Notes from the Margin Cyclic numbers and Latin squares by Samin Riasat (University of Waterloo)
An interesting property of the number is the following:
(165432), (135)(246), (123456), (14)(25)(36)
Volume X • 2016
N = 142857
respectively. These permutations are all powers of (123456) or (165432), in accordance with the fact that 3 and 5 are primitive roots modulo 7. (Recall that a number a is said to be a primitive root modulo n if every number relatively prime to n is congruent, modulo n, to some power of a.)
2N = 285714 3N = 428571 4N = 571428 5N = 714285 6N = 857142 and 7N = 999999 , all in base 10. The last equality implies that
1 = 0.142857. 7
We refer to N as the period of 1/7. Numbers such as N are often called cyclic numbers, since multiplying N by 1, 2, . . . , N 1 permutes the digits of N
cyclically. Moreover, written in an array the digits of these multiples form a Latin square:
Samin Riasat The worst part about being a pure mathematician and an engineering TA at the same time is when students ask about applications and you need to make something up on the spot.
1 2 4 5 7 8
4 8 2 7 1 5
2 5 8 1 4 7
8 7 5 4 2 1
5 1 7 2 8 4
7 4 1 8 5 2
Viewing these permutations as elements of the symmetric group S6 one notices that the permutations correspond, using the cycle notation for permutations, to
(1)(2)(3)(4)(5)(6), (153)(264),
In general, let p be an odd prime and b a positive integer with multiplicative order d modulo p. Then the base b expansion of 1/p has period length d with the period given by (bd 1)/p d 1 = mp , in base b. To see this, note that if b then m has at most d digits in base b, and
m m m m 1 = d = d + 2d + 3d + · · · . p b 1 b b b Now let b > p be a primitive root modulo p. We shall work in base b. By the above observation, 1/p has period L = (bp 1 1)/p of length p 1. Since (p 1)L < bp 1 1 , it follows that 1/p, 2/p, . . . , (p 1)/p have periods L, 2L, . . . , (p 1)L respectively, all of length p 1. Since b, 2b, . . . , (p 1)b are distinct modulo p and, by the assumption that b is a primitive root, are congruent to 1, b, . . . , bp 1 in some order, it follows that 1/p, 2/p, . . . , (p 1)/p are the fractional parts of 1/p, b/p, . . . , bp 1 /p in some order. So each digit of L occurs as the rightmost digit of some aL with 1 a p − 1 . It follows
Preamble
by Kyle MacDonald (McMaster University)
In one of the first lectures I attended at university (“Engineering Profession and Practice”), the professor told a story that reflected many stereotypes of the academic career. A foolish applicant to a faculty position at an unnamed university asked his interviewer, “What are the hours like?” The interviewer replied, “Very flexible. You can work whichever hundred hours per week that you like.”
Editor Kyle MacDonald
McMaster University
Editor at Large Kseniya Garaschuk
University of British Columbia
In this volume of Notes from the Margin, our contributors discuss the nonmathematical aspects of life in a mathematics department. Our Editoratlarge discusses the gruelling university entry exams in the former Soviet Union. Another contributor illustrates the interview process for new faculty members in a mathematics department. One piece reflects on a mathematician’s home life, and two more rejoice in the puzzles and the pure mathematics that provide every department with its raison d’etre. For my part, I am grateful to our readers and to the CMS for giving me an opportunity, through Notes from the Margin, to participate in service to the Canadian mathematics community. Every hour of studying I have given up to proofread articles or argue about the minutiae of our workflow has paid dividends in experience and exposure to people far cleverer than me. As ever, I encourage you to submit your work at studenteditor@cms.math.ca. Our inbox is always wide open, and we welcome anything written by mathematics students about mathematics, with the definitions of “student” and “about” to be taken as suitably broad. Cover photo credit: Newcomb, Alyssa. ABC News. “Math Nerds Will Geek Out at This Prime Discovery.”
that L, 2L, . . . , (p 1)L are distinct cyclic permutations of the digits of L. But L, 2L, . . . , (p 1)L are nonzero and distinct modulo b, so they end in different digits. Therefore the digits of L are all distinct and nonzero.
L 2L .. . (p
1)L
Diagram 1
1 2 4 5 7 8
4 8 2 7 1 5
2 5 8 1 4 7

8 7 5 4 2 1
5 1 7 2 8 4
Diagram 2
1 2 4 5 7 8 8 7 5 4 2 1
4 8 2 7 1 5 5 1 7 2 8 4
2 5 8 1 4 7 7 4 1 8 5 2
Diagram 3
2
7 4 1 8 5 2
then taking shows that every digit except , and appears in the period of . This is illustrated in the following example with and .
So we’ve shown that the base b digits of diagram 1 form a (p − 1) ⇥ (p − 1) Latin square.
One interesting feature of the Latin square is the following. Divide the square into its left and right halves, and place one half above the other. The two halves then become mirror images of one another. This result is known as Midy’s theorem. Why is this true? It is a fact that if is a primitive root modulo then has multiplicative order modulo , where is the Euler totient function (the number of positive integers less than and coprime to ). Since is a primitive root modulo , we have , i.e., for any . So one obtains by performing a halflength rotation, i.e., a reflection on the digits of . This proves the theorem. So if a digit appears in the period of then the digit does too. Consequently never appears in the period of , and if is odd, then the digit also never appears since the digits have to be distinct. As a result, if is a primitive root modulo
References [1] J. H. Conway, R. K. Guy The Book of Numbers New York: SpringerVerlag, 1996.
[2] S. Manber, N. Ordulu, K. Yessenov Decimal expansions of the inverses of prime numbers http:// people.csail.mit.edu/kuat/courses/decimalexpansions.pdf [3] Wikipedia, Midy’s theorem
p
Are you smarter than a Russian 12th grader? by Kseniya Garaschuk (University of British Columbia)
As you might suspect, the mechanism of entering a university depends on the country where said university is located. In Belarus (where I am originally from), just like in many other countries of the former Soviet Union, the higher education is free — if you’re good enough, that is. Your “goodness” is determined by the entrance exams and the competition is fierce. By grade 9 of the 12year secondary education, you already know which faculty you will be applying to and you start preparing for the entrance exams: you take evening classes, do endless exercises similar to problems appearing on previous years’ exams, and might even switch high schools to one that has a specialized class for the subject of your choice. Hard sciences are always popular. In my year, the number of students writing the exams for the Faculty of Mechanics and Mathematics of Belarusian State University was 17 times the number of places available. Let me make it clear: you had to beat out 16 other students on the entrance exams in order to get in. You can only apply to one faculty within one toptier public university. If you don’t get in, you can try less prestigious universities and/or private universities where you pay for your education (they strategically hold their entrance exams later in the summer). Or you can try again next year, unless you are male, in which case your mandatory military service will delay your second attempt by 2 years. You get the idea — the stakes are high. So what do these entrance exams look like? Here are the problems from 2009 written entrance exam to the Faculty of Mechanics and Mathematics of Moscow State University, the top university in the country. You can find the archive here: http://new.math.msu.su/admission 1. Find all values of x for which the functions
y= 11 + 24x
17x2 ,
y = arccos (x2
p
11 + 24x
y = arccos (x2
2x + 1),
17x2 , 2x + 1),
3. How many fish did Andrey catch, if we know that he caught 55% of all salmon and cod caught by the boys? 4. A circle of radius 2 with its centre on the base of an isosceles triangle is tangent to two of the triangle’s sides. One of the tangency points is connected to the opposite vertex of the triangle by a line segment. This line segment is divided by the altitude dropped to the base of the triangle in a 4 : 3 ratio, starting from the vertex. Find the area of the triangle.
Kseniya Garaschuk Be a good department citizen, particularly after you get tenure: go to department meetings, actually contribute to the committees, participate in outreach, go to department tea. While the research distinguishes department members as scientists, it is the service that distinguishes them as community members.
5. Find all values of x 2 [19, 29] which satisfy the inequality
a6 + 8a5 − 2 1, ax where a is a root of y
17
+ 2y 11 + 4y 5 = 1 .
If you score high enough on the written exam, you proceed to the oral stage, where you draw 2–3 random problems, are given a short time to solve them, and then get to discuss your solutions with the hiring committee. Yes, very intimidating. Just to give you some idea of what the problems might be like, here are two from 2009: 1. Solve the equation
⇣ cos 4x + sin 2x
a⇡ ⌘ = sin 3x, 64
where a is the smallest twodigit natural number which, when written to the right of 20092009, produces a tendigit number divisible by 36.
2. Find all values of c such that the set of solutions 2 y = arccos (x(x, 2xof+the1), y = log3 (x2 ) y) system
y = log3 (x2 )
y = log3 (x2 )
are defined and at least one of them is equal to zero. 2. Andrey and Boris are tallying up the results of their fishing trip where they caught salmon and cod. Boris, whose total catch consists of no more than 100 fish, has 25% fewer salmon than Andrey, but Andrey has 25% fewer cod than Boris.
forms a line segment. I can’t help but wonder, should I give this exam to my students here in Canada next time I teach a firstyear course? Will it be a sobering experience for them as they realize what it takes to get a higher education in another country? 3
Campus Interviews: Theory and Applications by Anonymous
Candidate Schedule for Dr. School of Hard Knocks No person’s Land At a certain point, you just know it’s time to get a job. You’ve paid your dues, you’ve built your own research interests, your own research program, you’ve taught some courses and informally supervised students and you think, “I definitely deserve to be faculty.” And it always helps when you know a guy at a place that has a position opening up. You get a phone or Skype interview, or you go through the “meat market” at the Joint Math Meetings and you get a couple of campus interviews. You can teach, you can give a research talk and you can mathitup at a conference with your friends and colleagues. So you’re set. Your flight is booked, and you’re preparing for the interview, making your slides, researching the department. And now you realize that you have no idea what you’re doing or even what a campus interview looks like; at least I didn’t. I knew the broad brush strokes: a research talk, a teaching talk, a meeting with the Dean and the Chair of the department. But that doesn’t prepare you for a real campus interview. It’s not intended to be a secret, but we are only ever given a vague snapshot of them, so here’s a sample of a real campus interview, based off of one of my own.
There was some randomization and the names have been changed to protect the innocent.
Day 0 Evening: Arrive in No Person’s Land, get picked up for the airport and treated for dinner. Have them buy you a beer and ask you friendly but probing questions.
Day 1 08:0008:30 – Meet Dr. SoandSo for breakfast. Lights. Camera. Action. 08:3009:00 – Meeting with Human Resources. Feel free to ask about benefits, moving allowance, compensation and pay bands but this is not the time for negotiation. 09:3010:00 – Meeting with Dr. Smirnov (Data Science). 10:0010:30 – Meeting with Dr. Smith (Differential Equations). 10:3011:30 – Meeting with the Department Head. You will talk about the recent past and future of
The AARMSCMS Student Poster Session at the 2015 CMS Winter Meeting in Montreal brought together presenters from all over the country, from different disciplines and different levels of education. At the CMS banquet, poster session winners celebrated their achievements with many brilliant mathematicians.
La session de présentation par affiches AARMSSMC de la Réunion d’hiver SMC 2015 à Montréal a permis de réunir des participants de disciplines variées et de tous les niveaux scolaires. Au banquet de la SMC, les gagnants de la session par affiche ont célébré leur réalisation en présence de mathématiciens célèbres.
We invite you to participate in the next poster session that will be held at the 2015 CMS Summer Meeting in Edmonton, June 24th‑27th. CMS now offers a lower registration fee for poster session participants. More information about participating will be posted on our website (studc.math.ca) and our Facebook page (CMS Studc) in the coming months. Studc is planning a number of other exciting student events to be held during the meeting, including a student workshop and a social. Check the meeting website cms.math.ca/Events/summer16/ for the most uptodate information.
Nous vous invitons à participer à la prochaine compétition qui aura lieu lors de la Réunion d’été SMC 2015 à Edmonton, du 24 au 27 juin. La SMC offre maintenant une réduction sur les frais d’inscription pour les participants de la session de présentation par affiches. D’autres informations sur l’événement seront mises en ligne sur notre site web (studc.math.ca) et sur notre page Facebook (CMS Studc) dans les mois à venir. Studc prépare actuellement d’autres activités excitantes destinées aux étudiants qui prendront part à la réunion, y compris des ateliers ainsi qu’une activité sociale. Consultez le site web de la réunion, cms.math.ca/Reunions/ete16/, pour obtenir les informations les plus récentes.
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The Student Committee is accepting applications to fund student events across Canada. We sponsor regional conferences, student socials, seminars and other events. Visit http://studc.math.ca for more information and to find the application form.
the department, how you fit in the plan, how your position came to be. If there is a group in your area, you will be asked how you complement it. If there are one or two areas, then you should reinforce how you complement the department as a whole and how you will attract and develop colleagues to work with. 11:3012:00 – Talk to the students of one of the related groups outside of the department. Their supervisors will ask them if they liked you (Hint: The answer better be yes). 12:0013:00 – Lunch. 13:0013:30 – Campus tour. Look impressed and ask questions. Ask about the history of that cool building over there. 13:3014:00 – Stop for coffee, meet Dr. Suresh (Dynamical systems) on the way and have him ask you about your plans to apply for grants. 14:3015:30 – Meet with Dean of Undergraduate Studies. Convince her you are a good teacher, researcher, colleague, are interested in student wellbeing and that you will contribute to the atmosphere. She will try to convince you that her university is worth working at, the community is collaborative and the city is livable. 15:3016:00 – Meet with Dr. IndustryLiaison. Convince him you are interested in working with industry. 16:0017:00 – Give a teaching talk (or a research talk oriented at students). 17:0017:30 – Quick chat with Dr. GuyyouKnow. Take a second to get a drink of water. 18:0020:30 – Dinner with the Department Head, Dr. GuyyouKnow, Dr. Who? There will usually be wine, and you should be as friendly and personable as you can be. The wine can help – and that’s part of the plan: be loose, but not too loose.
Day 2 08:0009:00 – Meet Dr. SoandSo for breakfast again in the hotel lobby. Remember how tired you were from yesterday? Time to ramp up quickly today. Make sure you’re friendly, personable, but take the moment
Le comité étudiant accepte les demandes de financement pour des événements destinés aux étudiants et organisés à travers le Canada. Nous finançons les conférences régionales, les séminaires, les activités sociales et d’autres types d’événements. Visitez le http://studc.math.ca pour plus d’informations et pour obtenir le formulaire de demande.
to not need to be very technical. 09:0010:00 – Meet with Associate Department Head. This meeting will likely be more technical in nature than the meeting with the Head – what are your research goals and plans, what are your expectations or wishes for courses to teach, etc. 10:0010:30 – The Heads of the groups in related departments have had a chance to talk to their students, so they might want to talk to you in person now. You don’t have to be an expert in everything, but… 10:3011:00 – Campus tour – oops, you were supposed to prepare for your talk! 11:0012:00 – Give a talk about your research. Prove your chops as a mathematician. 12:0013:00 – Lunch; don’t relax yet! There will be people at your talk who haven’t had a chance to talk to you yet, and you’re almost done now, so they will pick your brain while you’re trying to decompress. 13:0014:00 – Traditional Interview. 14:00 – Head home. Breathe. I’ll leave you with this: interviews are both great and hairraising. You have two days to talk about the research you really enjoy to a group of people who have already decided that you are worth flying across the country to talk to in person. They want to sell you on their department, their university and their city. They will take you to the faculty club, the best coffee shop in town and a fancy restaurant (or two) for dinner. Take the opportunity to take it all in and feel free to let your personality out just a little bit, because they want to know you’re a good colleague to have as well as a stellar mathematician. On the other hand, if you’re like me you hate being the center of attention and you will be for two days straight. You will be rushed, you will have some probing questions and someone, at some point, will throw you a curveball. It’s both exhausting and rewarding: be warned, enjoy and good luck!
5
The Circle Game
by Robert Dawson (Saint Mary’s University)
“One more round of skipping? Pleeeease, Daddy?” “Okay, one more.” I’d promised Jane that morning that Amanda would play outdoors like a normal eightyearold, not stay inside all day reading my university math textbooks. Robert Dawson
Padma and Lucas turned the rope while Amanda skipped. The words they chanted made me smile: “Archimedes studied math,
“Quote“.
Left his trousers in the bath. When the neighbors asked him why, Told them he’d invented pi.” Amanda, sola: “Three, point, one, four, one, five, nine, two,…” It’s getting dark. Amanda’s still going. She’d better be in bed by the time Jane gets home from her meeting. This story originally appeared in Speck Lit on September 11, 2015.
The fourth annual Statistical Society of Canada Student Conference will take place Saturday, May 28, 2016 at Brock University, St. Catharines, Ontario.
Le quatrième congrès étudiant de la Société statistique du Canada aura lieu le samedi 28 mai 2016 à l’Université Brock à St. Catharines, Ontario.
This exciting conference once again welcomes graduate and undergraduate students interested in statistics to participate in a full day of activities and academic sessions. In addition to contributed research sessions, this conference features workshops and panel sessions focusing on skill and career development. Students will have the opportunity to interact with our invited career speakers, who have years of experience working in statisticsrelated jobs in industry, academia and government. There will be prizes for presentations and posters, thanks to our sponsoring partners, including the CMS.
Nous invitons les étudiants du premier cycle et des cycles supérieurs ayant un intérêt pour la statistique à participer à cette journée complète d’activités sociales et académiques. En plus des séances où des étudiants présenteront leur recherche, des tables rondes et des ateliers axés sur les compétences et l’avancement professionnel seront organisés. Les étudiants auront aussi la chance d’interagir avec nos conférenciers invités, lesquels ont plusieurs années d’expérience en statistique appliquée en industrie, dans le milieu universitaire ou au gouvernement. Il y aura des prix pour les meilleures présentations orales et par affiches grâce au soutien de nos commanditaires, parmi lesquels la SMC.
We are pleased to announce that Dr Robert Tibshirani will deliver the keynote address at the SSC student conference! More information and registration at www.ssc.ca/en/meetings/2016/ student_conference
6
Nous sommes heureux de vous annoncer que le discours d’honneur du congrès étudiant de la SSC sera présenté par Dr Robert Tibshirani! Toutes les informations et le formulaire d’inscription sont disponibles sur le site du congrès à l’adresse www.ssc.ca/fr/congrès/2016/ congrès_étudiant
The Distractions Page Something’s Amiss Ten cent coins. . . . . . . . . . . . . . . . . . . . . . . . . . . Stomped Went around in circles. . . . . . . . . . . . . . . . . . . . . . . Sink Put into a bank account. . . . . . . . . . . . . . . . . . . Riddles Cleverly invented. . . . . . . . . . . . . . . . . . . . . Enlistments
Tyrone Ghaswala
Monarch or ruler. . . . . . . . . . . . . . . . . . . . . . . Deviation
I find entering grades excruciatingly boring. To combat this, I try to find anagrams of student’s names. Sally Sixpack? Pascal’s Kylix! Hey, it’s better than just entering grades without anagrams.
Tender and nostalgic. . . . . . . . . . . . . . . . . . . . . . . Denim
The Sierpinski carpet was first published a hundred years ago in 1916 by Waclaw Sierpinski, a year after the betterknown triangular Sierpinski sieve or gasket. How time flies.”
Enjoy these puzzles? Check out the upcoming MUMS Puzzle Hunt (your first puzzle is to use Google to find the MUMS Puzzle Hunt website). The Puzzle Hunt is a weeklong puzzle competition run out of the University of Melbourne in Australia. This year it begins on the 9th of May. Anyone from anywhere is welcome to participate, check out the website for more details and past puzzles.
The fifteenth annual Summer School sponsored by the Atlantic Association for Research in the Mathematical Sciences (AARMS) will take place at Dalhousie University in Halifax, Nova Scotia from July 11 until August 5, 2016. The school is intended for graduate students and exceptional undergraduate students from all parts of the world. Each participant is expected to register for two courses, each with five ninetyminute lectures per week. Course topics include but are not limited to category theory, stable polynomials, topology, special functions, and quantum computing. These are graduate courses, approved by Dalhousie, and we will facilitate transfer credit to the
extent possible. Dalhousie tuition will be waived for all participants and AARMS will cover all accommodation (dormitory) costs. Over the weekends there will be social activities and excursions to explore Nova Scotia. For more information on the courses, or to express interest in attending, visit our website, https://aarms.math.ca/the2016aarmssummerschool/, and use the linked forms to apply, or email the directors, Dr. Geoffrey Cruttwell, gcruttwell@mta.ca, and Dr. Dorette Pronk, pronkd@dal.ca.
You want to keep informed of the latest events and activities for math students? An easy solution: follow the CMS student committee in the social media. Search for CMS Studc on Facebook, Twitter and Google+. Vous désirez demeurer au fait des tous derniers événements et activités destinés aux étudiants en mathématiques? Une solution facile: suivez le comité étudiant de la SMC sur les réseaux sociaux. Cherchez CMS Studc sur Facebook, Twitter et Google+. 7
We are happy to announce that the Canadian Undergraduate Mathematics Conference will be hosted next year at the University of Victoria in beautiful British Columbia. CUMC 2016 at UVic will be an exciting five days of sharing ideas, networking, and celebrating the vast diversity within the field of mathematics and within the community of mathematicians. Delegates will have the opportunity to experience all that it has to offer, from spectacular beaches and parks to the charming Inner Harbour in the center of downtown. Furthermore, UVic boasts an active Department of Mathematics and Statistics whose faculty are engaged in exploring a wide range of fields, including graph theory, mathematical biology, and operator algebras. Our organizing committee is dedicated to ensuring that the event runs smoothly and is enjoyable for every participant. We hope to see you there!
The CMS Student Committee is looking for proactive mathematics students interested in joining the Committee in the roles of chair, regional representatives for Quebec and Ontario, and EnglishtoFrench translator. Francophone applicants are preferred for the positions of translator and regional representative for Quebec. If you are interested, or know someone who may be, please visit our website studc.math.ca for more information and an application form. If you have any questions, please contact us at chairstudc@cms.math.ca.
Notes from the Margin is a semiannual publication produced by the Canadian Mathematical Society Student Committee (Studc). The Margin strives to publish mathematical content of interest to students, including research articles, profiles, opinions, editorials, letters, announcements, etc. We invite submissions in both English and French. For further information, please visit studc.math.ca; otherwise, you can contact the Editor at studenteditor@cms.math.ca.
Visit us at http://studc.math.ca
© Canadian Mathematical Society 2016. All rights reserved.
Nous sommes heureux d’annoncer que le Congrès canadien des étudiants en mathématiques sera organisé l’année prochaine à l’Université de Victoria, dans la merveilleuse province de la ColombieBritannique. Au cours des cinq jours passionnants du CCÉM les membres de notre communauté auront la chance de partager leurs idées, de se réseauter et d’explorer la grande diversité à l’intérieur du domaine des mathémathiques. Les délégués auront également l’occasion de découvrir tout ce qui est offerte par la charmante ville de Victoria, tels que ses plages et ses parcs spectaculaires ainsi que le bien agréable port intérieur du centreville. En outre, l’Université de Victoria bénéficie d’un Département de mathématiques et de statistiques qui s’engage activement à de multiples domaines d’étude, y compris la théorie des graphes, la biologie mathématique, et les algèbres à opérateur. Le comité d’organisation se consacre à s’assurer que l’événement se déroule sans heurts et de façon convenable à tous. Nous espérons vous y retrouver!
Le comité étudiant de la SMC est à la recherche d’étudiants en mathématiques dynamiques qui seraient intéressés à joindre le comité étudiant pour combler les postes du président, représentants régionaux pour le Québec et pour l’Ontario, et du traducteur anglaisfrancais. Les applicants francophones seront préférés pour les postes du traducteur et du représentatif regional pour le Quebec. Si vous êtes intéressé ou connaissez quelqu’un qui pourrait l’être, veuillez consulter notre site web studc.math.ca pour plus d’informations et pour télécharger le formulaire d’application. Pour toute question, veuillez nous écrire à chairstudc@cms.math.ca.
Notes from the Margin est une publication semiannuelle produite par le comité étudiant de la Société mathématique du Canada (Studc). La revue tend à publier un contenu mathématique intéressant pour les étudiants tels que des articles de recherche, des profiles, des opinions, des éditoriaux, des lettres, des annonces, etc. Nous vous invitons à faire vos soumissions en anglais et en français. Pour de plus amples informations, veuillez visiter studc.math.ca; ou encore, vous pouvez contacter le rédacteur en chef à studenteditor@cms.math.ca.