[the, commutator] vol 2 issue 1 edition 1

[The, Commutator] Vol 2 Ed 1 (1)_Commutator Volume 2 Edition 1 20/03/2011 13:36

2 [The, Commutator] March 2011 www.the-commutator.com

Special thanks toCarl Chaplin, Dr. Lorna Love, Mrs Shazia Ahmed, Dr. Tom Leinster,

Prof. Stephen Senn, Prof. Peter Kropholler, Dr. Radostin Simitev, Dr. Tara Brendle, Dr. David Moore, Colin Pratt

Contents

University of GlasgowSchool of Mathematics and Statistics

University Gardens, University of GlasgowGlasgow, G12 8QA

http://www.gla.ac.uk/mathematicsstatisticstel: +44 (0) 141 330 5176

email: maths-stats-enquiries:glasgow.ac.uk

Paul Erdösthe mathematician who sawproofs under every roof

Francis Galtonand Regression to the mean

Letter from the editor

The Futurama Theorem

The Full Converse to Lagrange’s Theorem

Did you know...

MacSoc

Games and Puzzles

a counterexample

The Problem of Apolloniusand the high school student who pro-vided the 5th ever unique solution

activities and events

[email protected]

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21

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explained

Editor in ChiefHristo Georgiev

DesignerJaspal Puri

Finance ManagerRoss McKinstry


www.the-commutator.com March 2010 [The, Commutator] 3

Letter from the editor..ank you

First of all, I want to thank everybody involved in the project, especially Jaspal Puri, who is the mainreason behind the present issue’s great look and design. I would also like to thank the University ofGlasgow School of Mathematics and Statistics, the MacLaurin Society and the Institute of Mathematicsand its Applications. e IMA was founded in 1964 and now has over 5000 members. It is the pro-fessional and learned society for qualified and practising mathematicians. It supports the advancementof mathematical knowledge and its applications and promotes and enhances mathematical culture inthe United Kingdom, and elsewhere, for the public good.

Why is Maths interesting?Mathematics is largely a game which works in this way: we make up a set of rules and then we try

to see what consequences follow from the rules. Mathematics is a collection of extended, collaborativeand smaller games of ‘what if ’ played by mathematicians who make up sets of axioms and then explorethe consequences of following those axioms.

Mathematicians are interested not only in what happens when you adopt a particular set of rulesbut also in what happens when you change the rules. For example, one game is called ‘arithmetic.’ Wedefine what it means to be a number, i.e. the first number is zero, and if we have any number we candefine another number by adding 1 to it. Of course, to do this, we also have to define what we meanby ‘addition’. Now, having made up these rules, we start playing with them to see what happens. Onething that happens is that we notice certain patterns and we use these patterns to add new rules. Wealso make up new notations to make things easier to write.

Here is the important point: all of this is presented to students as if it has always been known. Butin fact it was all discovered by mathematicians playing around making up questions, trying to answerthem and then sharing their answers with other mathematicians who would use the answer to makeup new questions, and so on.

Of course, over time, the questions get more complicated and you have to understand a lot of whathas happened in the game already to even understand what is being asked. And at a certain point thequestions start to be about the very nature of games: what is mathematics? But it is still the same game.You invent objects (numbers, sets, groups, fields, points, lines, planes, manifolds, algorithms), andrules that have to be obeyed by the objects, and then the game begins.

What is the present issue about?We’ll be talking about the first mathematical theorem invented especially for TV series, the Futurama

eorem. e reader will also find out more about the bizarre life of one of the greatest mathematiciansof the 20th century, Paul Erdős. Another interesting figure presented on our pages is the great Victorianeccentric and scientist Francis Galton. Some of the most fascinating mathematical facts and the newestsolution to the Apollonius’ problem are covered in our Facts and News sections respectively. We presenta Counterexample to the Full Converse of Lagrange's eorem and an idea of what it is to be a memberof the MacSoc and more about its activities and events.

We keep the icing for the cake on our puzzle pages where the reader can test how good they are atsolving maths and logic problems. We look forward to receiving your solutions, suggestions, or com-ments on our editorial e-mail address. You can also get it touch with us through the Contact us formon our website.

“Any intelligent fool can makethings bigger, more complex,and more violent. It takes atouch of genius - and a lot ofcourage - to move in the oppo-site direction.”

Albert Einstein

Hristo Georgiev


4 [The, Commutor] March 2010 www.the-commutator.com

18-year-old Radko Kotev, a studentat the National High School of NaturalSciences and Mathematics in Sofia,Bulgaria, presented his solution as aproject at the European Union Contestfor Young Scientists (EUCYS) in Lis-bon, Portugal. Competing along with85 other student projects from acrossthe world, Kotev was awarded a prizeof €7,000 and a one-week visit to theInstitut Laue-Langevin (ILL) in Greno-ble where he presented his project inone of the world’s most modern re-search centres.

The notorious puzzle has severalvariants. Originally the complex geo-

metric problem was set by Euclid andwas further developed by his discipleApollonius of Perga (ca. 262 BC – ca.190 BC) - known as ‘The Great Geome-ter.’ The original solution, posed andsolved in his work ‘Tangencies,’ waslost forever in the fire that destroyedthe Library of Alexandria.

Kotev is the fifth mathematician tofind a valid solution to the enigma.‘Now we have powerful computers athand,’ he says. ‘I don’t think I wouldhave been able to achieve the same re-sults without one.’

The young mathematician spenttwo years working on the project. Allthe way though it, he was assisted by

his teacher.The solution appeared to

Kotev on an early morning.‘It was so strange,’ he says. ‘Iwent to bed that night,thinking about the problem,and in the morning I said tomyself, why not try this way.I took a sheet of paper, drewup what was needed andthe solution came very eas-

ily.’ He first showed the solution to his

parents who are both mathematiciansand then to his teacher. This wayRadko Kotev managed to solve a prob-lem that has challenged the hu-mankind for more than 2000 years.

‘I have never viewed mathematicsas an obligation,’ he says. ‘I have neverthought I should solve problems allthe time and write homework end-lessly. I have always solved problemsfor the fun of it. In fact, it is my hobby.’

Radko hopes to start his undergrad-uate degree in applied mathematics atthe University of Glasgow in Septem-ber 2011.

The four famous mathematicianswho provided straightedge and com-pass solutions to the Apollonius’ prob-lem are François Viète (1600),Jean-Victor Poncelet (1811), JosephDiaz Gergonne (1814), and Julius Pe-tersen (1879). In the 16th century Adri-aan van Roomen solved the problemusing intersecting hyperbolas but hissolution did not use only straightedgeand compass constructions. Themethod of van Roomen was simplifiedlater by Isaac Newton, who showedthat the Apollonius’ problem is equiv-alent to finding a position from the dif-ferences of its distance to three knownpoints, which has application in navi-

gation and positioning systems likethe GPS.

The Apollonius’ problem has stimu-lated much further work. Generalisa-tions to three dimensions –constructing a sphere tangent to fourgiven spheres – and beyond have beenstudied. The configuration of threemutually tangent circles has receivedparticular attention.

René Descartes gave an algebraic

formula relating the radii of the solu-tion circles and the given circles, nowknown as the Descartes’ theorem. Solv-ing Apollonius’ problem iteratively inthis case leads to Apollonian gasketwhich is one of the earliest fractals de-scribed in print.

Kotev is the fifth mathe-matician to find a validsolution to the enigma

He hopes to start his undergraduate degree in ap-plied mathematics at the University of Glasgow inSeptember 2011

Figure1. Radko Kotev’s solution tothe Apollonius problem.

The Problem of Apollonius is to construct a circle tangent to three given circles using straightedge and compass. The problem usuallyhas eight different solution circles that exist that are tangent to the given three circles in a plane. The given circles must not betangent to each other, overlapping, or contained within one another for all eight solutions to exist. The last known solution satisfyingall requirements dates back to the nineteenth century and was done by Joseph Diaz Gergonne. There are several methods involvingdifferent coordinate systems but they are not straightedge and compass constructions.

THE APOLLONIUS PROBLEMTHE HIGH SCHOOL STUDENT WHO GAVE THE LATEST SOLUTION

News The Apollonius Problem

Hristo Georgiev



Largest Known PrimesThe largest known primes are of the

form (2m - 1). The reason is that thereexist efficient ways to test whethersuch numbers are prime. Primes ofthis type are called a Mersenne primes. As of Sept 2010, the largest knownprimes were:243,112,609 - 1 (discovered in Aug 2008)242,643,801 - 1 (discovered in Jun 2009)237,156,667 - 1 (discovered in Sep 2008)The largest is over 2 million digitslong. These primes were all discov-ered in the last 3 years. The search forlarge primes has accelerated with thehelp of several hundred people acrossthe internet in a project called GIMPS(the Great Internet Mersenne PrimeSearch).

Pi DayThe number π in some form or an-

other has been around for thousandsof years. Ancient Babylonians, Egyp-tians, Greeks and Indian mathemati-cians estimated pi to varying degreesof accuracy. Greek mathematicianArchimedes (287-212 B.C.E.) calcu-lated π to 3.1419 in his On the Measure-ment of a Circle. In 480 C.E., Chinesescholar Zu Chongzhi used a previousscholar's algorithm to calculate thevalue of pi as 3.1415926 - the most ac-curate approximation of pi to beknown for the next 900 years.

The symbol for pi was first used in1706 by William Jones but was popu-larised later by the Swiss mathemati-cian and physicist Leonhard Euler in1737.

Pi Day is a holiday commemoratingthe mathematical constant π (pi). PiDay is celebrated on March 14 (or 3/14in month/day date format), since 3, 1and 4 are the three most significantdigits of π in the decimal form. In2009, the United States House of Rep-resentatives supported the designa-tion of Pi Day. March 14 is also AlbertEinstein’s birthday.

Pi Approximation Day is held on July22 (or 22/7 in day/month date format),since the fraction 22/7 is a common ap-proximation of π. Larry Shaw created Pi Day in 1989.The holiday was celebrated at the SanFrancisco Exploratorium, which nowa-days continues to hold Pi Day celebra-

tions with staff and public marchingaround one of its circular spaces thenconsuming fruit pies.

On March 12, 2009, the U.S. Houseof Representatives passed a non-bind-ing resolution (HRES 224) recognizing

March 14, 2009 as National Pi Day.Sometimes the so-called Pi Minute

is also commemorated. This one oc-curs twice on March 14 at 1:59 a.m.and 1:59 p.m. If π is truncated to sevendecimal places, it becomes 3.1415926,making the Pi Second occur on March14 at 1:59:26 a.m. (or 1:59:26 p.m.) If a24-hour clock is used, the Pi Second oc-curs just once yearly on March 14 at01:59:26. In 2015, Pi Day will reflectfive digits of π (3.1415) as 3/14/15 inmonth/day/year date format. There willalso be a Pi Second accurate to 10 dig-its (3.141592654) at 9:26:54 in thatyear's Pi Day.

MIT loves Pi! Each year the Massa-chusetts Institute of Technology releasesits admission decisions at 1:59pm onPi Day.

Citing the Digits of Pi

Akira Haraguchi (原口證), born in1946, is a retired Japanese engineercurrently working as a mental healthcounsellor and business consultant.He set the current world record,100,000 digits in 16 hours, starting at 9a.m (16:28 GMT) on October 3, 20stopping with digit number 100,000 at1:28 a.m. on October 4, 2006. The eventwas filmed in a public hall in Kisarazu,east of Tokyo where he had five-minute breaks every two hours to eatonigiri rice balls to keep up his energylevels. Even his trips to the toilet werefilmed to prove that the exercise waslegitimate. Haraguchi’s previousworld record (83,431) was performedfrom July 1 2005 to July 2 2005.

Despite Haraguchi’s efforts and de-tailed documentation, the GuinnessWorld Records have not yet acceptedany of his records because of the ruleallowing the time between two num-

bers to be no more than 15 seconds.The Guinness-recognised record forremembered digits of π is 67,890 dig-its, held by Lu Chao (吕超), a 28-year-old graduate student from China whospent roughly one year memorisingthe digits. The record was set in 2006when he was 24. It took him 24 hoursand 4 seconds to recite to the 67,890th

decimal place of π without an error -with no lunch time and no toiletbreaks. Lu Chao said he had got100,000 digits of pi, and had beengoing to recite 91,300 digits of them,but he made a mistake at the 67,891th

digit.Haraguchi views the memorisation

of Pi as ‘the religion of the universe,’and as an expression of his lifelongquest for eternal truth. Since child-hood, he has always wondered whysome people - especially those withphysical and mental disabilities - suf-fer. He consulted religion and philoso-phy books for answers, but only invain. Then he turned to nature and re-alised, he said, that nothing in nature -be it leaves, trees or mountain scenery- is linear or square. ‘I realised that na-ture is not made of straight lines …And I realised that all things in the uni-verse … rotate. Rotation became a keyconcept for me.’So when he learned that Pi is an end-less series of numbers with no patternor repetition, it made perfect sense tohim to take it as a symbol of life, hesaid.

He uses a system he developed,which assigns the syllabic Japanesesymbols Kana, to numbers, allowingthe memorisation of Pi as a collectionof stories.

For example 0 can be substituted byo, ra, ri, ru, re, ro, wo, on or oh, and 1 canbe substituted by a, i, u, e, hi, bi, pi, an,ah, hy, hyan, bya or byan. The same isdone for each number from 2 through9. Combining these characters, he hascreated a myriad of stories and poems,including a story about the legendary12th-century hero Minamoto no Yoshit-sune and his sidekick Benkei, who wasa Buddhist monk. When he recites dig-its, he explains that he ‘simultaneouslyinterprets’ his linguistic creations backinto numbers.

DID YOUKNOW...

Facts Did You Know...

Hristo GeorgievHristo Georgiev



Galton was born into a wealthy family, theyoungest of nine children and appears to havebeen a precocious child, in support of which, hisbiographer Forrest cites the following letter toone of his sisters:

My dear Adèle,I am four years old and can read any English book. Ican say all the Latin Substantives and adjectives andactive words besides 52 lines of Latin poetry. I can castup any sum in addition and multiply by2,3,4,5,6,7,8,(9),10,(11)I can also say the pence table, I read French a little andI know the clock.Francis Galton, February-15-1827

Apparently Galton was also a truthful child,since, having written the letter, he had realizedthat what he had claimed about the numbers 9and 11 was not quite true and had tried to oblit-erate them. And before you get too impressed, hisbirthday was 16 February so he was very nearlyfive!

Galton’s later progress in education was notquite so smooth. He dabbled in medicine and

F R A N C I S G A L T O N

AND REGRESSION TO

THE MEAN

This year marks the centenary of thedeath of the great Victorian eccentricand scientist Francis Galton (1822-1911),a cousin of Charles Darwin, who is animportant but also a curious figure inthe history of statistics but also in thatof many other sciences, in particular psychology and genetics.

Stephen Senn



then read mathematics at Cam-bridge but eventually had to takea pass degree. In fact he subse-quently freely acknowledged hisweakness in formal mathematicsbut this weakness was compen-sated by an exceptional ability tounderstand the meaning of data.Galton was a brilliant natural stat-istician.

Many words in our statisticallexicon were coined by Galton. Forexample, correlation and deviate aredue to him, as is regression and hewas the originator of terms andconcepts like quartile, decile andpercentile, and of the use of medianas the mid point of a distribution.Of course, words have a way of de-veloping a life of their own, sothat, unfortunately decile is increas-ingly being applied to mean tenth.There are, pretty obviously, tentenths of a distribution but thereare, slightly less obviously, onlynine deciles, since the deciles arethe boundaries between the tenths.To use decile to mean tenth, as forexample speaking of students ‘inthe top decile’ (according to theirexamination marks) is not onlypompous but wrong and meansthat yet another word will eventu-ally have to be invented to per-form the function that Galtoncreated decile to fulfil. In fact, I amtempted to say that to use decile tomean tenth is the mark of an imbe-decile but I digress…

To take another example, we nolonger use the term regression inquite the way Galton did, nowusually reserving it for the fittingof linear relationships. In Galton’susage regression was a phenome-non of bivariate distributions andsomething he discovered throughhis studies of heritability. How-ever the use of regression in Gal-ton’s sense does survive in thephrase regression to the mean – apowerful phenomenon it is thepurpose of this article to explain.Galton first noticed it in connec-tion with his genetic study of thesize of seeds but it is perhaps his1886 study of human height thatreally caught the Victorian imagi-nation. Galton had compared theheight of adult children to the

heights of their parents. For thispurpose he had multiplied theheights of female children by 1.08.For as he put it

In every case I transmuted the femalestatures to their corresponding maleequivalents and used them in theirtransmuted form, so that no objectiongrounded on the sexual difference ofstature need be raised when I speak ofaverages.

It is interesting to note, by theby, that he also consideredwhether 1.07 or 1.09 might not bea better factor to use but remarked

The final result is not of a kind to beaffected by these minute details, for ithappened that, owing to a mistaken di-rection, the computer to whom I firstentrusted the figures used a somewhatdifferent factor, yet the result cameout closely the same.

The year being 1886 the computerin question was, of course, ahuman and not an electronic assis-tant! The more interesting point,however, is that Galton providesan early example of what is nowrecognised as a general scientificphenomenon. Scientists neverseem to fail the robustness checksthey report. It is interesting tospeculate why.

Galton’s data consisted of 928adult children and 205 ‘parent-ages’ that is to say father and wifecouples. (The mean number ofchildren per couple was thus justover 4.5.) He represented theheight of parents using a singlestatistic ‘the mid-parent,’ thisbeing the mean of the height of thefather and of his wife’s height mul-tiplied by 1.08. Of course, as pre-viously noted, for the femalechildren the heights were alsomultiplied by 1.08. For the malechildren they were unadjusted.Figure 1 is a more modern graphi-cal representation of Galton’s data.(The data have been taken fromthe very useful website providedby the University of Alabama inHuntsville.) Galton had groupedhis results by intervals of one inchand in consequence, if a given

Feature Francis Galton



child’s recorded height were plottedagainst its recorded ‘mid-parent’height, many points would be over-plotted. I have added a small amountof ‘jitter’ in either dimension to sepa-rate the points, which are shown inblue. The data are plotted two ways:child against mid-parent on the leftand mid-parent against child on theright. The thin solid black diagonalline in each case is the line of equality.If a point lies on this line then childand mid-parent were identical inheight. Also shown in red in each caseare two different approaches onemight use to predicting ‘output’ from‘input’. The dashed line is the leastsquares fit, what we now (thanks toGalton) call a regression line. The thickblack line is a more local fit, in fact aso-called LOWESS (or locallyweighted scatterplot smoothing) line.The point about either of these two ap-proaches irrespective of whether wepredict child from mid-parent or viceversa is that the line that is producedis less steep than the line of exactequality. The consequence is that wemay expect that an adult child is closerto average heights than its parents butalso paradoxically, that parents arecloser to average height than is theirchild.Figure 1 is a more modern graph-ical representation of Galton’s data.(The data have been taken from thevery useful website provided by theUniversity of Alabama in Huntsville.)Galton had grouped his results by in-tervals of one inch and in consequence,if a given child’s recorded height wereplotted against its recorded ‘mid-par-ent’ height, many points would beover-plotted. I have added a small

amount of ‘jitter’ in either dimensionto separate the points, which areshown in blue. The data are plottedtwo ways: child against mid-parent onthe left and mid-parent against childon the right. The thin solid black diag-onal line in each case is the line ofequality. If a point lies on this line thenchild and mid-parent were identical inheight. Also shown in red in each caseare two different approaches onemight use to predicting ‘output’ from‘input’. The dashed line is the leastsquares fit, what we now (thanks toGalton) call a regression line. The thickblack line is a more local fit, in fact aso-called LOWESS (or locallyweighted scatterplot smoothing) line.The point about either of these two ap-proaches irrespective of whether we pre-dict child from mid-parent or vice versa isthat the line that is produced is lesssteep than the line of exact equality.The consequence is that we may ex-pect that an adult child is closer to av-erage heights than its parents but alsoparadoxically, that parents are closerto average height than is their child. This particular point is both deep andtrivial. It is deep because the first timethat students encounter it (I can still re-member my own reaction) they as-sume that it is wrong: its truth is wellhidden. Once understood, however, itbecomes so obvious that one isamazed at how regularly it is over-looked.

So, let’s leave Francis Galton for themoment and consider another exam-ple, this time a simulated one. Figure2 shows simulated values in diastolicblood pressure (DBP) for a group of1000 individuals measured on two oc-

casions: at baseline and atoutcome(blue circles) or inconsistent,tentatively tensive perhaps(orangestars). The distributions at outcomeand baseline are very similar with,means close to 90mmHg and a spreadthat can be defined by that Galtonianstatistic the inter-quartile range as beingclose to 11mmHg on either occasion.In other words, what the pictureshows is a population that all in all hasnot changed over time although, sincethe correlation (to use another Galton-ian term) is just under 0.8 and there-fore less than one, there is, of course,variability over time for many individ-uals.

However, in the setting of manyclinical trials, Figure 2 is not a figurewe would see for the simple reasonthat we would not follow up individ-uals who were observed to be nor-motensive at baseline. Instead whatwe would see is the picture given inFigure 3. Of the 1000 subjects seen atbaseline 285 had observed DBP valuesin excess of 95mmHg. We did notbother to call the other 715 back andconcentrated instead on those wedeemed to have a medical problem. Ifnow, however, we compare the out-come values of the 285 subjects wehave left to the values they showed atbaseline we will find that mean DBP atoutcome is more than 2mmHg lowerthan it was at baseline. What we havejust observed is what Francis Galtoncalled regression to the mean. It is a con-sequence of the observation that on av-erage extremes do not survive:extremely tall parents tend to havechildren who are taller than averageand extremely small parents tend to

Figure 1: Galton’s height data. Two scatter plots showing the regression phenomenon



Figure 2: Simulated diastolicblood pressure for 1000 patientsmeasured on two occasions.Blue circles, ‘Normotensive,’ reddiamonds, ‘hypertensive’ onboth occasions, orange stars, ‘in-consietent.’

Figure 3: Diastolic blood pres-sure on two occasions for pa-tients observed to behypertensive at baseline.

Figure 4: Patients from Figure 2who were normotensive at base-line but hypertensive at out-come.

have children who are smaller thanaverage but in both cases the childrentend to be closer to the average thanwere their parents. If that were notthe case the distribution of heightwould have to get wider over time.Of course there can be changes insuch distributions over time and it isthe case that people are taller nowthan in Galton’s day but this is sepa-rate phenomenon in addition to theregression to the mean phenomenon.

However, regression to the mean isnot restricted to height nor even to ge-netics. It can occur anywhere whererepeated measurements are taken. Inour blood pressure example there hasbeen an apparent spontaneous im-provement in blood pressure. Appar-ently many patients who werehypertensive at baseline became nor-motensive. It is important to under-stand here that this observed‘improvement’ is a consequence ofthis stupid (but very common) way oflooking at the data. It arises becauseof the way we select the values. Whatis missing because of our selectionmethod is bad news. We can only seepatients who remain hypertensive orwho become normotensive. The pa-tients who were normotensive but be-came hypertensive are shown inFigure 4. If we had their data theywould correct the misleading picturein Figure 3 but the way we have goneabout our study means that we willnot see their outcome values.

Does it happen that scientists get

fooled by Galton’s regression to themean? All the time! Right this mo-ment all over the world in dozens ofdisciplines, scientists are foolingthemselves, either by not having acontrol group that would also showthe regression effect or if they do havea control group by concentrating onthe differences within groups be-tween outcome and baseline ratherthan the differences between groupsat outcome. It is regression to themean that is a very plausible explana-tion for the placebo effect since entryinto clinical trials is usually only byvirtue of an extreme baseline value.This does not matter as long as youcompare the treated group to theplacebo group since both groups willregress to the mean. It does meanhowever, that you have to be verycareful before claiming that any im-provement in the placebo group is dueto the healing hands of the physicianor psychological expectancy.

To prove that would require a threearm trial: an active group, a placebogroup and a group given nothing atall. Then all three groups would havethe same regression to the mean im-provement and differences betweenthe placebo and the open arm couldbe judged to be due to a true placeboeffect. Not surprisingly very few suchtrials have been run. However, analy-sis of those that have been run sug-gest that only in the area of paincontrol do we have reliable evidenceof a placebo effect. But regression to

the mean is not just limited to clinicaltrials. Did you choose dangerous in-tersections in your region for correc-tive engineering work based on theirrecord of traffic accidents? Did youfail to have a control group of similarblack spots that went untreated? Areyou going to judge efficacy of your in-tervention by comparing before andafter? Then you should know thatFrancis Galton’s regression to themean predicts that sacrificing achicken on such black spots can beshown to be effective by the methodsyou have chosen. Did you give failingstudents a remedial class and didthey improve again when tested? Areyou sure that subsequence means con-sequence? What have you over-looked?

A Victorian eccentric who died 100years ago, although no great shakesas a mathematician, made an impor-tant discovery of a phenomenon thatis so trivial that all should be capableof learning it and so deep that manyscientists spend their whole careerbeing fooled by it.

.

Figure 1: Galton’s height data. Two scatter plots showing the regression phenomenon

Stephen Senn is a professor at theUniversity of Glasgow and the au-thor of ‘Dicing with Death: Chance,Risk and Health’, published by Cam-bridge University Press.

Feature Francis Galton



Paul Erdős, unarguably one ofthe most eccentric and definitelythe most prolific mathematician ofthe 20th century wrote and co-au-thored 1,475 academic papers.Many of which monumental andall of them substantial. Moreover,this enormously large number ofmathematical works, in fact morethan any other author has everwritten, is yet outshined by theiramazing quality: ‘There is an oldsaying,’ said Erdős ‘Non numeran-tur, sed ponderantur’ They are notcounted but weighed.

Born in Budapest, Hungary onMarch 26, 1913 to Jewish parents,Anna and Lajos, Erdős’s love for

numbers started at a very youngage - due mainly to both parentsbeing high-school mathematicsteachers. At three he could multi-ply three-digit numbers in his headand calculate how many seconds aperson had lived. A year later hediscovered negative numbers. ‘Itold my mother,’ he said,‘ that ifyou take two hundred and fiftyfrom a hundred you get minus ahundred and fifty.’ Paul had twosisters, aged three and five, whodied of scarlet fever just days be-fore he was born. This naturallyhad the effect of an over protectiveupbringing and is why Erdős waskept home from school until the

P A U LERDÖSThe MathematicianWho Saw ProofsUnder Every Roof

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age of 10. Perhaps it was due to thisthat Paul tied his shoes himself for thefirst time at 11. He never cooked nordrove a car, although he was familiarenough with the theoretical part ofdriving. He didn't have a license anddepended on a network of friends -known as “Uncle Paul sitters”- to givehim lifts from one place to another. Henever boiled water for tea and he ad-mitted having buttered his first pieceof toast at the age of 21.

As a high school student the youngmathematician became a keen solverof the problems proposed monthly inthe Hungarian Mathematical andPhysical Journal for SecondarySchools, KöMaL, Középiskolai Matem-atikai és Fizikai Lapok. Eventually hepublished several articles in it aboutthe problems in elementary planegeometry. Despite the restrictions onJews entering universities in Hungary,seventeen-year-old Erdős was allowedto enter in 1930 as a winner of a na-tional examination. He studied for hisdoctorate at the University PázmányPéter in Budapest where as a univer-sity fresher he became popular in themathematical circles with an amaz-ingly simple proof of Chebyshev’s the-orem (which says that a prime canalways be found between any positiveinteger and its double). In four yearshe completed his undergraduate work

and earned a Ph.D. in mathematics.Then it was time for Erdős’s legendaryjourney in the world of numbers to fi-nally begin. Erdős structured his lifeto maximize the amount of time hehad for mathematics. He had no wifeor children, no job, no hobbies, noteven a home to tie him down. Hiswhole wardrobe fit into a small suit-

case with plenty of room left for hisradio. The only possessions that mat-tered to him were his mathematicalnotebooks. He completed ten of themby the time he died. He proved theo-rems and solved problems in England,United States, Israel, China, Australia,and 22 other countries. Inspired,Erdős moved from one university or

research centre to the next. His mottowas ‘Another roof, another proof’.Very often he would show up unan-nounced on the doorstep of a fellowmathematician declaring ‘My brain isopen!’ and stay as long as his col-league served up interesting mathe-matical challenges. In many cases hewould ask the current host aboutwhom he should visit next. His work-ing style has been humorously com-pared to traversing a linked list.

Erdős had his own peculiar mathe-matical vocabulary describing the sur-rounding world. He believed thatGod, whom he affectively called theS.F. or “Supreme Fascist,” had a trans-finite book containing the shortest butmost beautiful and elegant proofs forevery mathematical problem. The S.F.

was often accused of hiding Paul’ssocks and Hungarian passports andsometimes keeping the best mathemat-ical proofs for himself. Children wereErdős’s “epsilons”, music was just“noise”, married people were “cap-tured” and giving a mathematical lec-ture was “preaching”. When he saidsomeone had “died”, Erdős meant

that the person had stopped doingmathematics. When he said someonehad “left”, the person had actuallydied. The highest compliment hecould pay to a colleague’s work was tosay, ‘That’s straight from The Book’.

After the death of Paul’s mother,Ronald Graham, another one of theworld's best-known mathematicians,computer theorists and technology vi-sionaries, took the responsibility oflooking after the Hungarian mathe-matician. Graham, at the time a direc-tor of the Mathematical SciencesResearch Centre at AT&T Bell Labora-tories, was in charge of Erdős’s moneyand mathematical papers - previouslythe work of his mother. The articles,more than a thousand in number,were kept in a closet full of filing cabi-nets. Near the closet there was a signin capital letters saying: ‘ANYONEWHO CANNOT COPE WITH MATH-EMATICS IS NOT FULLY HUMAN.AT BEST HE IS A TOLERABLE SUB-HUMAN WHO HAS LEARNED TOWEAR SHOES, BATHE, AND NOTMAKE MESSES IN THE HOUSE.’Graham also handled Erdős’s mail cor-respondence which involved sendingout 1,500 letters a year, only few ofthem non-mathematics related. ‘I amin Australia’ a typical letter wouldbegan. ‘Tomorrow I leave for Hungary.Let k be the largest integer …’

A fascinating fact about Ron Gra-ham is that he holds a world record.He is in the Guinness Book of WorldRecords for using (in 1977) the largestnumber ever in a mathematical proof.So large there isn't even a notation forit and now known as "Graham's num-ber."

A great testimony for Erdős’s lovefor maths came when he had to havesurgery on one of his eyes to treatcataracts. He wanted, during the oper-ation, to read a maths book with theother eye. The physicians objected for

Basic FactsBorn: 26th March 1913, Budapest, Austria-Hungary

Died: 20th September 1996 (aged 83), Warsaw, Poland

Nationality: Hungarian

Alma Mater : University Pázmány Péter, Budapest

Number of Academic Papers: 1,475

Notable Awards: Wolf Prize (1983/84), AMS Cole Prize (1951)

At three he could multiply three digitnumbers in his head.

Erdös structured his life to maximize the amount oftime he had for mathematics.

“God may not play dicewith the universe, butsomething strange isgoing on with the primenumbers”

Paul Erdös 1913 - 1996

Feature Biography



practical and scientific reasons both.As a compromise the doctorsarranged for a local mathematician tostay with Paul during the surgery andtalk maths.

In 1949 Erdős had his most satisfy-ing victory over the prime numberswhen he and Atle Selberg gave “TheBook” proof of the prime number the-orem (which is a statement about thefrequency of primes at larger andlarger numbers). In 1951 John vonNeumann presented the Cole Prize toErdős for his work in prime numbertheory. The Hungarian mathematicianhelped the founding of the Graph The-ory field and attended the first Interna-tional Conference on the subject in1959. During the next three decadesErdős continued to do important workin combinatorics, partition theory, settheory, number theory, and geometry- the diversity of the fields he workedin was unusual.

What little money Erdős received instipends or lecture fees he gave awayto relatives, colleagues, students, andstrangers. He could not pass a home-less person without giving him money.In 1984 he won the prestigious Wolfprize, the most lucrative award inmathematics. He contributed most ofthe $50,000 he received to a scholar-ship he established in Israel in thename of his parents. ‘I kept only sevenhundred and twenty dollars,’ Erdőssaid, ‘and I remember someone com-menting that for me even that was alot of money to keep.’ Whenever Erdőslearned of a good cause he promptlymade a small donation. In the late1980s Erdős heard of a promising highschool student named Glen Whitneywho wanted to study mathematics atHarvard but was a little short the tu-ition. Erdős arranged to see him and,convinced of the young man's talent,lent him $1,000. He asked Whitney topay him back only when it would notcause financial strain. A decade laterRonald Graham heard from Whitneywho at last had the money to repayPaul. ‘Did Erdős expect me to pay in-terest?’ Whitney wondered. ‘Whatshould I do?’ he asked Graham. Gra-ham talked to Erdős. ‘Tell him,’ Erdőssaid, ‘to do with the $1,000 what I did.’In 2009, Glen Whitney quit his job asan Algorithm Manager at the giantquantitative hedge fund RenaissanceTechnologies on Long Island to devotehimself to one of his biggest dreams:creating a world-class interactive Mu-

seum of Mathematics. The $20-million-capital project (abbreviated MoMath)is now within $1 million of reachingits goal. Negotiations concerning a lo-cation near Madison Square Park arein their final stages and Whitney ex-pects to cut the ribbon in the spring of2012.

The mathematics of Paul Erdős isthe mathematics of beauty and insight.He is the consummate problem solver,his hallmark is the succinct and cleverargument often leading to a solutionfrom “The Book”. He loves areas ofmathematics which do not require anexcessive amount of technical knowl-edge but give scope for ingenuity and

surprise. To Erdős, the proof had toprovide insight into why the resultwas true and not just a complicated se-quence of steps which would consti-tute a formal proof yet somehow failto provide any understanding. He didnot just want to solve problems, how-ever, he wanted to solve them in an el-egant and elementary way. Part of hismathematical success stemmed fromhis willingness to ask fundamentalquestions, to ponder critically thingsthat others had taken for granted. He

also asked basic questions outsidemathematics but he never remem-bered the answers and asked the samequestions again and again.

For decades Erdős vigorouslysought out new, young collaboratorsand ended many working sessionswith the remark, ‘We'll continue to-morrow if I live.’ With 485 co-authors,Erdős collaborated with more peoplethan any other mathematician in his-tory. Because of his prolific outputfriends created the Erdős number as ahumorous tribute. Erdős alone was as-signed the Erdős number of 0 (forbeing himself), while his immediatecollaborators (those lucky 485) could

claim an Erdős number of 1, their col-laborators have Erdős number at most2, and so on. Approximately 200,000mathematicians have an assignedErdős number, and the most famouscontemporary “computer personality”with a small Erdős number is BillGates, who has an Erdős number of 4.The great unwashed who have neverwritten a mathematical paper have anErdős number of infinity.The University of Glasgow’s represen-tatives with small Erdős number are

What little money Erdös received in stipends or lecture fees he gave away to relatives, colleagues

and strangers.

Paul Erdös with Ronald Graham and his wife Fan Chung Graham



Dr. Robert Irving, who has an Erdősnumber of 2, through Michael E. Saks.Dr. David Manlove, Dr. Lorna Love,Dr. Patrick Prosser, Dr. John D. Mc-Clure, Dr. Martin W. McBride all hav-ing an Erdős number of 3, through Dr.Irving.

Jerrold Grossman at Oakland Uni-versity in Rochester, Michigan, runsan Internet site called the Erdős Num-ber Project which tracks the covetednumbers. It reveals one very interest-ing observation: take the 485 mathe-maticians with Erdős number 1 andrepresent them by 485 points on asheet of paper. Draw an edge between

any two points whenever the corre-sponding mathematicians publishedtogether. The resulting graph, whichat last count had 1,381 edges, is theCollaboration Graph. Some of Erdős'scolleagues have published papersabout the properties of the Collabora-tion Graph, treating it as if it were areal mathematical object. One of thesepapers made the observation that thegraph would have a certain very inter-esting property if two particularpoints had an edge between them. Tomake the Collaboration Graph havethat property, the two disconnectedmathematicians immediately got to-

gether, proved something trivial, andwrote up a joint paper.

Defying the conventional wisdomthat mathematics was a young man’sgame, Erdős went on proving and con-jecturing until the age of 83 succumb-ing to a heart attack only hours afterdisposing of a nettlesome problem ingeometry at a conference in Warsaw.The epitaph Paul Erdős wrote for him-self was ‘Vegre nem butulok tovabb’;Finally I am becoming stupider nomore.

The Paul Erdős Prize (formerly Mathematical Prize)is given to Hungarian mathematicians not older than 40 by the

Mathematics Department of the Hungarian Academy of Sciences. Itwas established and originally funded by Paul Erdős.

The Paul Erdős Awardis established to recognise contributions of mathematicians whichhave played a significant role in the development of mathematical

challenges at the national or international level and which have been astimulus for the enrichment of mathematics learning. Each recipient ofthe award is selected by the Executive and Advisory Committee of theWorld Federation of National Mathematics Competitions on the rec-

ommendation of the WFNMC Awards Subcommittee.

The Anna and Lajos Erdős Prize in Mathematicsis a prize given by the Israel Mathematical Union to an Israeli mathe-matician (in any field of mathematics and computer science), “withpreference to candidates up to the age of 40.” The prize was estab-

lished by Paul Erdős in 1977 in honour of his parents, and is awardedannually or biannually. The name was changed from Erdős Prize in

1996, after Erdős's death, to reflect his original wishes.

PAUL ERDÖS: PRIZES

Hristo Georgiev



Futurama is an animated sci-fi sitcomcreated by Matt Groening and devel-oped by Groening and David X. Cohen.The series follows the adventures of apizza delivery boy, Philip J. Fry, fromNew York City who after being cryogeni-cally frozen by mistake just after mid-night on January 1, 2000, finds himselfwoken up 1,000 years later and startsworking for his only living relative, Pro-fessor Hubert J. Farnsworth. The distantbrilliant-minded grand-son is the ownerof Planet Express, a delivery companybearing the slogan “Our crew is replace-able, your package isn’t!” started to helpfund his research and Fry’s job is not toodifferent from his last one – he works asa cargo delivery boy.

In the 98th episode (the 10th episode ofseries 6) of Futurama called ‘The Pris-oner of Benda,’ Professor Farnsworthand Amy, a long-term intern from MarsUniversity, in order for each of them tobe able to live their biggest dreams, in-vent a machine (Figure 5) which allowstwo users to switch minds. It is, as theyfind out later, due to the brain’s naturalimmune response, unable to do so whenthese same two people already switchedtheir minds with each other before, inother words, the machine cannot be usedtwice in a row on the same pairing ofbodies. To try to return to their actualbodies, the rest of the crew of Planet Ex-press, and a few others are involved into

the mind switching game. Later on inthe episode, an equation for enoughswitches is invented and proved, so thateverything goes back to normal. One ofthe participants discovers that by includ-ing no more than two new people in thismind swap game, everyone can alwaysreturn to their original body.

A theorem, later called the FuturamaTheorem, based on group theory wasspecifically written and proven by KenKeeler.

Problem: Suppose that k people haveswapped bodies. Label these people sothat the 1st person is in the 2nd person’sbody, the 2nd person is in 3rd person’sbody, and so on, so that the last person(the kth person) must be in the 1st person’sbody:

Figure 1.

Solution: With two additional people xand y we can get everyone back to nor-mal using the following procedure:Let <a,b> represent the transposition thatswitches the contents of a and b. By hy-pothesis π is generated by distinctswitches on [n]. Introduce two ‘new bod-ies’ <x,y> and write:

Figure 2.

For any i=1 ... k-1 let σ be the series ofswitches:

Figure 3.

Note that each switch exchanges an ele-ment of [n] with one of <x,y> so they areall distinct from the switches within [n]that generated π and also from <x,y>. Byroutine verification:

Figure 4.

i. e. σ reverts the k-cycle and leaves x andy switched (without performing <x,y>).Now let π be an arbitrary permutationon [n]. It consists of disjoint cycles andeach can be inverted as above in se-quence after which x and y can beswitched if necessary via <x,y>, as wasdesired.

1

2

2

3 1

1

1

k k

k

n

n

f

f

f

f

=+

+r d n

1

2

2

3 1

1

1

k k

k

n

n

x

x

y

y

* f

f

f

f

=+

+r d n

1

1

2

2

n

n

y

y

x

x

* f

f

=r v d n

THE THEOREM

( ,1 ,2 , ( ,1 1

,1 2 , )( ,1 1 )( ,1 )

x x x i y

y y k x y

f

f

= +

+ +

G HG H G H G H

G H G H G H G H

v



David S. Cohen has a Bachelor’s Degreein physics from Harvard University anda Master’s Degree in theoretical com-puter science from University of Califor-nia, Berkeley. His most notableacademic publication (with ManuelBlum) concerned the computer scienceproblem of pancake sorting, which wasalso the subject of an academic paper byBill Gates. In addition, Cohen is creditedas a co-author on several papers by thecomputer vision researcher Alan Yuille. Before starting professional scriptwrit-ing Cohen used to write the humour col-umn for his high school newspaper andlater was a writer for the Harvard Lam-poon Magazine. He has won EmmyAwards for his work on both The Simp-sons and Futurama.

Ken Keeler has a Ph.D. in applied math-ematics from Harvard University. Hsdoctoral thesis was on Map Representa-tions and Optimal Encoding for Image Seg-mentation. He also published a paper onShort Encodings of Planar Graphs andMaps with Jeff Westbrook.After earning his doctorate in 1990,Keeler joined the Performance AnalysisDepartment at AT&T Bell Laboratories.

He soon left Bell Labs to write for DavidLetterman and subsequently for varioussitcoms, including The Simpsons andFuturama. Keeler has won Writer'sGuild and Emmy Awards.

Jeff Westbrook majored in physics andthe history of science at Harvard Univer-sity and he received his Ph.D. in com-puter science from Princeton Universityin 1989. The title of his doctoral thesiswas Algorithms and Data Structures forDynamic Graph Algorithms. He was anAssociate Professor in the Departmentof Computer Science at Yale Universityand also worked at AT&T Labs beforewriting for Futurama. He published anarticle entitled Short Encodings of PlanarGraphs and Maps with Ken Keeler.

J. Stewart Burns graduated magna cumlaude with a bachelor's degree in math-ematics from Harvard University in1992. His senior thesis was on The Struc-ture of Group Algebras. He received hismaster's degree in mathematics fromUniversity of California, Berkeley in1993. He has worked on dynamic graphproblems, in which the goal is to main-tain connectivity and planarity informa-

tion about graphs that are growing on-line, and various problems that can beattacked with competitive analysis. Hehas studied the problem of robot nav-igation in an unfamiliar environment,with co-authors Dana Angluin andWenhong Zhu, both of Yale University.

Sarah J. Greenwald has B.S. in mathe-matics from the Union College in Sch-enectady, NY and a Ph.D. inmathematics from the University ofPennsylvania. She is an associate pro-fessor at Appalachian State Universityand a 2005 Mathematical Associationof America Alder Award winner fordistinguished teaching, including heruse of popular culture in the class-room, and the 2010 Appalachian StateUniversity Wayne D. Duncan Awardfor Excellence in Teaching in GeneralEducation. She is a moderator for bothThe Simpsons and Futurama.

Bill Odenkirk has a PhD in in organicchemistry from the University ofChicago in 1995.

Hristo Georgiev

Feature Futurama Theorem

FUTURAMA WRITERS AND STAFF

Suppose that Professor Farnsworth

and Amy knew of this theorem at the be-ginning: then they would have knownthey could switch back with the aid oftwo other people x and y, which is de-scribed below (move and mind←body

correspondence at each stage).

Initial arrangement: Prof. (mind) Amy (mind)Prof. (mind) ←Amy (body)Amy (mind) ← Prof. (body)

x (mind) ← x (body)

y (mind) ← y (body)

Step 1: Amy (mind) x (mind)Prof. (mind) ←Amy (body)Amy (mind) ← x (body)x (mind) ← Prof. (body)y (mind) ← y (body)

Step 2: Prof. (mind) y (mind)Prof. (mind) ← y (body)Amy (mind) ← x (body)

Amy (mind) ← x (body)x (mind) ← Prof. (body)y (mind) ←Amy (body)

Step 3: Prof. (mind) x (mind)Prof. (mind) ← Prof. (body)Amy (mind) ← x (body)x (mind) ← y (body)y (mind) ←Amy (body)

Step 4: Amy (mind) y (mind)Prof. (mind) ← Prof. (body)Amy (mind) ←Amy (body)x (mind) ← y (body)y (mind) ← x (body)

Step 5: x (mind) y (mind)Prof. (mind) ← Prof. (body)Amy (mind) ←Amy (body)x (mind) ← x (body)y (mind) ← y (body)

The fact that the produced theorem isan original work inspired by the plot ofthe episode is a great example of howcritical thinking can be used to solve alltypes of problems.

Figure 5. Prof. Farnsworth, Amy and their mind-switching machine


www.the-commutator.com

Here is the formal mathematical explanation of what happens in the episode along with a proposed more efficient solution.

20 [The, Commutator] March 2011

Feature Futurama Theorem



A COUNTEREXAMPLE TOTHE FULL CONVERSE OF LAGRANGE’S THEOREM

Lagrange's Theorem is a funda-mental result in finite group the-ory and states that the order of asubgroup divides the order of thegroup. Although there are partialconverses which hold in certaincircumstances (for example if wehave a finite cyclic group) the fullconverse is false. The usual coun-terexample given is the alternatinggroup which is the subgroup of

containing all even permuta-tions on 4 letters (geometrically,the rotation group of a regulartetrahedron). Although 6 divides

, the group has nosubgroup of order 6. Below wegive two proofs of this fact. Thefirst proof makes use of cosets andthe second requires the additionalconcepts of normal subgroups,quotient groups and isomor-phisms.

A4

S4

A412A4; ;=

Assume that is a subgroup of of order 6. Since

contains eight elements of order 3, it follows that any subgroup of of

order 6, and in particular, must contain at least two elements of order

3. Let be a general element of order 3. Then since i.e.

has index 2 in ,

at most two of the cosets , , and a are distinct.We must consider three cases:

(i)

(ii)

(iii)

In all three cases we have so must contain all eight elements of

order 3, contradicting our assumption that has order 6. Thereforehas no subgroup of order 6.

H A4

{ ,( )( ),( )( ),( )( ),( ),( ),

( ),( ),( ),( ),( ),( )

A e 12 34 13 24 14 23 123 132

124 142 134 143 234 243

4 =

A4

H

a : 2A H4; ;= H

A4

aH a H2 a H eH H3= =

H aH a H& !=

( )H a H a H a a H2 2 2 1& &! != =

-

[ ] [ ]aH a H aa e aH e a H H aH a H2 2 2& & &/! ! != = =

a H! H

H A4

Once again we assume that is a

subgroup of of order 6. Since

has index 2 in , it follows that

and , i.e. is a

normal subgroup of and the quo-

tient group is isomorphic to

. Since the quotient group hasorder 2, both its elements are self in-verse and therefore square to give theidentity element. Therefore, for all

we have

, i.e. for all

we have . If is a general element of order 3, then

. Hence mustcontain all eight elements of order 3,contradicting our assumption that

has order 6. We conclude that hasno subgroup of order 6.

H

A4

H A41 /A H Z4 2,

/A H4

Z2

a A4!

( )aH a H eH H2 2= =

a A4! a H2 ! a A4!

( )a a a H4 2 2 != =

H

A4

H

A4

H

H

A4

Second proof:First proof:

Colin Pratt

Mathematics Lagrange’s Theorem



What We’ve

Been Up To!

Our first event, was a

great success, we man-

aged to squeeze more

than a 100 of you lovely

people into our common

room which is probably

record number! We had

so much fun, we had to

go out and get even more

booze!

Whoever said we are not

an artistic bunch was

definitely wrong, as we

showed up in force in our

fancy dress taking Ashton

Lane by surprise, with free

shots and tons of geeky

chat, before descending on

Quids in the QMU for a

chance to show our moves

on the dance floor.

We have had some super

Maths movies on in our

common room such as:

Good Will Hunting and

21, with free juice and

popcorn whats not to

like!

Geek Chic Pub

Crawl

Cheese andWine

Night MOVIENIGHTS

Think you have what it takes to run MacSoc? If your interested email us at

[email protected] no later than the 1st of April!



The Grad Ball For those of you will be raduating

this is for you and is sure to be one special night!

Good luck to you all what ever the future may

bring as we will sadly miss you all.

Further information can be found by searching

MACSOC PHYSOC ASTROSOC

GRAD BALL 2011

on facebook.

Think you have what it takes to run MacSoc? If your interested email us at

[email protected] no later than the 1st of April!


The meaning of the word ’burokku’ inJapanese (ブロック) is ’block’, thus thename of the puzzle suggests its maingoal: to find all defined blocks, using theinitially given clues and then fill themup with appropriate numbers.

There are a few simple rules whichneed to be followed:• Each cell contains either a zero or a

positive integer.• Repetition of cell numbers within a

block is forbidden.• Repetition of cell numbers within a

row or a column is forbidden except thezeros placed outside a block.

In addition, the following list ofslightly more sophisticated rules definesthe general flavour of the puzzle:• Each circle contains a number that is

going to be referred to as a header num-ber.• The header number represents a prod-uct of the sum of all adjacent cells andthe number of connections between thecircle and its neighbour circles.• The connections between circles de-

fine and divide the grid into severalblocks.• Each cell may contain either a posi-

tive integer or zero.• Each sum of all cells within every

block must be equal.• If one of the connections defining a

block is double, then the sum of thisblock’s cells is doubled as well; if thereare two double connections, then thesum is doubled twice etc.

Basically, the main initial approach ofsolving each Burokku puzzle shouldconsist of guessing each circle's sumand its number of connections withneighbours by logical deduction basedon the given header number.

BUROKKU ブロック

AB x C

AB x C

AB x C

AB x C

D

E

E E

E

A - header numberB - sum of all four surrounding

cell numbersC - number of connectionsD - cell numberE - connections

Figure 1. A 1x1 Burokku puzzle with a legend

Hristo Georgiev

6x

8x

16x

27x

15x

20x

8x

2x

4x

63 x 2

38

4 x 2

164 x 4

279 x 3

155 x 3

205 x 4

84 x 2

21 x 2

1

41 x 4

1515 x 3

560

15 x 4

3311 x 3

7525 x 3

4214 x 3

2010 x 2

84 x 2

2010 x 2

6

126 x 2

60x

75x

42x

20x

8x

20x

15x

33x

12x

10 4

1 4

www.burokku.org

Burokku is a newly invented puzzle inspired by otherpopular formats like Sudoku, KenKen and Slitherlink.You have the unique opportunity to test your skills onthe first officially published in print 3x3 Burokku puz-zle on page 27. If you find it too easy, there are a coupleof dozen puzzle sets on the official website.

Figure 2.2. Solution to puzzle 2.1

Figure 2.1. A sample 2x2 puzzle

Figure 3.2. Solution to puzzle 3.1Figure 3.1. Another sample 2x2 puzzle

Here are some useful hints you mightwant to consider when working on aBurokku puzzle solution:• The first multiplier in each corner is

the actual number occupying the cornercell, because it turns out to be the onlycell taken into account when making thesum of all adjacent to the circle cells.• There are no circles with only one

connection, so all header numbers areformed by multiplying the adjacent cellssum by a number of connections ≥ 2.• If the header number in one of the cor-ners is 0, the cells sum of the same circleis always 0, while the number of connec-tions can be either 0 or a positive integer.

www.the-commutator.com24 [The, Commutator] March 2011


le if I add that x and y are the two positive integers thatbegin the longest possible generalised Fibonacci chain end-ing in a term of 1,000,000.

Problem 3: The blank column

A secretary, eager to try out a new typewriter, thought of asentence shorter than one typed line, set the controls for thetwo margins and then, starting at the left and near the topof a sheet of paper, proceeded to type the sentence repeat-edly. She typed the sentence exactly the same way each time,with a period at the end followed by the usual two spaces.She did not, however, hyphenate any words at the end of aline: When she saw that the next word (including whateverpunctuation marks may have followed it) would not fit theremaining space on a line, she shifted to the next line. Eachline, therefore, started flush at the left with a word of hersentence. She finished the page after typing 50 single-spacedlines. Is there sure to be at least one perfectly straight col-umn of blank spaces on the sheet, between the margins, run-ning all the way from top to bottom?

Problem 4: A chess problem

Your king is on a corner cellof a chessboard and your op-ponent’s knight is on the cor-ner cell diagonally opposite,as shown in Figure 1. Noother pieces are on the board.The knight moves first. Forhow many moves can youavoid being checked?

Problem 5: Who is telling the truth?

A boy and a girl are sitting on the front steps of their school.‘I’m a boy,’ said the one with black hair. ‘I’m a girl,’ said the one with red hair. If at least one of them is lying, who is which?

Problem 6: Arrange the superqueen

A ‘superqueen’ is a chess queen that alsomoves like a knight. Place four su-perqueens on a five-by-five board (Fig-ure 2) so that no piece attacks another. Ifyou solve this, try arranging 10 su-perqueens on a 10-by-10 board so thatno piece attacks another. Both solutionsare unique if rotations and reflections

are ignored.

Problem 1: The amazing code

Dr. Zeta is a scientist from Helix, a galaxy in another space-time dimension. One day Dr. Zeta visited the Earth to gatherinformation about humans. His host was an American sci-entist named Herman.

Herman: Why don't you take back a set of the EncyclopaediaBritannica? It's a great summary of all our knowledge. Dr. Zeta: Splendid idea, Herman. Unfortunately, I can'tcarry anything with that much mass.However, I can encode the entire encyclopaedia on thismetal rod. One mark on the rod will do the trick. Herman: Are you joking? How can one little mark carry somuch information?Dr. Zeta: Elementary, my dear Herman. There are less thana thousand different letters and symbols in your encyclopae-dia. I will assign a number from 1 through 999 to each letteror symbol, adding zeros on the left if needed so that eachnumber used will have three digits.Herman: I don't understand. How would you code the wordcat?Dr. Zeta: It's simple. We use the sort of code I just showedyou. Cat might be coded 003001020.

Using his powerful pocket computer, Dr. Zeta scanned theencyclopaedia quickly, translating its entire content into onegigantic number. By putting a decimal point in front of thenumber, he made it a decimal fraction. Dr. Zeta then placeda mark on his rod, dividing it accurately into lengths a andb so that the fraction a/b was equivalent to the decimal frac-tion of his code.

Dr. Zeta: When I get back to my planet, one of our comput-ers will measure a and b exactly, then compute the fractiona/b. This decimal fraction will be decoded, and the computerwill print your encyclopaedia.

Would this approach work in practice?

Problem 2: The twenty bank deposits

A Texas oilman who as an amateur number theorist openeda new bank account by depositing a certain integral numberof dollars, which we shall call x. His second deposit, y, alsowas an integral number of dollars. Thereafter each depositwas the sum of the two previous deposits. (In other words,his deposits formed a generalised Fibonacci series.) His 20th

deposit was exactly a million dollars. What are the values ofx and y, his first two deposits? The problem reduces to a Dio-phantine equation that is somewhat tedious to solve, but adelightful shortcut using the golden ratio becomes availab-

PUZZLES FROM MARTIN GARDNER’S COLLECTION

Figure 1.

Figure 2.

Martin Gardner was an American mathematics and science writer born on Oct 21, 1914 in Tulsa, OK, USA and died onMay 22, 2010 in Norman, OK, USA.He was specialising in recreational and popular mathematics, and science journalism,particularly through his ‘Mathematical Games’ column, which ran from 1956 to 1981, in Scientific American. Gardner alsowrote a ‘puzzle’ story column for Asimov's Science Fiction magazine in the late 1970s and early 1980s.

March 2010 [The, Commutator] 25www.the-commutator.com

Games & Puzzles


Problem 7: Guess the four digits

A B C D

D C B A

? ? ? ?

1 2 3 0 0

ABCD are four consecutive digits in increasing order. DCBAare the same four in decreasing order. The four dots repre-sent the same four digits in an unknown order. If the sum is12,300, what number is represented by the four questionmarks

Problem 8: The coloured socks

Ten red socks and ten blue socks are all mixed up in adresser drawer. The twenty socks are exacty alike except fortheir colour. The room is in pitch darkness and you wanttwo matching socks. What is the smallest number of socksyou must take out of the drawer in order to be certain thatyou have a pair that match?

Problem 9: Weighty problem

If a basketball weighs 10.5 ounces plus half its own weight,how much does it weigh?

Problem 10: No change

‘Givee me change for a dollar, please,’ said the customer.‘I’m sorry,’ said Miss Jones, the cashier, after searchingthrough the cash register,‘but I can't do it with the coins Ihave here.’‘Can you change a half dollar then?’Miss Jones shook her head. In fact, she said, she couldn'teven make change for a quarter, dime, or nickel! ‘Do you have any coins at all?’ asked the customer.‘Oh yes,’ said Miss Jones. ‘I have $1.15 in coins.’

Exactly what coins were in the cash register?

Problem 11: The bicycles and the fly

Two boys on bicycles, 20 miles apart, began racing directlytoward each other. The instant they started, a fly on the han-dle bar of one bicycle started flying straight toward the othercyclist. As soon as it reached the other handle bar it turnedand started back. The fly flew back and forth in this way,from handle bar to handle bar, until the two bicycles met.If each bicycle had a constant speed of 10 miles an hour, andthe fly flew at a constant speed of 15 miles an hour, how fardid the fly fly?

Problem 12: Corner to corner

Given the dimensions (in inches) shown in the illustration,how quickly can you compute the length of the rectangle’sdiagonal that runs from comer A to corner B (Figure 3)?

Problem 13: The threepennies

Joe: ‘I’m going to tossthree pennies in the air.If they all fall heads, I’llgive you a dime. If theyall fall tails, I’ll give youa dime. But if they fallany other way, you haveto give me a nickel.’Jim: ‘Let me think aboutthis a minute. At least

two coins will have: to be alike because if two don't match,the third will have to match one of the other two. And iftwo are alike,then the third penny will either match the other two ornot match them. The chances are even that the thirdpenny will or won't match. Therefore the chances mustbe even that the three pennies will be all alike or not allalike. But Joe is betting a dime against my nickel that theywon’t be all alike, so the bet should be in my favor. Okay,Joe, I’ll take the bet!’

Was it wise for Jim to accept the bet?

Problem 14: The two tribes

An island is inhabited by two tribes. Members of one tribealways tell the truth, members of the other always lie. A missionary met two of these natives, one tall, the othershort. ‘Are you a truth-teller?’ he asked the taller one. ‘Oopf,’ the tall native answered. The missionary recognised this as a native word meaningeither yes or no, but he couldn't recall which. The shortnative spoke English, so the missionary asked him whathis companion had said. ‘He say “yes,” ’ replied the short native, ‘but him big liar!’

What tribe did each native belong to?

Problem 15: Series of numbers

What is the basis for the order in which these ten digitshave been arranged: 8-5-4-9-1-7-6-3-2-0

Figure 3. Find AB.

You can find the solutionsto all problems and puz-zles in the Solutions sec-tion of our website orthrough scanning the QRcode on the right withyour mobile device whichwill lead you directly tothe URL.



Choose one of the white pieces. A valid move is one that capturesany black piece acquiring the properties of whatever it has just cap-tured. White pieces cannot be captured. Only the first white pieceyou choose is removed from the chessboard. The goal is to capturethe black king in as few moves as possible.

Chess Puzzle

Fill in all missing numbers and lines, refering to the rules on page24. The sums of all defined regions must be equal.

Burokku

25

34

38

39

26 32 34 44 46

Fill in the numbers 1through 16, so that thenumbers on the right andon the bottom correspondexactly to each of the col-umn, row and diagonalsums.

Microsums

a b c d e f g h

a b c d e f g h

8

7

6

5

4

3

2

1

8

7

6

5

4

3

2

1

1 5 3

2 4 1

7 8 4

There is a digit in each of the cells.You may choose to either add an-other digit to the front or to theback and make a two-digit num-ber or leave the one-digit numberas it is. The goal is for the sum ofeach row and column to be ex-actly 100.

One Hundred

Fill in each squarewith an intergerfrom 1 to 6 indicat-ing the height ofeach building. Nonumber may appeartwice in any row orcolumn. The num-bers along the edgeof the puzzle indi-cate the number ofbuildings whichyou would see fromthat direction ifthere was a series of skyscrapers with heights equal the entries inthat row or column.

4 1 2

3

5

3

2 2

3

2 1 2

Skyscrapers

16x

30x

42x

78x

88x

84x

96x

32x

30x

20x

18x

12x

36x

30x

12x

6x

Moves:1.___________________________________________________2.___________________________________________________3.___________________________________________________4.____________________________________________________5.____________________________________________________6.___________________________________________________7.___________________________________________________8.___________________________________________________9.___________________________________________________10.__________________________________________________11.__________________________________________________12.__________________________________________________13.__________________________________________________14.__________________________________________________15.__________________________________________________16.__________________________________________________

Chess Puzzle Solution

Write down your solution in the spaces provided on the bottom rightof the page. Note that your solution does not have to be of exactly 16moves as the number of the spaces.

March 2010 [The, Commutator] 27www.the-commutator.com

Games & Puzzles


Arrange the weights so that the planetary system is in equilibrium,with equal sums of 42. The following criteria must be met:1. The four weights along each radius must add up to the sum.2. The four weights around each orbit (the solid lines) must add upto the sum.3. The four weights along the four spirals (the dashed lines) mustadd up to the sum.4. You may use only positive integers for weights and each of themmay not be used more than once, in other words, repetition of num-bers is not allowed.

Planetarium

0 3 8 6 5 D 7

D 1 3 E

2 9 A 3

5 A 2 C D B 1

4 0 3 6 8 2 9 B F D C

B 4 0 6 1

F E 3 A 0 7 2

6 2 3 4 B 5

B 6 7 1 A C

C F 3 2 D B 0 A

D A 1 6 7 8 C 3 2

4 0 8 C B A 6

9 8 6 3 D E B

F D 7 A B 6 2 8

0 D 4 2 8 E 6

1 F 4 9 C

Fill the 16×16 grid with hexadecimal digits so that each column,each row, and each of the sixteen 4×4 sub-grids contains all of thedigits 0,1,2,3,4,5,6,7,8,9A,B,C,D,E,F.

Hex Sudoku

Fill in the circles with operations and the squares with distinct non-negative integers as operands, so that the arithmetic tree bears theconstant 48. The result of each operation can be written next to thecorresponding circle.You may use any of the following operations as many times as youneed:•binary: +, -, ×, /, ^ (xy), Mod•unary: !, √, Inv (x-1), Sqr (x2), log10x, sin(sinxo), cos (cosxo), tan (tanxo)

Arithmetic Tree

Fill in with the numbers 1 through 5, so that each column, eachrow and each stream contains all the numbers exactly once.

Strimko

4

3

1

5

1024

+×

×

^

3

6

×

Games and Puzzes pages are prepared by Hristo Georgiev.All the puzzle sets are designed especially for the presentissue, while Burokku® ブロック is a newly invented puzzle.

Games & Puzzles


48


[the, commutator] vol 2 issue 1 edition 1

Documents

commutator volume

new rules

institute of mathematics

greatest mathematicians

practising mathematicians

new questions

particular set of rules

university of glasgow