, turn them inside out, and put them back on so that you, the rope, and your pants are all exactly as they were at the start, with the exception of your pants being inside out.

scalable

The property of a network that allows you to add nodes or increase its size easily.

cantors theorem

There are more real numbers than natural numbers

infinitude of primes

There is an infinite number of primes.

one-to-one correspondence

Used to compare two sets in which one element matches one and only on element in the other set. ex. (1,2,3..), (2,3,4..) is there a correspondence? the second one is bigger than the first: the first set is n and the second is n+1

watsamattawith U?

Watsamattawith University (WU) is a fine institution, but a paradoxical place. They have comfortable dorm rooms, yet all the students sleep in class; their track team streaks from place to place, yet their cheeks are red with embarrassment as they lose every meet; every student is vegetarian, yet their dining facility is named Holstein Hall; and their student senate is called the House of Representatives. Go figure! And go figure, indeed—for that is exactly what the registrar of WU did in computing the average GPA of the current graduating class. Every year she computes the average GPA of the male students, the female stu- dents, and then all the students from that graduating class. This year she noticed something most peculiar. The average GPA of the male students in the current graduating class was higher than the average GPA of the male students from last year's graduating class and the average GPA of the female students in the current graduating class was higher than the average GPA of the female students from last year's graduating class. Sounds great for the current graduates. Unfortunately, she discovered that the average GPA of the entire graduating class was actually lower than the average GPA of last year's class. What!? Given that there were no errors in the registrar's computations, is it possible that such a phenomenon could occur or is this scenario so ridicu- lously impossible that merely asking the question deserves the response: Watsamattawith U?! If such a scenario is possible, explain how by describ- ing an example where the GPAs of the males goes up, the GPA of the females goes up, but the GPA of all the students goes down. Otherwise respond with . . . well, you know. answer:uppose there are 300 students in each of the graduating classes. In last year's class, 150 were male and 150 were female. Suppose that the average GPA of last year's men was 2.0 and the average GPA of last year's women was 3.5. Then the average GPA for last year's graduating class was 2.75. One way to arrive at that answer is to replace all the men's GPAs with the male average and all the women's GPAs with the female average and then average all 300 GPAs as follows: (2.0 x 150 + 3.5 x 150)/300 = 825/300 = 2.75 Now assume that this year's graduating class is comprised of 200 men and only 100 women. If the average GPA of the male students rose to 2.1 (higher than last year's average) and the average GPA of the female students became 3.6 (again higher than last year's average), then the GPA for this year's class can be found by computing: (2.l x 200 + 3.6 x 100)/300 = (420 + 360)/300 = 780/300 = 2.6! Amazing . . . until we realized we had the ability to change the proportion of men to women. By having a higher proportion of the poorer male students, even though the GPA of the males increased from 2.0 to 2.1, since there were more males this year, they dragged down the GPA of the student body. The number of male students compared to female students was a quantity that could be changed, but we might not have thought about that possibility. Hidden features of a statistical situation like this are sometimes called "lurking variables."

The fountain of knowledge

While on safari, Trey Sheik suddenly found himself alone in the Sahara Desert. Both hours and miles passed as he wandered aimlessly through the desert, but he stumbled upon an oasis as he was nearing dehydration. There, sitting in a shaded kiosk beside a small pool of mango nectar, was an old man named Al Dente. Big Al not only ran the mango bar but was also a travel agent and could book Trey on a two-humped camel back to Kentucky. At the moment, however, Trey desired nothing but a large drink of that beautifully translucent and refreshing mangoade. Al informed Trey that he sold the juice only in 8-ounce servings and the cost for one serving was $3.50. Trey frantically searched his pockets and discovered that he had exactly $3.50 (along with a lot of sand). Trey's excitement was soon shattered when Al casually announced that he did not have an 8-ounce glass - all he had was a 6-ounce glass and a 10-ounce glass, both with no markings on them. And, Al was quite adamant and would only sell Trey exactly 8-ounces of juice. answer:Fill the 10-ounce glass and pour juice from this glass into the 6-ounce glass until it is full - this leaves 4 ounces of juice in the 10-ounce glass. Next, empty the 6-ounce glass and transfer the 4 ounces of juice in the 10-ounce glass into the 6-ounce glass Fill the 10-ounce glass with mango juice and pour juice into the 6-ounce glass until it is full. Since it already had 4 ounces of juice in it, you can only add 2 ounces. This leaves you with 8 ounces of juice in the 10-ounce glass.

rigid symmetry

a motion of the plane the preserves the pattern and does not shrink stretch or distort

prime numbers

a number greater than one that cannot be expressed as the product of the two smaller natural numbers

rational number

a number that can be written as a fraction a/b or -a/b in which a and b are natural numbers or a=0

triangle with pythagorean theorem

a=b=1

rational numbers and natural

all rational numbers have same cardinltiy as set of natural numbers

Pythagorean Theorem

a²+b²=c² where a and b represent the legs of a right triangle and c represents the hypotenuse

golden rectangle

captures the notion of rectangless, if we divided the length of the shorter side by the longer we would come up with the golden ratio

set of natural numbers

collection of positive integers so basic and natural to our way of thinking

ratio

dividing one number into the other

prime factorization of natural numbers

every number greater than 1 can be expressed as a product of a prime number or is a prime number

a natural bond b/w

fibonacci numbers

Cantor's Diagonal Argument

for any given correspondence from the natural numbers to the reals we can construct a new real number that does not appear on the list

Leonardo of Pisa

founded the fibonacci sequence

why do numbers of spirals seem to be consecutive

growth and packing

the pinwheel pattern

has no rigid symmetry

Prime Number Theorem

how many primes there are up to n for any positive integer n

the ping pong ball condrum

illustrates the finite versus the infinite, there is a one to one pairing b/w the intervals of half times and the numbered balls

section 2: Pythagorean Theorem

in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse (the other two sides)

real number are more

infinite than natural

square root of 2

is an irrational number

square using pythagorean theorem section 2

length of hypothenuse of the original right triangle multiplied by itself or c squared

real numbers are left out of

natural numbers which makes them less infinite

quantitative estimation

not an exact number, something harder to count so we estimate

natural numbers

numbers that are so natural to us

the power of reasoning

often after we learn a principle of logical reasoning we see many instances where it applies

dividing natural numbers

one method of writing natural numbers is to divide and then see if there is a remainder

symmetry

pattern lies in the regularity with which it appears over the whole plane

Georg Cantor

proved that the set of real numbers has more elements than the set of natural numbers, some infinities are more infinite than others

rational and irrational numbers form

real numbers

patterns that have symmetry of scale have

rigid symmetry

rational numbers

sets of all ratios of numbers is the set of all integers (fractions) in b/w these rational numbers are infinite sets

section 2: geometry

study of shapes

to find new patterns

take an abstract pattern and look at them by themselves

1:1 correspondence

the 1 in natural numbers is paired with any number in a new set

the golden ratio

the fixed value of consecutive fibonacci numbers approach and that it can be expressed in a remarkable way as an endless fraction within a fraction within a fraction-captures art and aritecture equation is (1+square root of 5)/2

Damsel in Distress

the notorious knight after capturing the damsel took the bridge away the moat is 20 ft across and has no draw bridge the good knight stumbled upon two sturdy beams of walking across but they were only 19 ft by 8 inches wide answer: Use one beam to span the corner and the second beam to cross onto the ground.

relative sizes

the quotients of consecutive fibonacci numbers, each fibonacci number can be written as a sum of the previous two

if two sets have the same cardinality

there is a 1 to 1 correspondence

uniqueness of scaling

there is only one way to group the pinwheel triangle into super tiles to create a pinwheel super pattern in the plane

if the set of natural numbers has the same cardinality w/ the number 1 removed

there is the same cardinality of all integers and rational numbers

natural numbers

they are simply a sequence of counting numbers each successive one bigger than its predecessor.

Karl Gauss and A.M. Legendre

they noticed that even though primes do not appear to occur in any predictable pattern the proportion of primes is related to the so called natural logarithm a function relating to exponents

unending 1s

this fraction never ends 1+1/(1+!)/(1+...)) it only contains 1s

number personalities

unique characteristics and distinctions from other numbers

Irrational Power

use it to prove that other more exotic numbers are irrational