notes from the margin

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PreambleBy Kseniya Garaschuk (University of Victoria)

Isnt math, like, done?How many times have you heard this statement when you say you do mathematical research? This perceptionof the subject is not really surprising considering the mathematics an average person gets exposed to in school.The age and various disguises of our discipline are, in this case, its major handicaps.

Think about oldest rigorous scientic results. In physics, Newtons laws of motion date back just over 300 years.Modern chemistry was born at around the same time with Mendeleev establishing the fundamentals of theeld 150 years ago. Evolutional biology was articulated by Darwin not long before that. What about math?Any teenager can state Pythagorean Theorem, circa 500 BC. Having to go from ancient Greeks to calculus(thats over 2 millennia of material) in the span of a couple of years of high school, how can we hope to showstudents the mathematics of today?

Moreover, math in our everyday lives is not visible to the naked eye of an untrained human. Take Alice, a highschool student interested in sciences and gardening. She sees biology at work as she plants tomatoes in herbackyard; she supports heavy branches in the ght against gravity recalling lessons from physics; she witnessesthe power of chemistry using detergent to wash tomato stains o her clothes. However, math in all of theabove escapes Alices attention even though she can use it to calculate everything from frequency of fertilizerapplication based on its decay rate to the market price of her product that will yield maximum revenue. And

math is the furthest thing from Alices mind when she sends Bob a picture of her beautiful tomato stand (viaa secure channel, of course).

Submit your articles to student-editor@cms.math.cato be published on the pages of the Margin. Lets show everybody that

math is not done, that it is a thriving eld whose theories support our everyday practices, that is it indeed everywhere. Lets

prove to them that they should want to grow up to be mathematicians.

Some images in this issues are courtesy of Wikimedia Commons. The cover image is due to Shinichi Sugiyama.

EditorKseniya Garaschuk

Ive always wanted tobe tall, but that didntwork out. I compensate

by wearing heelsand being loud.

Other Contributors

Lo BelzileMcGill University

the complete graph onkvertices. In fact, we may takethis as our denition ofk-colouring.

Now, using our homomorphism denition, we cangeneralize to dierent sorts of colourings. In particular,we will consider colourings of oriented graphs, thatis simple graphs with an orientation assigned to eachof the edges, now called arcs. IfGand Hare orientedgraphs, we say that Gadmits a homomorphism toHif there exists : V(G) V(H) such that for alluv A(G)we have that (u)(v) A(H). As such,Ghas an orientedk-colouringifGadmits a homomorphismto an oriented complete graph onkvertices.

Translating this denition using homomorphisms, backto one akin to our original definition of properk

-colouring, we get the following equivalent denition:A proper oriented k-colouring of G is a functionc:V(G) {0, 1, 2, 3 . . . , k 1}such that if uv A(G),then c(u)= c(v)and for all uv,xy A(G)ifc(u) = c(y),

then c(x)= c(v). Or, less formally: given an orientedgraphG, aproper oriented colouring with kcoloursis alabelling of the vertices with elements from{0, 1, 2, 3 . . . , k 1} such that no pair of adjacentvertices receive the same colour and if there is an arcwith the tail coloured iand the head coloured j, thenthere is no arc with the head colourediand the tail

coloured j. While this seems intuitive, without ourhomomorphism denition of proper k-colouring, thissecond requirement for an oriented colouring seemingly

comes from nowhere.Consider the oriented graph in Figure 2. If we colour x0with 0 andx1with 1, we cannot colourx2with colour 0,as we did in Figure 1. In fact, 5 colours are required foran oriented colouring of this oriented graph. From this,we immediately notice that not every oriented planargraph has a proper oriented 4-colouring. In fact, explicitexamples exist of oriented planar graphs which require18 colours. Even worse even though there is no knownexample of an oriented planar graph which requiresmore than 18 colours, the best known upper bound forthe family of all oriented planar graphs is 80 (See TheOriented Colouring Pagefor more information).

Though the introduction to this article gave thereader the tantalizing promise of tenuous application,I apologise, dear reader; you have been had. By way ofan apology, please accept the following joke:

What is a discrete mathematicians favourite colour? 0.

References:

1. The Oriented Colouring Page, http://www.labri.fr/perso/sopena/pmwiki/index.php?n=TheOrientedColoringPage.TheOrientedColoringPage

Figure 2: 4-colouring of anoriented planar graph.

Figure 1:

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Know your math If you dare!by Karl-Philippe Tremblay et Franois Lagac (Universit du Qubec Montral)

We dont always realize that we can question what we do and thinkregarding mathematics. And we often avoid doing so. After all, questioningthe meaning of things usually only brings more questions and uncertainties.

Or maybe we feel that once we open mathematics pandora box, we willnever see the subject the way we use to.

These questions leave traces, they infect us by aectinghow we perceive things around us. Wouldnt it be thesame with mathematics? Especially if we questionthe way we know (our) mathematics, whatever (our)mathematics is

As an example, how can we know that the math weare doing and understanding is done and understoodthe way it is supposed to? After all, we have goodreason to believe that we do not perceive mathematicslike mathematicians from other centuries did, thepythagoreans for instance. So what tells us that weunderstand a theory the way it was understood bythe person who created it? What if we dont?

Those questions emerged for us as we learned aboutepistemology: the study of dierent ways of seeing andunderstanding the world including mathematics. Doesmathematics exist? Most people may answer this withsomething like: Hold my gigantic Calculus textbook,doesnt that look real enough? But is this answersatisfying? Could math be reduced to a book? Learned

by reading it? And if math really exist, does it mean itis somewhere waiting to be discovered? Does it existindependently of us or is it ourproduction, somethingwe created and developed? We all heard those questions,

but what do they mean concretely? For instance, manyof us have once taught mathematics one way or another:

so what did we teach? Our mathematics or the realones?How do we know, as teachers, that what we want studentsto know is actually correct? How do we know if theyunderstand it the way we do? What if they dont? Is itstill the same math? A worse or a better version of it?

For us, teaching means exposing our ways of seeingmathematics to students, hence our ways of seeing

the world. This revealing happens whether we wantit or not because we necessarily have a particularway of doing mathematics with our students. Sincedoing so inuences the way theywill see mathematics,dont we owe it to them to question our beliefs?Should we not at least give them the opportunitytoquestion their vision of mathematics, whether thatopportunity leads them to see math as an external andxed reality, or, not unlike art, as their very creation?Maybe questioning what math is and how its donecan help with the mathophobia our society seemsto be plagued with.

We believe that positioning math as questionablecan help change peoples perception that math issomething that only those good enough can grasp.And that turning it to something like what we canget away with can make it something more personal,more alive. That is, to help people see how math is notas a thing that is around them whether they want itor not, but as something they can (have to?) take partof (in order for it to exist). As for us, questioningour beliefs led us to see math as something we do, asopposed to something we merely listen to. It does,however, leave (at least!) one question open: What

does (knowing) mathematics truly mean? You answerthis if you dare!

Art is what you can get away

with Marshall McLuhan andwhat about math? Could math

be what we can get away with?

Karl-Philippe Tremblay

tudiant la matrise et papade deux merveilleux (et trsjeunes) enfants. Je mintresseau sens que lon donne auxmathmatiques. Plus prcis-ment, je me pose la questionsuivante : est-ce que faire lesmathmatiques diremmenne reviendrait pas faire desmathmatiques direntes?

Franois Lagac

Masters student in math edu-

cation, my interest is in howour perception of maths-physics relation inuences whatwe consider to be maths (orphysics). When Im not think-ing about that, I enjoy playingwith my girl, as you may see!

The CMS Student Committee is looking for proactive mathematics students interested in joining the Committeein the role of Notes from the Margin editor. If you are interested, or know someone who may be, please visitour web-site http://studc.math.ca for more information and an application form. If you have any questions,please contact us at chair-studc@cms.math.ca.

Le comit tudiant de la SMC est la recherche dtudiants en mathmatiques, dynamiques et intresss joindrele comit au poste dditeur de la revue Notes from the Margin. Si vous tes intress, ou connaissez quelquunqui pourrait ltre, veuillez consulter notre site web http://studc.math.ca an dobtenir plus dinformations et

le formulaire dapplication. Si vous avez des questions, veuillez nous crire chair-studc@cms.math.ca.3

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The intuition you build from experimenting andproving basic facts is arguably paramount tobecoming a competent mathematician. Of course, thisis nothing but a verbose restatement that mathematicsis not a spectator sport; however, its for this reason thatI was so intrigued by the result well develop below.

Since elementary school, weve encountered right-angled triangles whose sides have integer length.3:4:5, 5:12:13, 8:15:17 and 9:40:41 are all examples ofPythagorean triples. But theres another interestingcommonality: every Pythagorean triple must haveat least one even number. While this fact can bederived independently using Euclids formula( m2 n2 : 2mn : m2 + n2, m > n , m , n N ) ,we can build on this intuition to establish a moregeneral result.

Proposition 1 There do not exist four points,a,b,c,d R2 such that the (Euclidean) distancebetween each pair is an odd integer.

In particular, observe that if we require opposingsides of the quadrilateral abcdto be parallel, thenour observation about Pythagorean triples followstrivially. But in keeping with the original theme, weshow that the simplicity and creativity of the proofgiven below is far more illuminating than the resultof the proposition, itself.

Proof.Suppose, in order to derive a contradiction,that there exist four points, a,b,c,d, such that allpairwise distances are odd integers. Without lossof generality, we may assume that = by rigidlytranslating a,b,c,dso that dlies at the origin. Thena, b, c, ab, bc and c a are oddintegers. Thinking of the points as vectors in R ,dene the matrix of inner products,B, by

B :=

a, a a, b a, cb, a b, b b, cc, a c, b c, c

Now, note that if m is an odd integer, thenm

2 1 (mod 8)and hence the cosine law implies

that 2a, b = a2+b2 a b2 1 (mod 8).

An identical result for the other pairs of distinct pointsshows that2Bis congruent, modulo 8, to the matrix

R :=

2 1 1

1 2 1

1 1 2

.

B ecause det(R) = 4, we k now t hatdet(2B) 4 (mod 8) . In particular, et(2B) is

nonzero, soBhas full rank, i.e. rank(B) = 3. But onthe other hand, by letting

A :=

a1 b1 c1a2 b2 c2

,

we write B = A A, so that rank(B)rank(ATA) max{rank(A), rank(AT)}= 2 ,a contradiction.Therefore, there do not exist fourpoints in 2such that all pairwise distances are odd.

We were able to use our knowledge of linear algebrato prove something non-obvious about the structureof quadrilaterals in the plane. But what does modulo 8have to do with quadrilaterals? And how does matrixalgebra emerge so naturally with our denition ofB?This example has made me think harder about newways of proving old facts; I hope it is able to inspire youto come up with fun, creative proofs in your own work!

Note: A related, more general (and more creative)version of the result above was proven by Anningand Erds [2]. They have shown that for any positiveintegern, it is possible to ndnpoints in the plane, notall collinear, such that their pairwise distances are allintegers (and that this result fails for =). If youre

looking for a bit of a challenge and a good read, youcan nd the reference to the original paper below!

References

[1] Matouek, J., 33 Miniatures, vol. 53. AMS Bookstore, 2010.[Adapted from original work].

[2] Anning, N. H. and Erds, P., Integral distances, Bulletin ofthe American Mathematical Society 51, 8 (1945), 598600.

The estranged hypotenuseby Aaron Berk (University of Toronto)

In mathematics, the theorems we learn in class are sometimes more usefulthan the techniques we use to prove them. But other times (as Im sureyouve realized from your analysis, or linear algebra homework), it is the

proof techniques, themselves, that are more useful than the basic results youdevelop along the way.Aaron Berk

Though brilliant maybe a sucient conditionfor becoming a successfulmathematician, I canassure you that sedulousis far more necessary.

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Introduction

Of the many card games in existence, the trick-takingfamily is one large sector, crowned by Bridge, self-styled King of them all. The relevant rules are simple:

There are four players traditionally named afterthe compass directions; NorthSouth and EastWest arepartnersin this enterprise;

The entire standard playing deck is dealt, 13 cardsto each player; aces high, deuces low;

The play consists of 13 atomic tricks, each player inclockwise rotation playing one card from their hand;

First player (the leader) may play anything;subsequent playersmustfollow suit if able, elsemay discard anything;

Highest card of the suit led wins the trick; winnerof the previous trick leads to the next.

The objective is generally to maximize the aggregatenumber of tricks you and your partner win in theplay; the bidding and the existence of trumps are left

to some other time.The 9-card t

We now turn to the actual combinatorics of a 9-cardt, that is when NorthSouth hold 9 of the 13 cards ina suit, say spades. We shall compute the probabilitiesof various breaks (holdings for EastWest). We willassume the cards are dealt uniformly at random, wecan see the hands of NorthSouth, and we knownothing a priori about what EastWest holds.

To generate the correct probabilities, we will deal allthe 26 unknown cards to EastWest, 4 spades and 22non-spades. Traditionally 31 means either 1 or 3 spadesdealt to East, so we will double these probabilities up. Asall cards are ultimately distinguishable, we can separateout specic holdings: there are a total of 16 = 24waysthe spades can be distributed as each individual cardcan independently be given to East or West. This tablebelow lists the probabilities to 4 decimal places:

Break Probability Decimal

22 22

11

4

2/

26

13 0.4070

31 222

12

4

1

/26

13

0.4974

40 222

13

/26

13

0.0957

Decimal

0.0678

0.0622

0.0478

Specific ways Probability4

2

= 6

22

11

/26

13

24

3

= 8

22

10

/26

13

24

4

= 2

22

9

/26

13

The decision

Now consider a 9-card t missing only the king andthree irrelevant low cards:

Do it with nesse:by Jerey Tsang (University of Guelph)

In the lore of trick-taking card games, many tables of probabilities and odds havebeen published. We examine here the decision to nesse or drop in a 9-card tmissing the king, especially how the probabilities changeas cards hit the table.

You want to keep informed of the latest events and activitiesfor math students? An easy solution: follow the CMS studentcommittee in the social media. Search for CMS Studc onFacebook, Twitter and Google+.

Vous dsirez demeurer au fait des tous derniers vnements etactivits destins aux tudiants en mathmatiques? Une solutionfacile: suivez le comit tudiant de la SMC sur les rseaux sociaux.Cherchez CMS Studcsur Facebook, Twitter et Google+.

Jerey Tsang

Diractionslitandp

hylumtree,

Demandcurvesriseandrun,

Iambicfootandfugalp

hrase,

Inmathconvergetoone.

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The object is to not lose a trick to the king. For ourpurposes, we completely ignore all non-spade plays,hence the unbalanced number of cards.

There are two approaches available. The rst is to playthe ace and hope the king dropsunder it. The second,thenesse, leads the queen and plans to play low fromNorth, hoping West holds the king and is unable

to play it (under pain of immediate capture by theace). If the queen holds, the jack (ten, nine) is led torepeat the nesse. Given there are only 4 spades out,even if West held them all his king can be picked upwithout loss.

The chance of success for the drop is easy to calculate.There are two successful cases: the king alone withEast or West, hence 2 x 0.0622 0.1243, just shy of 1 in8. The nesse is not harder to analyze: as mentionedearlier, it succeeds i West holds the king regardlessof the rest of his hand, an exact half chance a priori.

The updateWe turn to a slightly harder problem. Let us say welead the queen from South and West plays the 2.At this time, we can compare going up with the ace(playing East for singleton king) and playing low(going ahead with the nesse).

(Horizontal cards are the ones played). Since Westis now known to hold the 2, half of the original 16holdings are now impossible.

The drop works if West holds 432 (East K), succeeding0.0622/0.5 0.1243 of the time, curiously unchanged

as a 1 in 8 underdog; the nesse works if West holdsK432, K42, K32 or K2, succeeding (0.0478 + 2 x 0.0622+0.0678)/0.5 of the time, losing ground to exactly 48%.

The reveal

How can it be that the mere act of West followingsuit decreasesthe chance of success? Look at it in adierent way. When the queen was led from South,

one of ve mutually exclusive and exhaustive eventscould happen West could play any of the 4 spades,or show out(discard).

If West discards, you are instantly doomed, Eastholding the guarded king behind your ace; if the kingpops up, all your troubles are over as you obviouslycapture with the ace. In a strict game-theoretical sense,the other three events are essentially equals.

So how did the nesse lose ground? The 40 breakwith East is a priori less likely than the 31 breakwith West holding the bare king (not to mention the

possibility West somehow makes the horrible play ofthe king with low cards to spare). Thus, we ruled outmore of the successful cases than the failing cases,decreasing the remaining probability of success.

Finally, students of the theory of vacant spaces(a heuristic) may notice that 48% is exactly 12/(12+13).At the time of decision, West has shown up withthe 2, which as she would not play the king even ifshe had it, still gives no knowledge about the rest ofher hand. Thus, with 12 unknown cards to Easts 13,the probability she now holds the king is 12 in 25.Its that simple.

The author would like to thank Nate Munger, whoselament about an ill-mannered opponent (who ewace unsuccessfully and tried to argue Natemade theerror) inspired this exercise.

For further information, including a free Learn toPlay Bridge interactive e-book (from the downloadversion), visit Fred Gitelmans excellent Bridge BaseOnline (www.bridgebase.com).

University of Manitoba is hosting the 2014 CMS Summer Meeting, June 6 th-9 th. Studc isplanning a number of exciting student events to be held during the meeting, including a postersession, a student workshop and and a social. Check the meeting website cms.math.ca/Events/summer14/.efor the most up-to-date information.

University of Manitoba est lhte de la Runion dt SMC 2014qui aura lieu du 6 au 9 juin.Studc prpare actuellement plusieurs activits excitantes destines aux tudiants qui prendrontpart la runion: il y aura une session de prsentation par aches, des ateliers ainsi quuneactivit sociale. Consultez le site web de la runion,cms.math.ca/Reunions/summer14/.f,

pour obtenir linformation la plus rcente.

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Good luck.

This is a kind of puzzle inspired byvarious puzzle hunt competitions.

There are no instructions and it is yourjob to work out what to do and then doit. The only rule is that you have to ndan answer, which is a word or a shortphrase. The title is sometimes helpfulin either working out what to do orconrming that you have the correctanswer. Sometimes it is useless. You areallowed to use absolutely any tools atyour disposal (including the internet).

Indeed an operation, one of four, in maths.

Not italicised or lined.Didnt I say margarine was better?

Eat this romantic fruit.

Xihe, Vishnu, Zeus for example.

Brick road leads to magical what city?

Yellow and blue in combination.

Little in aesthetics, very ugly and gross.

Out of avour, very weak and pathetic.

Native marsupial to the bizarre land down under.

Gets items into orderly arrangement.

Extinct, sometimes wooly, member of the Elephantidae.Sheep (eg) leftovers in dish.

The poor demoted ex planet.

When a recovering addict returns to drugs.

Outer space beckons when on this.

Really big busy capital city of Ontario.

Day of the week, referred to aectionately as hump.

Striped quadraped from African plains.

Lengthy Secrets

by Tyrone Ghaswala (University of Waterloo)

The Distractions Page

Tyrone Ghaswala

I am a PhD student at theUniversity of Waterloo byday, and puzzle enthusiast

by night. When I wasyounger, I wanted to beMichael Jordan when Igrew up. Secretly I still do.

La session de prsentation par aches AARMS-SMCde la Runiondhiver SMC 2013 Ottawa a permis de runir des participants dedisciplines varies et de tous les niveaux scolaires - incluant 2 prsentationsdlves du secondaire! Au banquet de la SMC, les gagnants de la sessionpar ache ont clbr leur ralisation en prsence de mathmaticiensclbres, notamment John Conway. Ils ont galement pu apprcier laprestation de Daniel Richter, le crieur publique de la capitale nationale.

Nous vous invitons participer la prochaine comptition qui aura lieulors de la Runion dt SMC 2014 Winnipeg, du 6 au 9 juin.Vousaurez lopportunit de prsenter votre ache et mme la possibilit de

remporter des prix! De surcrot, la SMC ore maintenant une rductionsur les frais dinscription pourles participants de la sessionde prsentation par aches.Dautres informations surlvnement seront mises enligne sur notre site web(studc.math.ca) et surnotre page Facebook

(CMS Studc) dansles mois venir.

The AARMS-CMS Student Poster Sessionat the 2013 CMS Wintermeeting in Ottawa brought together presenters from all over the countryfrom dierent disciplines and dierent levels of educations including2 presentations from high school students! At the CMS banquet, postersession winners celebrated their achievements with many brillianmathematicians, including John Conway, and enjoyed the performanceof Daniel Richter, the town crier.

We invite you to participate in the next poster session that will be heldat the 2014 CMS Summer Meeting in Winnipeg, June 6th-9th.You wil

have a chance to present your poster and even compete for

prizes! As a further incentive, CMS now oers a lowerregistration fee for poster session participants

More information about participatingwill be posted on our website (studcmath.ca) and our Facebook page (CMS

Studc) in the coming months.

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Canadian Mathematical Society 2014. All rights reserved.

Visit us athttp://studc.math.ca

Notes from the Marginis a semi-annual publication produced

by the Canadian Mathematical Society Student Committee(Studc). The Margin strives to publish mathematical content

of interest to students, including research articles, proles,opinions, editorials, letters, announcements, etc. We invite

submissions in both English and French. For furtherinformation, please visit studc.math.ca; otherwise, you cancontact the Editor at student-editor@cms.math.ca.

Notes from the Marginest une publication semi-annuelle produite par

le comit tudiant de la S ocit mathmatique du Canada (Studc).La revue tend publier un contenu mathmatique intressant pour lestudiants tels que des articles de recherche, des proles, des opinions,

des ditoriaux, des lettres, des annonces, etc. Nous vous invitons

faire vos soumissions en anglais et en franais. Pour de plus amplesinformations, veuillez visiter studc.math.ca; ou encore, vous pouvezcontacter le rdacteur en chef student-editor@cms.math.ca.

The Student Committee is accepting applicationsto fund student events across Canada. We sponsorregional conferences, student socials, seminars andother events. The next application deadline is March 15th.Visit http://studc.math.ca for more information and to ndthe application form.

Le comit tudiant accepte les demandes de nancement pour desvnements destins aux tudiantset organiss travers le Canada.Nous nanons les confrences rgionales, les sminaires, les activitssociales et dautres types dvnements. La prochaine date limite pourdposer une demande est le 15 mars. Visitez lehttp://studc.math.ca pourplus dinformations et pour obtenir le formulaire de demande.

SUMMpar Renaud Raqupas (McGill Universit)

Les 10, 11 et 12 janvier dernier sest tenue lUniversit Concordia la cinquimedition des Sminaires universitaires en mathmatiques Montral (SUMM).Cest avec enthousiasme et curiosit que plus de 102participants venus du Qubec comme de lOntario yont partag intrts, ides, nourriture et camaraderies.Tout comme la provenance des participants laconfrence, les sujets discuts y taient trs varis:thorie des nombres, thorie des catgories, topologie,algbre, thorie des noeuds, gomtrie direntielle,physique mathmatique, etc.

Dveloppe autour de 19 exposs tudiants,4 confrences plnires de professeurs des universitsde lle de Montral et 2 langues ocielles, les activits

des SUMM ont satisfait lapptit mathmatique etsocial de ses participants de lcole Polytechniquede Montral, lUniversit Concordia, lUniversitde Montral, lUniversit du Qubec Montral,lUniversit Laval, lUniversit McGill, lUniversitdOttawa et lUniversit de Waterloo. Principalementfinancs par les dpartements et associationstudiantes de ses universits organisatrices(Concordia, McGill, UdeM et UQAM), par lISM et parle STUDC/CMS, les SUMM attirent chaque annede plus en plus dtudiants de lextrieur, lesquelsviennent de plus en plus loin: que cela continue!

Renaud Raqupas

Au dbut, javais enviede partager avec les gensle plaisir que je trouvais faire des maths, maisces eorts suscitaient peude passion... Maintenant,

je ne donne plus trop dedtails et je suppose queles gens continuent decroire que japprends compter plus rapidement.