xevex january 2014

Volume IV, Issue One

December 2013/January 2014

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The Mathematics Magazine of Ramaz

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XeVeX



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Inside This Issue

Profiles:

The Story of George Dantzig (10)

Fermat’s Enigma (12)

Math & the Past:

Bible Code (4)

Square Roots and the Talmud (6)

The Konigsberg Bridge Problem (9)

Math & Problem Solving:

Mathematics of a Basketball Shot (5)

Cryptography (7)

Fermi Problem (11)

Mathematical Fun:

Math Art (7)

Crossword (10)



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One of the most important things for investors, traders, and people who are handling their retirement accounts is choosing the best stock. There are many different ways to evaluate a stock mathematically. One of them is by looking at its P/E ratio, which is one of the more simple evaluations.

P/E is a way to evaluate how much an investor is paying for every dollar of earnings. The P stands for price and the E stands for earnings. Usually, the price of a single share of a company is divided by earnings per share, or EPS, for the entire year. The average P/E of companies on the New York Stock Exchange is around 12. Often there are good explanations as to why a company would have a higher or lower P/E than the average. The way the EPS is derived is by taking the total earnings over a period of time, and dividing it by the number of outstanding shares. For example, Apple has 899.74 Million shares, and its earnings over 2012 were $35.657 Billion. If you divide Apple’s earnings by the outstanding shares, you get an EPS of $39.63. On Friday November 29, 2013, the closing price for a share of Apple stock was $556.07. If you divide the price, $556.07, by the earnings of $39.63, you get a P/E ratio of14.03. If the same procedure is followed

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for Google, another technology company, you would derive that the P/E ratio is 30.44. This number is much higher than that of Apple. Following this valuation, one would think that it would be a good idea to short, or bet against, Google stock, because of how expensive it is on a P/E basis, or, at least that Apple was a bargain.

The problem is if you evaluate it strictly on a P/E basis, you would short Google, while really that might not be the best idea. For instance, a company may trade at a higher P/E because earnings are expected to rise a lot. It may also trade at a higher P/E because it is a more stable company,

so people are willing to pay more for something stable, than for something that will fluctuate a lot.

For some companies, earnings are expected to fall, so their P/E ratio will be very low. While it may seem like you are paying very little for that company, the P/E ratio will most likely even out, even if the stock does not rise. While P/E is a good way to start, it is not wise to evaluate stocks using only this method, because there is often more to the story than just the current ratio of price to earnings. Much is based on the future expectations or opportunities as well.

Evaluating a Stock DJ Presser `16



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The Bible Code, otherwise known as Torah code, is a theory that there is a purported set of secret messages within the text of the Torah. This hidden code has been described as a method by which specific letters from the text can be selected to reveal an otherwise obscure message. The theory was popularized in Michael Drosnin’s book The Bible Code and in the movie The Omega Code. The hidden references are revealed by turning the text into a string of letters without spaces and looking for words formed by equidistant letter sequences. For instance, computers might select every ninth Hebrew letter and register a "hit" when a "coded" word intersects with a Bible verse containing related words. Major Bible scholars have ignored the code, noting that no one has a letter-by-letter version of the Bible as originally written. The oldest surviving manuscripts have slight variations. The theory was put forward in a 1994 article in the Institute of Mathematical Statistics, in which three Israeli scholars reported on tests using the Book of Genesis that produced intersections between names of famous rabbis and their birth or death dates. Opponents of the bible code such as Dror Bar-Natan and Gil Kalai, who teach mathematics at Jerusalem's Hebrew University; Maya Bar-Hillel,

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a psychology professor at the same school, and Brendan McKay, a computer scientist at Australian National University, claimed that “such letter configurations can be found in any long text.” For example, there is also a Koran code, which is why some Muslims claim superiority of the Koran. It seems anyone can have some sort of superhuman, magical code in their holy book from

God. There are even messages like “There is no God” and “Satan is Jehovah” in Bible code. Here’s what happens when we don’t have to deal with vowels: Using English as an example, let’s say we find an “R” in the code. Just an “R” in a strand of letters. Since I get to add the vowels at will, here is a sample of what I can do with just an “R” and my

choice of vowels: AIR, ARE, EAR, IRE, OAR, OR, ORE And if I add an “S” after the “R” (R and S are, after all, common letters in English): AIRS, EARS, OARS, ORES, OR IS, OR AS, ARISE, RISE, ROSE, RAISE Bible code is incredibly subjective and easy to manipulate. In fact, it isn’t just the case that I can manipulate this data. Since there are no vowels, and I am trying to make words from this mess, I must manipulate this data.

Bible Code Eddie Mattout `15



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If you were downtown on the evening of December 5, 2013 at the Flatiron Building, you would have seen an interesting gathering. Is the historic Flatiron Building really based on 5-12-13 triangle? Hundreds of New Yorkers were surrounding the landmark building, holding glow sticks in an effort to measure the sides of the triangular building. The Museum of Math will demonstrated how indeed, the Flatiron Building, is

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based on a 5-12-13 triangle. The museum also demonstrated a proof of the Pythagorean Theorem using three large squares, projected onto the side of the building.

It’s no wonder the Math Museum chose 12-5-13 to demonstrate this. After all, we all know that we’ll have to wait another 92 years until the date includes another Pythagorean triplet! Good catch Mo-Math!

Is the Flatiron Building a 5-12-13 Right Triangle? Sarah Ascherman `16

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The effectiveness that one is capable of shooting a basketball with is largely dependent upon practice, however mathematics play a large role as well. Many mathematical calculations can be taken into account in order to ascertain the best way to shoot a basketball. For starters, the diameter of the hoop’s rim is 18 inches, and the basketball itself has a 9.4-inch diameter. Therefore, there is nearly enough room for two basketballs to fit in the hoop. This may make it seem fairly easy to score the basketball, however the angle at which the ball approaches the rim determines the true diameter of the rim. For example, if the ball approaches the rim at 0 degrees, then its as if the rim is a straight line and the ball has access to none of its diameter. On

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the other hand, if the ball approaches the rim at 90 degrees, it will have access to the full 18-inch diameter of the rim. For the ball to be capable of

going in, it must be shot at an angle of at least 30 degrees. Using parabolic trajectory, mathematics and basketball experts have determined that 45 degrees (access to 13 inches of the rim) is the optimal angle for a jump shot to have the highest chance of going in. While this may seem counterintuitive, due to the smaller rim access, 45 degrees is the optimal angle because of the force required to shoot with a greater arc. The larger the angle the ball is shot at, the more force is required. When a player

shoots with greater force, he compromises the accuracy of the shot, thereby taking away the advantage of the greater rim diameter.

The Mathematics of a Basketball Shot Sammy Merkin `15



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To the surprise of many, including myself, the Amoraim during the time of the Gemara and the Rishonim that comment on the Talmudincluding Rashi and Tosfot, knew a lot about mathematics.

The Gemera on Succah 7 cites the opinion of Ravi that the minimum size of a square Succah must be 4 by 4 Amot (There are differing opinions as to the size of an Ama in today’s measurements. They range from 18-24 inches). The Gemara then discusses what is the minimum size of a round Succah The Gemara discusses the relationship between the circumference of a circle and its diameter. The Talmud approximates to be 3 and states that the ratio of the circumference of a circle to its diameter is 3:1. The Talmud then states that the ratio of a side of that square to its diagonal is 1:1.4 or 5.7. With this logic, the Gemara concludes that the diagonal of a square 4 Amot on each side will be 5.6 and if this square is inscribed in a circle the circumference of the circle will be 16.8.

Note the imprecision of 1.4. We know that 1.4 is close to the accurate length of the diagonal of a square of 1 x 1, but really is not close enough when dealing with bigger numbers, as will be noted by Tosfot. As derived by the Pythagorean Theorem, the real length of the diagonal would be

, which is closer to 1.414, which is an irrational

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number. I believe that is why the Gemara just

approximated to one decimal place.

However, the Ba'ali Tosfot note this inaccuracy in their last commentary on that Daf . They do so by setting up an example. They describe a diagram of a square, 10 Amot by 10 Amot, with an area of 100 square Amot. They divide this square into 4 equal quadrants, each being 5 x 5. They then cut the diagram further, by making a square inscribed in the bigger square, as shown in the diagram.

Mathematically speaking, this new inscribed square’s area has to be 50 square Amot because it has to be half of the original square’s area of 100 square Amot. However, if one uses the approximated (1.4 x 5) length of the new square, and squares that length to get the area of the square, as the Gemara would, they would notice that they don’t get 50, rather they get 49- (7 x 7). Thus, the Ba'ali Tosfot conclude that the 1.4 is less than the actual number, and the approximation does in fact lead to important inaccuracies in practical matters. So, their conclusion is that the ratio of a diagonal of a square to its length is slightly above 1.4:1, which is a very impressive fact, considering how long ago they lived.

in the Talmud Ben Rabinowitz `16

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Volume V, Issue One December 2012

Calculator Art Michael Weisberg `17

Cryptography Zachary Metzman `16

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If you love math and you love Chanukah, why not combine them? Using your TI-84 calculator you can make a chanukiah! Step 1: Press the Y= button on the top of your calculator. Step 2: Type in the following functions on your calculator: Y1= X2 + 10 Y2= X2 + 20 Y3= X2 + 30 Y4= X2 + 40 In Step 2, we created the branches of the chanukiah. Meanwhile, the Y-axis will serve as the shaft. In Step 3, we will create the base of the chanukiah. Step 3: Type in the following function on your calculator: Y5= -X2 + 9.5 To optimize your chanukiah hit the window button on the top of your calculator and change

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the Ymin to -25 and the Ymax to 125. The following steps will cover the process of saving your chanukiah. Step 4: Hit the 2nd button and then PRGM

Step 5: Move two spaces to the right (STO) and scroll down to number 3: “Store GDB.” Press ENTER. At this point you will have to name your GDB (Graph Database). The following steps go through the process of reopening your chanukiah: Step 6: Repeat step 4 Step 7: Scroll down to number 4: “Recall GDB” and press ENTER. Step 8: At this point, you will have to give the name of your

GDB. Step 9: Press Graph HAPPY CHANUKAH!

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The use of cryptography exploded during Caesar's rule. According to Suetonius, Caesar used a shift cipher, known as the Caesar Cipher, to encrypt messages of military significance.

The Caesar Cipher requires at least two parties to

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agree in person on a shift number. A shift number is the fixed distance of letters down the alphabet that will substitute each letter in the unencrypted message. Below is an example of a shifted alphabet and a Caesar Cipher written with the shifted alphabet.



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The top is the regular alphabet and below it is the alphabet shifted five spots to the right.

Unencrypted: Hello my name is John Doe.

Encrypted: Mjqqt rd sfrj nx otms itj.

The Caesar Ciphers was used up until the 9th century, when the Arab mathematician, Al-Kindi, discovered a technique to crack Caesar Ciphers. Al-Kindi realized that in all languages there are letters that occur more frequently than other letters in that language. So in a Caesar Cipher, which is merely just changing the representation of a letter, the ratio of the occurrence of letters will be equal to that of the language it is written in. Let’s say ‘f’ occurs most frequently in a Caesar Cipher. Then it is most likely that ‘f’ represents ‘e’ because ‘e’ is the most commonly used letter in the English language. Once you are able to decrypt one letter, you are able to find the shift number thus cracking the encrypted message.

Years later, a new form of cryptography, the one-time pad, was derived. Instead of using one shift number for a whole message, as done in the Caesar Cipher, each character in a message is given a random number from 1 to 26 as a shift number. Since each character has a random shift number, there will be an equal occurrence of each letter in the encrypted message thus making the one-time pad method uncrackable.

Both of these forms of cryptography have a problem; they require parties to exchange information in person. What happens if parties are unable to meet each other, then how will they securely communicate with each other?

Using the modulo operation, in 1976, Whitfield Diffie and Martin Hellman developed a one way function that allows parties to agree on a key publicly. The modulo operation is finding the

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remainder of a generator divided by modulus, commonly written as (generator) mod (modulus).

Diffie-Hellman Key Exchange requires the at least two parties to share a prime modulus and generator publicly. Each party then raises the generator to a random number and then publicly sends the result of the operation to the other party. Once each party has received the other’s result, they replace the original generator with that number and raise it to the same number they raised the original generator to. When each party computes the operation the result will be the same. Each party can use this number in any form of cryptography they want.

Persons A and B share 3^x mod 17 publicly.

Person A: 3^ 13 mod 17 and sends the result which is 12 to person B publicly.

Person B: 3^15 mod 17 and sends the result which is 6 to person A publicly.

Once they receive each other’s result:

Person A: 6^13 mod 17 = 10

Person B: 12^ 15 mod 17 = 10

The both arrive at the same number because:

12 = 3^13 mod 17 therefore 12^15 mod 17 = 3^13^15 mod 17

6 = 3^15 mod 17 therefore 6^13 mod 17 = 3^15^13 mod 17

and

3^15^13 mod 17 = 3^13^15 mod 17

Diffie-Hellman Key Exchange is important because it laid the basis for the discovery of many other forms of public key cryptographies, like RSA, allowing the use of the internet to grow geometrically.



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Koningsburg was a city in Prussia, but today it is called Kaliningrad and it is in western Russia.

The Pregel River flowed through Koningsburg and seven bridges crossed the branches of the river. A famous problem was whether it was possible to take a walk through each part of the city so as to crossover every bridge only once. No one succeeded in doing this. This problem was given to a famous Swiss mathematician called Leonhard Euler (1707-1783) who in the process of his solution, invented the branch of mathematics now known as graph theory

Euler replaced each land mass with an abstract "vertex" or node and each bridge with a connection, an "edge" which helps us see which pair of vertices (land masses) are connected by that bridge. The resulting structure is called a graph.

An Euler path is a path that transverses each edge once and only once. Euler shows that the

The Konigsberg Bridge Problem Skyler Levine `15

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existence of a path depends on the degrees of the vertices. The degree of a vertex is the number of edges touching it. A necessary condition for an Euler path or walk is that the graph be connected and have zero or two vertices of odd degree. The network representing the Koninsburg bridge problem has 4 odd vertices and therefore does not have an Euler path.

Look at these networks:

The one on the left has 4 vertices with a degree of 2. It has an Euler path starting at any vertex and ending at the same vertex. The next one has vertex A and B of degree 3(odd) and the other four vertices are of even degree (two). This network has an Euler path starting at A or B and the ending point must be the other odd vertex. The network on the right does not have an Euler path, because it has 4 odd vertices.

In the Koningsburg bridge problem all 4 vertices are odd and therefore, the graph does not have an Euler path. Koningsburg was bombed during World War II. Two of the seven bridges did not survive. Now there are five bridges only. The Euler path does exist now as only 2 vertices are odd (degree 3).The path although possible now, is impractical for tourists since it must begin in one island and end on the other island.

The applications of graph theory are many in science and engineering. Graphs have other applications such as molecular structures in Chemistry, computer networks and electrical networks.



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The Story of George Dantzig Zachary Metzman `16

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George Dantzig was a mathematical scientist, whose fame originated from a mistake he made as a graduate student at UC Berkeley. One day he was late to class. When he arrived, George copied two problems from the board assuming they were for homework. A few days later, Dantzig handed in the completed homework problems noting that the problems "seemed to be a little harder than usual". Six weeks later, on an early

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Sunday morning, Dantzig was visited by his Professor Neyman who told him that the two problems that he handed in as homework were actually two famously unsolved statistic problems. A year later, when Danzig had to choose a thesis topic in order to earn his PhD, Neyman, said he would accept the two problems he solved.

Math Crossword Sarah Ascherman `16



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Question: How many characters do all American teenagers and young adults (15-25) type/text every year?

Solution: Let’s begin with how many characters all Americans text every year.

Let’s assume that the average teenage sends about 30 text messages everyday (more than 10 less than 100) or 1000 text messages every month which results in the average American person between ages of 15 and 25 sending 12,000 texts every year.

Now we must estimate how many characters are in the average text message. I am going to estimate around 10 (geometric mean: over 1 less than 30).

Lastly we must figure out how many people are in the age group of 15- 25. If there are about 300 million people in America and the life expectancy is 75 years old,

3x108 / 75 = 4x106 people in in each year of life. Since there are 10 years of life in between 15-25,

there are 4x107 people between the ages of 15 and 25.

So the number of characters texted every year=

(4x107 people) x(12000 texts/year-person) x(10

A Fermi Question Eddie Mattout `15

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char/text) is about 5 x1012 characters a year.

Now, let’s compare this to the number of characters typed by Americans in the same age group.

Let’s assume that the average person types about 50 words per minute, and that there are about 500 words per page, and about 5 characters in each word.

Next we must figure out how much time a person spends typing. If the average person spends about 5 hours a day in front of a computer, and types for one and a half of those hours.

(1.5 hours/day) x (24 hrs/ day) x (60 min/hour) x

(50 words/ min) (5 char/ word) (365 days/ year)=

2 x 108 char/yr.

If we multiply this by the number of American people in the age group of 15-25 we get:

(2 x 108 char/yr) x (4x107 people) = 8 x1015



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This year, Rabbi Stern’s 11th grade Honors Pre-Calculus class is reading the book Fermat’s Enigma by Simon Singh. The book is centered on Andrew Wiles’ solution to Pierre de Fermat’s final theorem, that xn + yn ≠ zn where n represents any number greater than two. While the book’s focus is on Wiles’ solution, Singh also takes the reader through the failed proofs and mathematical milestones that all contributed to Andrew Wiles’ eventual success in 1995.

Singh begins his tale by detailing Pierre de Fermat’ life in 17th century France, a time when Cardinal Richelieu held control over Paris’s government and the plague threatened the health of everyone in Europe. Fermat was a mid-level statesman who devoted most of his spare time to mathematics. Fermat had little formal mathematical schooling, but was considered the “prince of amateurs” because he was “so really great that he should count as a professional.” Interestingly enough, Fermat actually discovered the rules of probability along with his friend Pascal, rules that have an extraordinary effect on how we live our lives. Fermat first published his final theorem in 1637, and 28 years later, in 1665,

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Fermat died.

The book goes on to discuss the numerous mathematicians who attempted and failed to solve the last theorem before Wiles finally succeeded in 1995. Wiles’ life story is an extremely interesting one. He first encountered Fermat’s final theorem in 1963 when he was ten years old in a library on Milton Road when he was a child in Cambridge, England. The Last Problem by Eric Temple Bell interested Wiles’ because, unlike any other puzzle book, The Last Problem contained no solution to its problem. Wiles was finally able to solve the theorem by isolating himself for seven years in order to devote all his time and effort to finding a solution.

The thing that surprised me most about Fermat’s Enigma is how much I am actually enjoying it. Every two weeks we are assigned another chapter to read and each time I always to find out what happens next in the pursuit of the solution to arguably the greatest challenge that mathematics has ever known.

Fermat’s Enigma Avi Goldman `15

Faculty Advisor:

Dr. Koplon

Editors:

Brandon Cohen `14

Dan Korff-Korn `14

Layla Malamut `14

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