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PI DAY EDITION Editors Michael Rosenberg Benjamin Kaplan Brandon Cohen Dan Kor-Korn
Faculty Adviser Dr. Renee Koplon
Contributors Sarah Ascherman Jacob Berman Brandon Cohen Benjamin Kaplan Ma hew Levy Skyler Levine Eddie Ma out Zachary Metzman Sammy Merkin Benjamin Rabinowitz Michael Rosenberg Tess Solomon
Math Disabilities Tess Solomon '16
Learning disabilities are conditions that typically are caused by the brain’s inability to receive and process information “normally”. However, having such a condition by no means implies that someone who suffers from a learning disability is unable to learn. As has been studied extensively in recent years, many people who have learning disabilities simply learn at a slower pace or in different ways from people without them. There are several disabilities that specifically apply to absorption and retention of mathematical skills and principles. Related disabilities involve what has been called the “emotional” ability to learn them. These disabilities mostly fall under the descriptive names, dyscalculia and math anxiety. Dyscalculia describes a wide range of learning disabilities that affect absorption and retention of material. According to studies done in the United Kingdom, between 3.6 and 6.5% of the population there is dyscalculic. No international study has been done to estimate how common it is in other countries or regions. There are two main types of difficulties that can contribute to a diagnosis of dyscalculia. The first is a difficulty processing visual-spatial relationships, relating to the brain’s ability to process images it receives from the eye. People with this disability have difficulty visualizing patterns or estimate distances. The second is language-processing difficulties, relating to the brain’s ability to process sound it receives from the ear. This can make it difficult to understand math word problems or retain math vocabulary. Usually, dyscalculia is noticed and hopefully identified when the person is in elementary or middle school. A trained professional identifies dyscalculia and then tells the student his or her specific
weaknesses and how they should proceed. In many states, those between the ages of 1-26 are entitled to free testing by their local public education service district. Many times, the answer involves getting help outside of the classroom to work on specific issues the student might be having. Working with the student outside of the classroom also takes away the pressure of moving on to new topics too quickly, before the student fully understands them. Another option, or sometimes an additional option, employs alternative methods of teaching and learning. For example, sometimes students learn more easily if they are provided with examples before being taught a general rule. Math anxiety is a different malady. Professor Mark H. Ashcraft, Ph. D., describes math anxiety as being “a feeling of tension, apprehension, or fear that interferes with math performance.” As opposed to dyscalculia, where a student is typically incapable of doing the math problem in the same way as a student without dyscalculia, students with math anxiety will try to avoid situations in which they will have to perform calculations. According to a study at the University of Chicago, math anxiety actually has nothing to do with the math itself. It is the anticipation of solving the problem, or the fear of not being able to solve it, that actually causes the anxiety. The area of the brain that is triggered when someone has math anxiety is the same place in the brain where bodily harm is registered. Also, the effects of math anxiety reach beyond numbers. A correlation has been found between math anxiety and low confidence and motivation. Treatment of math anxiety falls into two categories. The first essentially relates to what attitude is taken by the school or teachers towards the student. 1
The expectations of parents and teachers are not usually the cause of math anxiety, but they can perpetuate anxious feelings if they set unrealistic expectations for the students. Teachers who accommodate different learning styles and have a positive attitude in what they teach have been found to be more successful in helping students overcome their anxiety. The other category of treatment is therapy, where the reasons for the anxiety are addressed directly and remedial help in math can be given. Therapists introduce new coping devices to the students, which can help them channel their anxiety elsewhere. For
example, relaxation techniques such as meditation have also been found to help anxiety. In conclusion, our society is increasingly math and science dependent. Our individual and collective abilities to succeed depend importantly on greater and greater numbers of us excelling in math. By understanding the root causes of two of the most prevalent learning disabilities associated with math — dyscalculia and math anxiety — we as a society have brought ourselves one important step closer to achieving individual and societal mastery in math.
Mathematics in The Simpsons Sammy Merkin '15
The writing staff for the renowned animated television show, The Simpsons, includes an abundance of brilliant men with impressive backgrounds in mathematics. While it may seem like these men have wasted their adeptness for mathematics, they have not, as many episodes of the show include some very sophisticated mathematics. Frequently in the background of scenes during the show complex mathematical solutions are written out or implied. For example, in a 1998 episode “The Wizard of Evergreen Terrace,” written by David X. Cohen, a Harvard graduate, a near miss solution for Fermat’s Last Theorem was written on a blackboard in the background. This was noticed by none other than Simon Singh the writer of Fermat’s Enigma, a required reading for Rabbi Stern’s Honors Pre-Calculus Class. In fact, Singh recently published his fifth book titled: The Simpsons and Their Mathematical Secrets, an in-depth analysis of the underappreciated mathematical brilliance of the show. This also shows why it makes sense that Mr. Greene is teaching a minicourse on The Simpsons. Another example of math in The Simpsons appears in a 2006 episode when three numbers appear on the screen asking fans at a baseball game
to guess the attendance. While the numbers seen arbitrary and ordinary, they are far from it. The first option, 8,128, is a perfect number, meaning all of its divisors add up to equal the number itself. The second number 8,208, is a narcissistic number because the sum of each of the numbers raised to the fourth power equals the number itself. The final option 8,191 is not only prime number, but a Mersenne prime number, named after the 17th century French Mathematician Marin Mersenne. When David X. Cohen was asked why a brilliant mathematical mind was suited for comedy writing he replied: The process of proving something has some similarity with the process of comedy writing, inasmuch as there is no guarantee you are going to get to your ending. When you’re trying to think of a joke out of thin air, there is no guarantee that there exists a joke that accomplishes all the things you’re trying to do and is funny as well. Similarly, if you’re trying to prove something mathematically, it is possible that no proof exists. And it is certainly very possible that no proof exists that a person can wrap their minds around.
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Pay some Interest: Annuity or Perpetuity Ma hew Levy '16
A perpetuity is a type of annuity, and while both share a common ideology, their differences are timebending (literally). Annuities are a type of fixedincome investments from which you receive periodic interest payments over a series of time (often to ensure a long term steady cash flow) and receive interest on your principal investment. While there are many different variables, interest payments are most commonly made monthly, quarterly, semi-annually, or annually. While your return can also vary based on whether you receive payment at the beginning or end of the period, there is one major difference between annuities and perpetuities: your principal. Perpetuities give you interest payments forever on your investment, but never return your principal investment explicitly, compared to an annuity which gives interest for a period of time and then repays your investment. You might be asking yourself, why would I put my money into a perpetual investment? The one thing that all investors look for is upside: how much money can I make (no, this is not a strictly Jewish reference)? What might be mindtwisting is that you will receive interest payments forever on a perpetual investment. This means that you will make money (off of interest) as long as your n’shama — forever. However, our physical bodies and uses for money aren’t infinite, meaning we have to find the point in time that a perpetual investment would overtake an annuity (in terms of return). If, for example, Saul and David each wanted to invest $1,948 into a fixed income investment with an annual interest rate of 1.8%, which should they invest in?
P=
r(PV) . 1 − (1 + r)−n
P = Payment PV = Present Value r = rate per period n = number of periods He would in tern receive roughly $214.60 per year for ten years, or $2,146 which is approximately 10% return over 10 years. 2. David, preparing for the future (of Israel), invests his money into a perpetuity with the same rate and principal investment. Using the perpetuity payment formula: P = PVr P = Payment PV = Present Value r = annual interest rate David will receive more than $35 forever.
Assuming that David and Saul do not reinvest their periodic payments (from the annuity), it would take Saul 61.21 years to match David’s return of $2,146, but will continue to receive the $35.06 per year forever. The question one must ask himself is: is it worth the wait? In my opinion, sometimes it’s better to wait, as you appreciate it more when you get your 1. Saul invests his money into a 10 year annuity “investment” back! (compounded yearly). Then: 3
Fibonacci Representation of Naturals with Zeckendorf Michael Rosenberg '15
Introduction There exist many different ways to represent numbers, whether they be irrational reals, naturals, complex, etc. One such way of representing natural numbers is by the sum of Fibonacci numbers. More specifically, Zeckendorf ’s theorem proposes that every natural number has a unique representation as a sum of non-consecutive Fibonacci numbers.
Proof We will prove Zeckendorf ’s theorem. To do so, we will need to prove both the existence of such a representation, then the uniqueness of it. Existence We will prove the representation’s existence using mathematical induction. We begin with 1,2, and 3 as they are Fibonacci numbers themselves. Suppose all n ≤ k have a Fibonacci representation for some upper bound k. If k + 1 is a Fibonacci number, then it has a trivial Fibonacci representation and we are done. Otherwise, there must exist two consecutive Fibonacci numbers such that Fj < k + 1 < Fj+1 . Let a = (k + 1) − Fj . Since a < k, a must have a Fibonacci representation. Furthermore:
tion. Firstly, we require a lemma. Lemma: The sum of any non-empty set of distinct, nonconsecutive Fibonacci numbers not containing F0 or F1 whose largest member is Fj is strictly less than Fj+1 . Proof: We will use a∑ proof by induction on j. The base case is F2 = 1. {1} = 1 < F3 = 2. Now assume the theorem is true for all j ≤ n. Let P be a set of distinct non-consecutive Fibonacci numbers whose greatest element is Fn+1 . Let P′ be the set of all elements of P besides Fn+1 (set-theoretically denoted as P′ = P \ Fn+1 ). The next greatest element of P can be no larger than Fn−1 , as no consecutive Fibonacci numbers ∑ may be in P. Thus, by inductive hypothesis, P′ := p < Fn . And if p < Fn then p + Fn+1 < Fn+2 = Fn+1 + Fn .
Now that the lemma has been proven, we may prove the uniqueness of the Fibonacci representation. Assume that there exist distinct sets S and T such that their elements non-consecutive Fi∑ are∑ bonacci numbers and S = T = n where n is some natural number. Let S′ = S \ T, that is the set of all elements in S that are not in T. Similarly, Fj + a = k + 1 let T′ = ∑ T \ S. The sum still ∑of these two sets ∑ are ′ ′ equal, as S = n − (S ∩ T ) = T . De< Fj+1 ( ) note the largest element of S′ as Fs and the largest by property of Fibonacci = Fj + Fj−1 element of T′ as Ft . Note that Fs ̸= Ft . Without numbers loss of generality, assume that Fs < Ft . It is obvia < Fj−1 ∑ ′ ′ ous that T ≥ Ft as Ft is ∑an ′element of T . From the lemma, we know Fs+1 ≤∑ FT . But ∑ that ∑S < ′ ′ Since a < Fj−1 , it must be the case that Fj is not this implies that < T =⇒ S′ ̸= ∑ ′ ∑ S ∑ a consecutive Fibonacci number to any of the numT =⇒ S ̸= T, thus contradicting the bers in the Fibonacci representation of a. Therefore assumption. ■ k = a + Fj fits the criteria and k has a Fibonacci representation. References Uniqueness
http://www.proofwiki.org/wiki/Zeckendorf% We will prove the uniqueness of the Fibonacci rep- 27s_Theorem resentation of a number using a proof by contradic4
Pi in the Torah Jacob Berman '16
For many years people have wondered if anywhere in the Torah, there was any mention of the irrational number we now know as pi. Pi is the number that is used to find a circle’s circumference or area. Its value is 3.1415… and 22/7 is often used to approximate it. In Jewish texts, the use of gematria, or letters that represent numbers, is often used to make connections from the text to numbers. Every specific Hebrew letter has a value, and Rabbis often interpret lines of the Bible as numeric codes. There have been many different explanations of words in the Bible that signify the numbers of pi. The Vilna Gaon, a Lithuanian scholar, offers an interpretation in the Bible, which represents many of pi’s digits. In the Book of Kings the Hebrew word for perimeter is used. The word is written “Kava” but read “Kav”. If you
take the value of Kava, which is 111, and divide it by the oral form Kav, whose value is 106, and the multiply by 3 you get 3.1415094…! That is the value of pi to the ten thousandth column or the fourth digit! Another famous connection in the Bible to pi, are said by God. The first words of the first commandment is “I am God” or Anochi and then God’s name, Yehova. The value of Anochi is 81 divided by God’s name, which is 26. This answer is approximately the value of pi, or 3.1. The fact that there are connections between a scroll written thousands of years ago, to a recently understood number such as pi, is astounding. There are many other connections with pi and the Bible, some weak and others fascinating but these are the primary connections.
Simpson's Paradox Skyler Levine '15
Simpson’s Paradox is a statistical phenomenon that occurs when data is aggregated. This data may reveal a trend that contrasts with that of its subgroups. The concept was illustrated by Simpson in a paper written in 1951.
Surgeon A B
# Patients # Survived % Survived 100 95 95% 80 72 90%
Sometimes data can be misleading and the story From this analysis, which surgeon should we on the surface can take people in the wrong direction. To make more sense of Simpson’s paradox, let’s choose to treat us? It would seem that surgeon A is look at the following example. In a certain hospital the safer bet. But is this really true? From further research into the data we found there are two surgeons, surgeon A and surgeon B, that of the 100 patients that surgeon A treated, 50 with the following statistics: were high risk, of which three died. The other 50 were considered routine, and of these 2 died. This means that for routine surgery, a patient treated by surgeon A has a 96% survival rate. 5
Now we look more carefully at the data for sur- players had the following stats: geon B and find that of the 80 patients 40 were high risk, of which seven died. The other 40 were rouHitter A tine and only one died. This means that a patient has 97.5% survival rate for a routine surgery with surPitcher Type At-Bats Hits Average geon B. Righty 300 90 .300 Lefty 200 50 .250 Now which surgeon seems better? If your surgery is to be a routine one, then surgeon B is acTotal 500 140 .280 tually the better surgeon. However if you look at all surgeries performed by the surgeons, A is better. Another example of Simpson’s Paradox is evident in educational testing. Between 1981 and 2002, the national average for the verbal Scholastic Aptitude Test (SAT) score appeared to remain relatively stable at 504 points (Bracey 32).However, during that same time period, the average verbal scores for all racial and ethnic subgroups increased by between eight and twenty-seven points. Bracey attributed this to changing demographics of SAT test takers. Over this time period, the number of white students taking the SAT fell while the number of minority students rose. Performance improved across all racial and ethnic groups, but minority (excluding Asian American) students’ average verbal scores remained below the national average. Higher numbers of increasing but below average scores resulted in a national average that not only failed to reflect subgroups’ improved verbal scores; they failed to reflect any change at all.
Hitter B Pitcher Type Righty Lefty Total
At-Bats Hits Average 100 32 .320 300 78 .260 400 110 .275
Hitter B has a higher batting average against both righties and lefties, but Hitter A has a higher overall average; therefore, I would advise that one inserts him as pinch hitter. As is evident, Simpson’s Paradox applies to many cases in several different fields. References
Bracey, G.W (2004). Simpson’s paradox and other statistical mysteries. American School Board Journal, 191, 32-33. Examples of Simpson’s paradox appear not only Courtney Taylor. What is Simpson’s Paradox? in medical research and educational testing but also Goltz, Heather Honore & Mathew Lee Smith in sports rating. For example, suppose two baseball (2010). Simpson’s Paradox in Research.
History of Math Education Sammy Merkin '15
Elementary mathematics was a part of the education system in most ancient civilizations, including Ancient Greece, Ancient Egypt, and the Roman Empire. In ancient civilizations, an education was only available to high-class males. Documents from Mesopatamia, dating back to 1800 BCE, were found with multiplication and division, as well as meth-
ods for solving quadratic equations. The Rhind Papyrus, is an Ancient Egyptian math textbook that dates back to 1650 BCE, however it is most likely a copy of an even older document. Math continued to be taught throughout the middle ages. In medieval Europe, mathematics was taught to apprentices where it was practical in their 6
trade. All geometry that was taught was based upon Euclid’s Elements. The first mathematics textbook written in English and French was The Grounde of Artes, published by Robert Recorde in 1540. Unfortunately, during the Renaissance period, mathematics was set aside in favor of the study of philosophy. However, in the seventeenth century there was a revival of mathematics studying and Mathematics chairs were set up in Aberdeen, Oxford, and Cambridge universities. It was uncommon for mathematics to be taught anywhere outside of universities. The Industrial Revolution in the 18th and 19th centuries caused mathematics to become a foundational skill that most people needed. Practical applications of mathematics, such as telling time and counting money required simple arith-
metic to be taught everywhere. The institution of public education systems allowed for math to be taught at a young age everywhere. Today, many countries, such as England, have standards for what and how much mathematics must be taught at schools. However, in the United States the government releases recommendations of what should be A page from the original taught in each grade but book The Elements by Eudoes not enforce these clid ideals. Despite this, AP Statistics and AP Calculus are among the most popular APs with 150,000 and 385,000 respectively.
The Perfect Bracket Benjamin Kaplan '16
Since the year 1939, every year around March there has been the NCAA Men’s Division I Basketball tournament. As of now 68 Division I teams qualify for the single-elimination tournament, which consists of 7 different rounds. The tournament and the events surrounding it, including brackets, have received the name March Madness. A bracket is a form of gambling in which people try to guess and decide who they believe will win each game. This practice has grown exponentially leading to people trying to get a perfect bracket. This year, as well as last year, there is a perfect bracket challenge by QuickenLoans. The challenge appears quite simple: make a perfect bracket and receive a large prize totaling $1 billion. However, if we look at the mathematics, it is not that simple. To try and determine the probability, let’s assume each team has an equal chance of winning each game. So to determine that, it would be 263 , as there are 63
games. If you do that out, there are 9.2 quintillion ways to fill out a bracket. So if you still think you are going to win the Billion Dollar Perfect Bracket Challenge, you may want to reconsider. Now to put those odds in perspective, the odds of winning the lottery are 1 in 175 million, and the odds of being struck by lightning are 1 in 3000. Now obviously, your odds are slightly better if you know basketball, as you probably will not bet on the 64th seed to upset the 1st seed. According to a DePaul math professor, John Bergen, if you know something about the NCAA basketball tournament the odds are more like 1 in 128 billion. Using that number, according to Chris Chase, someone would get a perfect bracket once every 400 years if every person in the US filled out a bracket. Everyone fantasizes and theorizes about how to achieve this phenomenon but it does not seem likely to occur in the near future.
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Ulam Spiral Zachary Metzman '16
Prime numbers are very important and frequently used in everyday life. They are required in many forms of cryptography. Normally, when large prime numbers are used in a form of cryptography, the encryption is strong. It has always been a challenge to find large prime numbers even with computers. It is so difficult for a computer to find new prime numbers that many individuals who want to test the durability of their computer search for primes. The largest prime number known today is 17425170 digits long, making it 4446981 digits longer than the second largest prime number known. Huge gaps like this make it difficult for computers to find primes. In addition, the general occurrence of prime numbers was not known until mathematician Stanislaw Ulam acciShortly after his discovery, he was able to use dentally discovered a pattern. Ulam was at a presen- a computer to generate a spiral up to 65,000. The tation of a “long and very boring paper”, when he be- Ulam spiral became famous when Martin Gardner gan drawing a spiral of numbers like this: published an article about it in Gardner’s Mathematical Games column. One can use the equation of a heavily dotted diagonal to generate all the numbers in it, giving computers fewer numbers to search through in order to find a new prime. For example, the diagonal that contains 3, 13, 31… has the equation 4x2 − 2x + 1 where x is the xth number in the diagonal. So if you want to find the 10th number in that diagonal, He then circled all the prime numbers in the plug in 10. Many mathematicians have tried to exspiral and realized that the prime numbers were in plain the reasoning behind these patterns but none semi-consecutive rows, diagonals, and columns. Be- of them have been able to prove their conjectures, low is an Ulam spiral where each dot is a prime num- further showing the significance and mysteriousness ber: of the Ulam spiral.
Go fried Leibniz Sarah Ascherman
Gottfried Wilhelm von Leibniz may be best known for his discovery of infinitesimal calculus— mathematics concerning functions, tangents to curves, as well as maxima and minima. However, you may not know that Leibniz was also a renowned
philosopher. Additionally, he studied physics, theology, history, biology, candle making, legal affairs and linguistics! Leibniz was born in Leipzig, Saxony, in 1646 and died in 1716, living around the same time as Isaac Newton, who also independently 8
discovered infinitesimal calculus. By the age of 16, Leibniz had attended university and had earned his bachelor’s degree in philosophy. In the area of philosophy, Leibniz strongly believed in rationalism. Four years later, at the age of 20, Leibniz wrote his first book on philosophy, “On the Art of Combinations.” In the study of mathematics, Leibniz not
only discovered infinitesimal calculus, but also developed mathematical notations that we still use in our ∫ classrooms to this day, such as the integral sign, . So next time you’re in math class learning calculus or using the integral sign, you’ll know where it comes from! P.S. On a sweeter note, Leibniz even has cookies named after him!
Measurements in the Talmud Ben Rabinowitz
On 109a-b in Pesachim, the Talmud goes through great explanation of how we come to the measurement of the reviis—the unit that we use when measuring the minimum amount of wine we need for the four cups of wine on Pesach, and kiddush on shabbos. It is stated that a mikvah’s minimum dimensions are 1 × 1 × 3 amos. We use a Mikvah to purify ourselves and our utensils. The next statement of the Talmud is that that measurement—3 cubic amos— 40 se’ah, which, as mentioned before, is the mincontains forty se’ah of water. Earlier in the sugya, R’ imum volume of a Mikvah, is 3, 840 reviios. Chisda said that the reviis of the Torah, which they So, if we equate the two values which we reused for measuring liquids as well, was 2 × 2 × 2.7 ceived: fingerbreadths. (The way he states it is 2 × 2 × (2 + 1/2 + 1/5) fingerbreadths.) 1 × 1 × 3 amos = 24 × 24 × 72 fingerbreadths The following dimensional analysis is performed = 41, 472 cubic fingerbreadths to achieve the measurement of the reviis in terms of fingerbreadths, to reinforce R’ Chisda’s statement: = 41, 472 fingerbreadths3 = 40 se’ah = 3, 840 reviios. Lastly, we show that
1 amah 1 tefach 1 amah = × 6 tefachim 4 fingerbreadths = 24 fingerbreadths 1 se’ah 1 kav 1 lug 1 se’ah = × × 6 kabim 4 lugin 4 reviios = 96 reviios.
fingerbreadths3 41, 472 fingerbreadths3 = 10.8 , 3, 840 reviios reviios and 10.8 = 2 × 2 × 2.7, thus proving R’ Chisda’s point.
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Illegal Numbers and Colors Zachary Metzman '16
Computers do not use letters; they use binary numbers to represent symbols. If it is illegal to distribute a piece of information in its normal form, then it is illegal to distribute it in any other form. If copyrighted information is distributed even in binary form, that is illegal. Similarly, a group of colors can be illegal because binary can be converted to hexadecimal which in turn can be represented by colors. In recent years, a feud involving illegal numbers happened between Advanced Access Content System Licensing Administrator (AACS LA) and the website Digg. In 2006, a forum user, muslix64, published encryption keys to DVD players on Doom9. The encryption keys gave users the ability to copy or pirate DVDs. The next year, the AACS LA issued Digital Millennium Copyright Act violation notices to various websites that hosted the encryption keys. Many websites took the keys down but Digg was defiant. The founder of Digg told the AACS LA, “We hear you, and effective immediately we won’t delete stories or comments containing the code
and will deal with whatever the consequences might be.” Infuriated, the AACS LA attempted to remove all keys from the internet. Paradoxically, the num- The first 16 bytes of an enber of keys online went cryption key represented in from 9,410 Google re- colors sults to 300,000 Google results in one day, making it impossible to remove the keys from the internet. Digg embedded the information in songs and converted the keys to numbers and groups of colors in case a court would force Digg to take the keys down. After the wide spread of the numbers, the American BAR Association published a paper questioning illegality of publishing the keys. The paper concluded that it is risky to distribute the keys and not worth the AACS LA’s time and effort to pursue the distributors.
Fibonacci's Sequence Zachary Metzman '16
Although the Fibonacci Sequence is named after the The are many patterns in Fibonacci’s sequence. Italian mathematician, it appeared in Indian mathe- For example, a Fibonacci number Fn is divisible by matics before it appeared in the book Liber Abaci, by Fm if and only if n|m (that is, when n divides m). Fibonacci. The book states: Therefore, since F6 = 8, Fn |8 occurs only when n|6. This is to say that every 6th Fibonacci number is diSuppose a newly-born pair of rabbits, one visible by 8. This divisibility rule applies for all Fimale, one female, are put in a field. Rabbits bonacci numbers. are able to mate at the age of one month so The ratio Fn/Fn−1 approaches the Golden Ratio as that at the end of its second month a female n increases. The Golden Ratio is one of the most can produce another pair of rabbits. Suppose that our rabbits never die and that the frequently seen ratios in nature and is used in defemale always produces one new pair (one sign and architecture. Many proportions of flowmale, one female) every month from the ers are in the Golden Ratio. Also, the number of second month on. How many pairs will petals of most flowers is usually a Fibonacci number. there be in one year? The Golden Ratio can also be found in posters, playThe solution to the riddle is the first two num- ing cards, and wide-screen televisions. Some peobers in the sequence are 0 then 1 and the rest of the ple even claim that parts of the Pantheon are in the numbers are the sum of the previous two numbers Golden Ratio. (1, 2, 3, 5, 8, 13, 21,…). 10
Puzzle
Trivia • The first non-trivial non-prime in 4x2 − 2x + 1 is 57 (x = 4) ∫ +∞ √ 2 • −∞ e−x dx = π
A magic square is a grid where the sum of every row, column, and diagonal is equal. This number is known as the magic number. See if you can fill in the grid below.
√ • Stirling’s approximation states that nn e−n 2πn ≈ n!. ! That is to say that limn→∞ nn e−nn√ = 1. 2πn
9 7
6 4
13
16
1
2
5
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Image Sources Cover Mandelbrot Photo https://www.youtube.com/watch?v=j1pjw4qxjM4
Kid & Math (p.1) http://gattissolutions.com/wp-content/uploads/2012/01/kid_math-492x369.jpg
Simpsons (p.2) http://web.carteret.edu/keoughp/Jeyl/MAT-161pics/circles_simpsons.jpg
Investing (p.3) http://www.trojaninvestingsociety.com/wp-content/uploads/2012/09/investingstockphoto.jpg
Torah (p.5) http://chabadic.com/media/images/803/xNTw8031430.png
March Madness (p.7) http://anchormd.com/wp-content/uploads/2014/03/March-Madness-2013.jpg
Dot Spiral (p.8) [By Grontesca at the English Language Wikipedia] (All uncredited images are public domain)
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