X e V e X

PI DAY EDITIONEditors

Michael RosenbergBenjamin KaplanBrandon CohenDan Korff-Korn

Faculty Adviser

Dr. Renee Koplon

Contributors

Sarah AschermanJacob Berman

Brandon CohenBenjamin Kaplan

Ma hew LevySkyler Levine

Eddie Ma outZachary Metzman

Sammy MerkinBenjamin Rabinowitz

Michael RosenbergTess Solomon

Math DisabilitiesTess Solomon '16

Learning disabilities are conditions that typicallyare caused by the brain’s inability to receive andprocess information “normally”. However, havingsuch a condition by no means implies that some-one who suffers from a learning disability is unableto learn. As has been studied extensively in recentyears, many people who have learning disabilitiessimply learn at a slower pace or in different waysfrom people without them.

There are several disabilities that specifically ap-ply to absorption and retention of mathematicalskills and principles. Related disabilities involvewhat has been called the“emotional” ability to learnthem. These disabilitiesmostly fall under the de-scriptive names, dyscalculiaand math anxiety.

Dyscalculia describes awide range of learning dis-abilities that affect absorp-tion and retention of ma-terial. According to stud-ies done in the United King-dom, between 3.6 and 6.5%of the population there is dyscalculic. No interna-tional study has been done to estimate how com-mon it is in other countries or regions. There aretwo main types of difficulties that can contribute toa diagnosis of dyscalculia. Thefirst is a difficulty pro-cessing visual-spatial relationships, relating to thebrain’s ability to process images it receives from theeye. People with this disability have difficulty vi-sualizing patterns or estimate distances. The sec-ond is language-processing difficulties, relating tothe brain’s ability to process sound it receives fromthe ear. This canmake it difficult to understandmathword problems or retain math vocabulary.

Usually, dyscalculia is noticed and hopefullyidentified when the person is in elementary or mid-dle school. A trained professional identifies dyscal-culia and then tells the student his or her specific

weaknesses and how they should proceed. In manystates, those between the ages of 1-26 are entitledto free testing by their local public education ser-vice district. Many times, the answer involves get-ting help outside of the classroom to work on spe-cific issues the student might be having. Workingwith the student outside of the classroom also takesaway the pressure of moving on to new topics tooquickly, before the student fully understands them.Another option, or sometimes an additional option,employs alternative methods of teaching and learn-ing. For example, sometimes students learn more

easily if they are providedwith examples before beingtaught a general rule.

Math anxiety is a differ-ent malady. Professor MarkH. Ashcraft, Ph. D., de-scribes math anxiety as be-ing “a feeling of tension, ap-prehension, or fear that in-terferes with math perfor-mance.” As opposed todyscalculia, where a studentis typically incapable of do-

ing the math problem in the same way as a studentwithout dyscalculia, students withmath anxiety willtry to avoid situations in which they will have to per-form calculations. According to a study at the Uni-versity of Chicago, math anxiety actually has noth-ing to do with the math itself. It is the anticipa-tion of solving the problem, or the fear of not be-ing able to solve it, that actually causes the anxiety.The area of the brain that is triggered when some-one has math anxiety is the same place in the brainwhere bodily harm is registered. Also, the effects ofmath anxiety reach beyond numbers. A correlationhas been found between math anxiety and low con-fidence and motivation.

Treatment of math anxiety falls into two cate-gories. The first essentially relates to what attitude istaken by the school or teachers towards the student.

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Theexpectationsof parents and teachers arenotusu-ally the cause of math anxiety, but they can perpet-uate anxious feelings if they set unrealistic expecta-tions for the students. Teachers who accommodatedifferent learning styles and have a positive attitudein what they teach have been found to be more suc-cessful in helping students overcome their anxiety.The other category of treatment is therapy, wherethe reasons for the anxiety are addresseddirectly andremedial help in math can be given. Therapists in-troduce new coping devices to the students, whichcan help them channel their anxiety elsewhere. For

example, relaxation techniques such as meditationhave also been found to help anxiety.

In conclusion, our society is increasingly mathand science dependent. Our individual and col-lective abilities to succeed depend importantly ongreater and greater numbers of us excelling in math.By understanding the root causes of two of themostprevalent learning disabilities associated with math— dyscalculia and math anxiety — we as a societyhave brought ourselves one important step closer toachieving individual and societal mastery in math.

Mathematics in The SimpsonsSammy Merkin '15

The writing staff for the renowned animated tele-vision show, The Simpsons, includes an abun-dance of brilliant men with impressive backgroundsin mathematics. While it may seemlike these men have wasted theiradeptness for mathematics, theyhave not, as many episodes of theshow include some very sophisti-cated mathematics. Frequently inthe background of scenes during theshow complex mathematical solu-tions are written out or implied. Forexample, in a 1998 episode “The Wizard of Ever-green Terrace,” written by David X. Cohen, a Har-vard graduate, a near miss solution for Fermat’s LastTheorem was written on a blackboard in the back-ground. This was noticed by none other than SimonSingh the writer of Fermat’s Enigma, a required read-ing for Rabbi Stern’s Honors Pre-Calculus Class.In fact, Singh recently published his fifth book ti-tled: The Simpsons and Their Mathematical Secrets,an in-depth analysis of the underappreciated math-ematical brilliance of the show. This also shows whyit makes sense that Mr. Greene is teaching a mini-course on The Simpsons.

Another example of math in The Simpsons ap-pears in a 2006 episode when three numbers ap-pear on the screen asking fans at a baseball game

to guess the attendance. While the numbers seenarbitrary and ordinary, they are far from it. Thefirst option, 8,128, is a perfect number, mean-ing all of its divisors add up to equal the number

itself. The second number 8,208,is a narcissistic number because thesum of each of the numbers raisedto the fourth power equals the num-ber itself. The final option 8,191is not only prime number, but aMersenne prime number, named af-ter the 17th century French Math-ematician Marin Mersenne. When

DavidX.Cohenwas askedwhyabrilliantmathemat-ical mind was suited for comedy writing he replied:

The process of proving something hassome similarity with the process of com-edy writing, inasmuch as there is no guar-antee you are going to get to your ending.Whenyou’re trying to thinkof a jokeout ofthin air, there is no guarantee that there ex-ists a joke that accomplishes all the thingsyou’re trying to do and is funny as well.Similarly, if you’re trying to prove some-thing mathematically, it is possible that noproof exists. And it is certainly very pos-sible that no proof exists that a person canwrap their minds around.

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Pay some Interest: Annuity or PerpetuityMa hew Levy '16

A perpetuity is a type of annuity, and while bothshare a common ideology, their differences are time-bending (literally). Annuities are a type of fixed-income investments from which you receive peri-odic interest payments over a series of time (often toensure a long term steady cash flow) and receive in-terest on your principal investment. While there aremany different variables, interest payments aremostcommonly made monthly, quarterly, semi-annually,or annually. While your return can also vary basedonwhether you receive payment at the beginning orend of the period, there is one major difference be-tween annuities and perpetuities: your principal.

Perpetuities give you interest payments foreveron your investment, but never return your principalinvestment explicitly, compared to an annuitywhichgives interest for a period of time and then repaysyour investment. You might be asking yourself, whywould I put my money into a perpetual investment?

The one thing that all investors look for is up-side: how much money can I make (no, this is nota strictly Jewish reference)? What might be mind-twisting is that you will receive interest paymentsforever on a perpetual investment. This means thatyouwill makemoney (off of interest) as long as yourn’shama — forever. However, our physical bodiesand uses for money aren’t infinite, meaning we haveto find the point in time that a perpetual investmentwould overtake an annuity (in terms of return).

If, for example, Saul and David each wanted toinvest $1,948 into a fixed income investment withan annual interest rate of 1.8%, which should theyinvest in?

1. Saul invests his money into a 10 year annuity(compounded yearly). Then:

P =r(PV)

1 − (1 + r)−n .

P = PaymentPV = Present Value

r = rate per periodn = number of periods

Hewould in tern receive roughly $214.60 peryear for ten years, or $2,146 which is approx-imately 10% return over 10 years.

2. David, preparing for the future (of Israel), in-vests his money into a perpetuity with thesame rate andprincipal investment. Using theperpetuity payment formula:

P = PVr

P = PaymentPV = Present Value

r = annual interest rate

David will receive more than $35 forever.

Assuming that David and Saul do not rein-vest their periodic payments (from the annuity), itwould take Saul 61.21 years to match David’s returnof $2,146, butwill continue to receive the $35.06peryear forever.

The question onemust ask himself is: is it worththe wait? In my opinion, sometimes it’s better towait, as you appreciate it more when you get your“investment” back!

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Fibonacci Representation of Naturals with ZeckendorfMichael Rosenberg '15

IntroductionThere exist many different ways to represent num-bers, whether they be irrational reals, naturals, com-plex, etc. One suchwayof representingnatural num-bers is by the sum of Fibonacci numbers. Morespecifically, Zeckendorf ’s theorem proposes that ev-ery natural number has a unique representation as asum of non-consecutive Fibonacci numbers.

ProofWe will prove Zeckendorf ’s theorem. To do so, wewill need to prove both the existence of such a rep-resentation, then the uniqueness of it.

Existence

We will prove the representation’s existence usingmathematical induction. We begin with 1,2, and 3as they are Fibonacci numbers themselves. Supposeall n ≤ k have a Fibonacci representation for someupper bound k. If k+ 1 is a Fibonacci number, thenit has a trivial Fibonacci representation and we aredone. Otherwise, there must exist two consecutiveFibonacci numbers such that Fj < k + 1 < Fj+1.Let a = (k + 1) − Fj. Since a < k, a must have aFibonacci representation. Furthermore:

Fj + a = k + 1< Fj+1

= Fj + Fj−1

( by property of Fibonaccinumbers

)a < Fj−1

Since a < Fj−1, it must be the case that Fj is nota consecutive Fibonacci number to any of the num-bers in the Fibonacci representation of a. Thereforek = a + Fj fits the criteria and k has a Fibonaccirepresentation.

Uniqueness

We will prove the uniqueness of the Fibonacci rep-resentation of a number using a proof by contradic-

tion. Firstly, we require a lemma.

Lemma:The sum of any non-empty set of distinct, non-consecutiveFibonacci numbers not containingF0 orF1 whose largest member is Fj is strictly less thanFj+1.

Proof:We will use a proof by induction on j. The base caseis F2 = 1.

∑{1} = 1 < F3 = 2. Now assume

the theorem is true for all j ≤ n. Let P be a set ofdistinct non-consecutive Fibonacci numbers whosegreatest element is Fn+1. Let P′ be the set of all ele-ments of P besides Fn+1 (set-theoretically denotedas P′ = P \ Fn+1). The next greatest element ofP can be no larger than Fn−1, as no consecutive Fi-bonacci numbers may be in P. Thus, by inductivehypothesis,

∑P′ := p < Fn. And if p < Fn then

p + Fn+1 < Fn+2 = Fn+1 + Fn.

Now that the lemma has been proven, we mayprove the uniqueness of the Fibonacci representa-tion. Assume that there exist distinct sets S andT such that their elements are non-consecutive Fi-bonacci numbers and

∑S =

∑T = n where n is

some natural number. Let S′ = S \ T, that is theset of all elements in S that are not in T. Similarly,let T′ = T \ S. The sum of these two sets are stillequal, as

∑S′ = n −

∑(S ∩ T ) =

∑T′. De-

note the largest element of S′ as Fs and the largestelement of T′ as Ft. Note that Fs ̸= Ft. Withoutloss of generality, assume that Fs < Ft. It is obvi-ous that

∑T′ ≥ Ft as Ft is an element of T′. From

the lemma, we know that∑

S′ < Fs+1 ≤ FT. Butthis implies that

∑S′ <

∑T′ =⇒

∑S′ ̸=∑

T′ =⇒∑

S ̸=∑

T, thus contradicting theassumption. ■

References

http://www.proofwiki.org/wiki/Zeckendorf%27s_Theorem

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Pi in the TorahJacob Berman '16

For many years people have wondered if any-where in the Torah, there was any mention of theirrational number we nowknow as pi. Pi is the numberthat is used to find a circle’s cir-cumference or area. Its value is3.1415…and 22/7 is often usedto approximate it. In Jewishtexts, the use of gematria, or let-ters that represent numbers, isoften used tomake connectionsfrom the text to numbers. Ev-ery specific Hebrew letter hasa value, and Rabbis often in-terpret lines of the Bible as nu-meric codes.

There have been many different explanationsof words in the Bible that signify the numbersof pi. The Vilna Gaon, a Lithuanian scholar, of-fers an interpretation in the Bible, which repre-sents many of pi’s digits. In the Book of Kingsthe Hebrew word for perimeter is used. Theword is written “Kava” but read “Kav”. If you

take the value of Kava, which is 111, and di-vide it by the oral form Kav, whose value is 106,

and the multiply by 3 youget 3.1415094…! That is thevalue of pi to the ten thou-sandth column or the fourthdigit! Another famous con-nection in the Bible to pi,are said by God. The firstwords of the first command-ment is “I am God” or Anochiand then God’s name, Yeho-va. The value of Anochi is 81divided byGod’s name, whichis 26. This answer is approxi-mately the value of pi, or 3.1.

The fact that there are connections between ascroll written thousands of years ago, to a recentlyunderstood number such as pi, is astounding. Thereare many other connections with pi and the Bible,some weak and others fascinating but these are theprimary connections.

Simpson's ParadoxSkyler Levine '15

Simpson’s Paradox is a statistical phenomenon thatoccurs when data is aggregated. This datamay reveala trend that contrasts with that of its subgroups. Theconcept was illustrated by Simpson in a paper writ-ten in 1951.

Sometimes data can bemisleading and the storyon the surface can take people in the wrong direc-tion. Tomakemore sense of Simpson’s paradox, let’slook at the following example. In a certain hospitalthere are two surgeons, surgeon A and surgeon B,with the following statistics:

Surgeon # Patients # Survived % SurvivedA 100 95 95%B 80 72 90%

From this analysis, which surgeon should wechoose to treat us? It would seem that surgeon A isthe safer bet. But is this really true?

From further research into the data we foundthat of the 100 patients that surgeon A treated, 50were high risk, of which three died. The other 50were considered routine, and of these 2 died. Thismeans that for routine surgery, a patient treated bysurgeon A has a 96% survival rate.

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Now we look more carefully at the data for sur-geon B and find that of the 80 patients 40 were highrisk, of which seven died. The other 40 were rou-tine and only one died. This means that a patienthas97.5%survival rate for a routine surgerywith sur-geon B.

Now which surgeon seems better? If yoursurgery is to be a routine one, then surgeon B is ac-tually the better surgeon. However if you look at allsurgeries performed by the surgeons, A is better.

Another example of Simpson’s Paradox is ev-ident in educational testing. Between 1981 and2002, the national average for the verbal ScholasticAptitude Test (SAT) score appeared to remain rel-atively stable at 504 points (Bracey 32).However,during that same time period, the average verbalscores for all racial and ethnic subgroups increasedby between eight and twenty-seven points. Braceyattributed this to changing demographics of SATtest takers. Over this time period, the number ofwhite students taking the SAT fell while the numberof minority students rose. Performance improvedacross all racial and ethnic groups, but minority (ex-cluding Asian American) students’ average verbalscores remained below the national average. Highernumbers of increasing but below average scores re-sulted in a national average that not only failed to re-flect subgroups’ improved verbal scores; they failedto reflect any change at all.

Examples of Simpson’s paradox appear not onlyin medical research and educational testing but alsoin sports rating. For example, suppose two baseball

players had the following stats:

Hitter A

Pitcher Type At-Bats Hits AverageRighty 300 90 .300Lefty 200 50 .250Total 500 140 .280

Hitter B

Pitcher Type At-Bats Hits AverageRighty 100 32 .320Lefty 300 78 .260Total 400 110 .275

HitterBhas ahigher batting average against bothrighties and lefties, but Hitter A has a higher over-all average; therefore, I would advise that one insertshim as pinch hitter.

As is evident, Simpson’sParadox applies tomanycases in several different fields.

References

Bracey, G.W (2004). Simpson’s paradox and otherstatistical mysteries. American School BoardJournal, 191, 32-33.

Courtney Taylor. What is Simpson’s Paradox?Goltz, Heather Honore & Mathew Lee Smith

(2010). Simpson’s Paradox in Research.

History of Math EducationSammy Merkin '15

Elementary mathematics was a part of the educa-tion system in most ancient civilizations, includ-ing Ancient Greece, Ancient Egypt, and the RomanEmpire. In ancient civilizations, an education wasonly available to high-class males. Documents fromMesopatamia, dating back to 1800 BCE, were foundwith multiplication and division, as well as meth-

ods for solving quadratic equations. The Rhind Pa-pyrus, is an Ancient Egyptian math textbook thatdates back to 1650 BCE, however it is most likely acopy of an even older document.

Math continued to be taught throughout themiddle ages. In medieval Europe, mathematics wastaught to apprentices where it was practical in their

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trade. All geometry that was taught was based uponEuclid’s Elements. The first mathematics textbookwritten in English and French was The Grounde ofArtes, published by Robert Recorde in 1540. Un-fortunately, during the Renaissance period, mathe-matics was set aside in favor of the study of philoso-phy. However, in the seventeenth century there wasa revival of mathematics studying and Mathematicschairs were set up in Aberdeen, Oxford, and Cam-bridge universities. It was uncommon for mathe-matics to be taught anywhere outside of universities.

The Industrial Revolution in the 18th and 19thcenturies caused mathematics to become a foun-dational skill that most people needed. Practi-cal applications of mathematics, such as tellingtime and counting money required simple arith-

metic to be taught everywhere. The institutionof public education systems allowed for math tobe taught at a young age everywhere. Today,many countries, such as England, have standards for

A page from the originalbook The Elements by Eu-clid

what and how muchmathematics must betaught at schools. How-ever, in the UnitedStates the governmentreleases recommenda-tions of what should betaught in each grade butdoes not enforce theseideals. Despite this, APStatistics and APCalculus are among the most pop-ular APs with 150,000 and 385,000 respectively.

The Perfect BracketBenjamin Kaplan '16

Since the year 1939, every year around March therehas been the NCAA Men’s Division I Basketballtournament. As of now 68 Division I teams qual-ify for the single-elimination tournament, whichconsists of 7 different rounds. The tournamentand the events surrounding it, in-cluding brackets, have received thenameMarchMadness. Abracket isa form of gambling in which peo-ple try to guess and decide whothey believe will win each game.This practice has grown exponen-tially leading topeople trying togeta perfect bracket.

This year, as well as last year, there is a perfectbracket challenge by QuickenLoans. The challengeappears quite simple: make a perfect bracket and re-ceive a large prize totaling $1 billion. However, ifwe look at the mathematics, it is not that simple. Totry and determine the probability, let’s assume eachteam has an equal chance of winning each game. Soto determine that, it would be 263, as there are 63

games. If you do that out, there are 9.2 quintillionways to fill out a bracket. So if you still think you aregoing towin the BillionDollar Perfect BracketChal-lenge, youmaywant to reconsider. Now to put thoseodds in perspective, the odds of winning the lottery

are 1 in 175 million, and the oddsof being struck by lightning are 1 in3000. Now obviously, your oddsare slightly better if you know bas-ketball, as you probably will notbet on the 64th seed to upset the1st seed. According to a DePaulmathprofessor, JohnBergen, if youknow something about the NCAA

basketball tournament the odds are more like 1 in128 billion. Using that number, according to ChrisChase, someone would get a perfect bracket onceevery 400 years if every person in the US filled outa bracket. Everyone fantasizes and theorizes abouthow to achieve this phenomenon but it does notseem likely to occur in the near future.

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Ulam SpiralZachary Metzman '16

Prime numbers are very important and frequentlyused in everyday life. They are required in manyforms of cryptography. Normally, when large primenumbers are used in a form of cryptography, the en-cryption is strong.

It has always been a challenge to find large primenumbers even with computers. It is so difficult fora computer to find new prime numbers that manyindividuals who want to test the durability of theircomputer search for primes. The largest prime num-ber known today is 17425170 digits long, making it4446981 digits longer than the second largest primenumber known. Huge gaps like this make it dif-ficult for computers to find primes. In addition,the general occurrence of prime numbers was notknown until mathematician Stanislaw Ulam acci-dentally discovered a pattern. Ulamwas at a presen-tation of a “long and very boring paper”, when he be-gan drawing a spiral of numbers like this:

He then circled all the prime numbers in thespiral and realized that the prime numbers were insemi-consecutive rows, diagonals, and columns. Be-low is anUlam spiral where each dot is a prime num-ber:

Shortly after his discovery, he was able to usea computer to generate a spiral up to 65,000. TheUlam spiral became famous when Martin Gardnerpublished an article about it inGardner’sMathemat-ical Games column.

One can use the equation of a heavily dotted di-agonal to generate all the numbers in it, giving com-puters fewer numbers to search through in order tofind a new prime. For example, the diagonal thatcontains 3, 13, 31… has the equation 4x2 − 2x+ 1where x is the xth number in the diagonal. So ifyou want to find the 10th number in that diagonal,plug in 10. Many mathematicians have tried to ex-plain the reasoning behind these patterns but noneof them have been able to prove their conjectures,further showing the significance andmysteriousnessof the Ulam spiral.

Go fried LeibnizSarah Ascherman

Gottfried Wilhelm von Leibniz may be bestknown for his discovery of infinitesimal calculus—mathematics concerning functions, tangents tocurves, as well as maxima and minima. However,youmay not know that Leibniz was also a renowned

philosopher. Additionally, he studied physics, the-ology, history, biology, candle making, legal affairsand linguistics! Leibniz was born in Leipzig, Sax-ony, in 1646 and died in 1716, living around thesame time as IsaacNewton, who also independently

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discovered infinitesimal calculus. By the age of 16,Leibniz had attended university and had earned hisbachelor’s degree in philosophy. In the area of phi-losophy, Leibniz strongly believed in rationalism.Four years later, at the age of 20, Leibniz wrote hisfirst book on philosophy, “On the Art of Combi-nations.” In the study of mathematics, Leibniz not

only discovered infinitesimal calculus, but also de-veloped mathematical notations that we still use inour classrooms to this day, such as the integral sign,∫. So next time you’re in math class learning calcu-

lus or using the integral sign, you’ll know where itcomes from!P.S. On a sweeter note, Leibniz even has cookies named after him!

Measurements in the TalmudBen Rabinowitz

On 109a-b in Pesachim, the Talmud goes throughgreat explanation of how we come to the measure-ment of the reviis—the unit that we use when mea-suring theminimumamount ofwineweneed for thefour cups ofwine onPesach, and kiddushon shabbos.

It is stated that a mikvah’s minimum dimensionsare 1× 1× 3 amos. We use a Mikvah to purify our-selves and our utensils. The next statement of theTalmud is that that measurement—3 cubic amos—contains forty se’ah of water. Earlier in the sugya, R’Chisda said that the reviis of the Torah, which theyused for measuring liquids as well, was 2 × 2 × 2.7fingerbreadths. (Theway he states it is 2× 2× (2+1/2 + 1/5) fingerbreadths.)

The followingdimensional analysis is performedto achieve the measurement of the reviis in terms offingerbreadths, to reinforce R’ Chisda’s statement:

1 amah =1 amah

6 tefachim× 1 tefach

4 fingerbreadths= 24 fingerbreadths

1 se’ah =1 se’ah6 kabim

× 1 kav4 lugin

× 1 lug4 reviios

= 96 reviios.

40 se’ah, which, as mentioned before, is themin-imum volume of a Mikvah, is 3, 840 reviios.

So, if we equate the two values which we re-ceived:

1 × 1 × 3 amos = 24 × 24 × 72 fingerbreadths= 41, 472 cubic fingerbreadths= 41, 472 fingerbreadths3

= 40 se’ah= 3, 840 reviios.

Lastly, we show that

41, 472 fingerbreadths3

3, 840 reviios= 10.8

fingerbreadths3

reviios,

and 10.8 = 2 × 2 × 2.7, thus proving R’ Chisda’spoint.

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Illegal Numbers and ColorsZachary Metzman '16

Computers do not use letters; they use binary num-bers to represent symbols. If it is illegal to distributea piece of information in its normal form, then it is il-legal todistribute it in anyother form. If copyrightedinformation is distributed even in binary form, thatis illegal. Similarly, a group of colors can be ille-gal because binary can be converted to hexadecimalwhich in turn can be represented by colors.

In recent years, a feud involving illegal numbershappened between Advanced Access Content Sys-tem Licensing Administrator (AACS LA) and thewebsite Digg. In 2006, a forum user, muslix64, pub-lished encryption keys to DVD players on Doom9.The encryption keys gave users the ability to copyor pirate DVDs. The next year, the AACS LA issuedDigital Millennium Copyright Act violation noticesto various websites that hosted the encryption keys.Many websites took the keys down but Digg wasdefiant. The founder of Digg told the AACS LA,“We hear you, and effective immediately we won’tdelete stories or comments containing the code

The first 16 bytes of an en-cryption key represented incolors

and will deal with what-ever the consequencesmight be.” Infuri-ated, the AACS LA at-tempted to remove allkeys from the internet.Paradoxically, the num-ber of keys online wentfrom 9,410 Google re-sults to 300,000 Googleresults in one day, making it impossible to removethe keys from the internet. Digg embedded the in-formation in songs and converted the keys to num-bers and groups of colors in case a court would forceDigg to take the keys down. After the wide spread ofthe numbers, the American BAR Association pub-lished a paper questioning illegality of publishingthe keys. The paper concluded that it is risky to dis-tribute the keys and not worth the AACS LA’s timeand effort to pursue the distributors.

Fibonacci's SequenceZachary Metzman '16

Although the Fibonacci Sequence is named after theItalian mathematician, it appeared in Indian mathe-matics before it appeared in the bookLiber Abaci, byFibonacci. The book states:

Suppose a newly-born pair of rabbits, onemale, one female, are put in a field. Rabbitsare able tomate at the age of onemonth sothat at the endof its secondmontha femalecan produce another pair of rabbits. Sup-pose that our rabbits never die and that thefemale always produces one new pair (onemale, one female) every month from thesecond month on. How many pairs willthere be in one year?

The solution to the riddle is the first two num-bers in the sequence are 0 then 1 and the rest of thenumbers are the sum of the previous two numbers(1, 2, 3, 5, 8, 13, 21,…).

The are many patterns in Fibonacci’s sequence.For example, a Fibonacci number Fn is divisible byFm if and only if n|m (that is, when n divides m).Therefore, since F6 = 8, Fn|8 occurs only when n|6.This is to say that every 6th Fibonacci number is di-visible by 8. This divisibility rule applies for all Fi-bonacci numbers.

The ratio Fn/Fn−1 approaches theGoldenRatio asn increases. The Golden Ratio is one of the mostfrequently seen ratios in nature and is used in de-sign and architecture. Many proportions of flow-ers are in the Golden Ratio. Also, the number ofpetals ofmost flowers is usually a Fibonacci number.TheGoldenRatio can also be found in posters, play-ing cards, and wide-screen televisions. Some peo-ple even claim that parts of the Pantheon are in theGolden Ratio.

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Puzzle

A magic square is a grid where the sum of every row,column, and diagonal is equal. This number is knownas the magic number. See if you can fill in the grid be-low.

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7 6

13 16 1 4

2 5 12

Trivia

• The first non-trivial non-prime in 4x2 − 2x + 1 is 57(x = 4)

•∫ +∞−∞ e−x2dx =

√π

• Stirling’s approximation states that nne−n√2πn ≈ n!.That is to say that limn→∞

n!nne−n

√2πn

= 1.

Image SourcesCover Mandelbrot Photohttps://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 LanguageWikipedia]

(All uncredited images are public domain)

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