[The, Commutator] vol.2 issue 1 by [The, Commutator]

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

Contents 3

Letter from the editor

The Problem of Apollonius

Did you know...

Francis Galton

Paul Erdös

The Futurama Theorem

The Full Converse to Lagrange’s Theorem

MacSoc

Games and Puzzles

and the high school student who provided the 5th ever unique solution

and Regression to the mean

the mathematician who saw proofs under every roof

explained

a counterexample

activities and events

Editor in Chief Hristo Georgiev Designer Jaspal Puri

Finance Manager Ross McKinstry

Special thanks to Carl 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

editor@the-commutator.com University of Glasgow School of Mathematics and Statistics University Gardens, University of Glasgow Glasgow, G12 8QA http://www.gla.ac.uk/mathematicsstatistics tel: +44 (0) 141 330 5176 email: maths-stats-enquiries:glasgow.ac.uk

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Letter from the editor.. ank you First of all, I want to thank everybody involved in the project, especially Jaspal Puri, who is the main reason behind the present issue’s great look and design. I would also like to thank the University of Glasgow School of Mathematics and Statistics, the MacLaurin Society and the Institute of Mathematics and its Applications. e IMA was founded in 1964 and now has over 5000 members. It is the professional and learned society for qualified and practising mathematicians. It supports the advancement of mathematical knowledge and its applications and promotes and enhances mathematical culture in the 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, collaborative and smaller games of ‘what if ’ played by mathematicians who make up sets of axioms and then explore the consequences of following those axioms. Mathematicians are interested not only in what happens when you adopt a particular set of rules but also in what happens when you change the rules. For example, one game is called ‘arithmetic.’ We define what it means to be a number, i.e. the first number is zero, and if we have any number we can define another number by adding 1 to it. Of course, to do this, we also have to define what we mean by ‘addition’. Now, having made up these rules, we start playing with them to see what happens. One thing that happens is that we notice certain patterns and we use these patterns to add new rules. We also 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. But in fact it was all discovered by mathematicians playing around making up questions, trying to answer them and then sharing their answers with other mathematicians who would use the answer to make up new questions, and so on. Of course, over time, the questions get more complicated and you have to understand a lot of what has happened in the game already to even understand what is being asked. And at a certain point the questions 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), and rules 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 mathematicians of the 20th century, Paul Erdős. Another interesting figure presented on our pages is the great Victorian eccentric and scientist Francis Galton. Some of the most fascinating mathematical facts and the newest solution to the Apollonius’ problem are covered in our Facts and News sections respectively. We present a Counterexample to the Full Converse of Lagrange's eorem and an idea of what it is to be a member of 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 at solving maths and logic problems. We look forward to receiving your solutions, suggestions, or comments on our editorial e-mail address. You can also get it touch with us through the Contact us form on our website.

Hristo Georgiev

“Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction.”

Albert Einstein

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News The Apollonius Problem

THE APOLLONIUS PROBLEM THE HIGH SCHOOL STUDENT WHO GAVE THE LATEST SOLUTION The Problem of Apollonius is to construct a circle tangent to three given circles using straightedge and compass. The problem usually has eight diﬀerent solution circles that exist that are tangent to the given three circles in a plane. The given circles must not be tangent to each other, overlapping, or contained within one another for all eight solutions to exist. The last known solution satisfying all requirements dates back to the nineteenth century and was done by Joseph Diaz Gergonne. There are several methods involving diﬀerent coordinate systems but they are not straightedge and compass constructions.

18-year-old Radko Kotev, a student at the National High School of Natural Sciences and Mathematics in Soﬁa, Bulgaria, presented his solution as a project at the European Union Contest for Young Scientists (EUCYS) in Lisbon, Portugal. Competing along with 85 other student projects from across the world, Kotev was awarded a prize of €7,000 and a one-week visit to the Institut Laue-Langevin (ILL) in Grenoble where he presented his project in one of the world’s most modern research centres.

his teacher. The solution appeared to Kotev on an early morning. ‘It was so strange,’ he says. ‘I went to bed that night, thinking about the problem, and in the morning I said to myself, why not try this way. I took a sheet of paper, drew up what was needed and the solution came very easily.’ He ﬁrst showed the solution to his

Kotev is the fifth mathematician to find a valid solution to the enigma metric problem was set by Euclid and was further developed by his disciple Apollonius of Perga (ca. 262 BC – ca. 190 BC) - known as ‘The Great Geometer.’ The original solution, posed and solved in his work ‘Tangencies,’ was lost forever in the fire that destroyed the Library of Alexandria. Kotev is the fifth mathematician to find a valid solution to the enigma. ‘Now we have powerful computers at hand,’ he says. ‘I don’t think I would have been able to achieve the same results without one.’ The young mathematician spent two years working on the project. All the way though it, he was assisted by

The four famous mathematicians who provided straightedge and compass solutions to the Apollonius’ problem are François Viète (1600), Jean-Victor Poncelet (1811), Joseph Diaz Gergonne (1814), and Julius Petersen (1879). In the 16th century Adriaan van Roomen solved the problem using intersecting hyperbolas but his solution did not use only straightedge and compass constructions. The method of van Roomen was simpliﬁed later by Isaac Newton, who showed that the Apollonius’ problem is equivalent to ﬁnding a position from the differences of its distance to three known points, which has application in navi-

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The Apollonius’ problem has stimulated much further work. Generalisations to three dimensions – constructing a sphere tangent to four given spheres – and beyond have been studied. The conﬁguration of three mutually tangent circles has received particular attention. René Descartes gave an algebraic

He hopes to start his undergraduate degree in applied mathematics at the University of Glasgow in September 2011 parents who are both mathematicians and then to his teacher. This way Radko Kotev managed to solve a problem that has challenged the humankind for more than 2000 years. ‘I have never viewed mathematics as an obligation,’ he says. ‘I have never thought I should solve problems all the time and write homework endlessly. I have always solved problems for the fun of it. In fact, it is my hobby.’ Radko hopes to start his undergraduate degree in applied mathematics at the University of Glasgow in September 2011.

The notorious puzzle has several variants. Originally the complex geo-

gation and positioning systems like the GPS.

Figure1. Radko Kotev’s solution to the Apollonius problem. formula relating the radii of the solution circles and the given circles, now known as the Descartes’ theorem. Solving Apollonius’ problem iteratively in this case leads to Apollonian gasket which is one of the earliest fractals described in print.

Hristo Georgiev www.the-commutator.com

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Facts Did You Know...

DID YOU KNOW... Largest Known Primes The largest known primes are of the form (2m - 1). The reason is that there exist efficient ways to test whether such numbers are prime. Primes of this type are called a Mersenne primes. As of Sept 2010, the largest known primes 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 digits long. These primes were all discovered in the last 3 years. The search for large primes has accelerated with the help of several hundred people across the internet in a project called GIMPS (the Great Internet Mersenne Prime Search). Pi Day The number π in some form or another has been around for thousands of years. Ancient Babylonians, Egyptians, Greeks and Indian mathematicians estimated pi to varying degrees of accuracy. Greek mathematician Archimedes (287-212 B.C.E.) calculated π to 3.1419 in his On the Measurement of a Circle. In 480 C.E., Chinese scholar Zu Chongzhi used a previous scholar's algorithm to calculate the value of pi as 3.1415926 - the most accurate approximation of pi to be known for the next 900 years. The symbol for pi was first used in 1706 by William Jones but was popularised later by the Swiss mathematician and physicist Leonhard Euler in 1737. Pi Day is a holiday commemorating the mathematical constant π (pi). Pi Day is celebrated on March 14 (or 3/14 in month/day date format), since 3, 1 and 4 are the three most significant digits of π in the decimal form. In 2009, the United States House of Representatives supported the designation of Pi Day. March 14 is also Albert Einstein’s birthday. Pi Approximation Day is held on July 22 (or 22/7 in day/month date format), since the fraction 22/7 is a common approximation of π. Larry Shaw created Pi Day in 1989. The holiday was celebrated at the San Francisco Exploratorium, which nowadays continues to hold Pi Day celebra-

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tions with staﬀ and public marching around one of its circular spaces then consuming fruit pies. On March 12, 2009, the U.S. House of Representatives passed a non-binding resolution (HRES 224) recognizing

March 14, 2009 as National Pi Day. Sometimes the so-called Pi Minute is also commemorated. This one occurs twice on March 14 at 1:59 a.m. and 1:59 p.m. If π is truncated to seven decimal places, it becomes 3.1415926, making the Pi Second occur on March 14 at 1:59:26 a.m. (or 1:59:26 p.m.) If a 24-hour clock is used, the Pi Second occurs just once yearly on March 14 at 01:59:26. In 2015, Pi Day will reflect five digits of π (3.1415) as 3/14/15 in month/day/year date format. There will also be a Pi Second accurate to 10 digits (3.141592654) at 9:26:54 in that year's Pi Day. MIT loves Pi! Each year the Massachusetts Institute of Technology releases its admission decisions at 1:59pm on Pi Day. Citing the Digits of Pi Akira Haraguchi (原口證), born in 1946, is a retired Japanese engineer currently working as a mental health counsellor and business consultant. He set the current world record, 100,000 digits in 16 hours, starting at 9 a.m (16:28 GMT) on October 3, 20 stopping with digit number 100,000 at 1:28 a.m. on October 4, 2006. The event was filmed in a public hall in Kisarazu, east of Tokyo where he had fiveminute breaks every two hours to eat onigiri rice balls to keep up his energy levels. Even his trips to the toilet were filmed to prove that the exercise was legitimate. Haraguchi’s previous world record (83,431) was performed from July 1 2005 to July 2 2005. Despite Haraguchi’s efforts and detailed documentation, the Guinness World Records have not yet accepted any of his records because of the rule allowing the time between two num-

bers to be no more than 15 seconds. The Guinness-recognised record for remembered digits of π is 67,890 digits, held by Lu Chao (吕超), a 28-yearold graduate student from China who spent roughly one year memorising the digits. The record was set in 2006 when he was 24. It took him 24 hours and 4 seconds to recite to the 67,890th decimal place of π without an error with no lunch time and no toilet breaks. Lu Chao said he had got 100,000 digits of pi, and had been going 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 lifelong quest for eternal truth. Since childhood, he has always wondered why some people - especially those with physical and mental disabilities - suffer. He consulted religion and philosophy books for answers, but only in vain. Then he turned to nature and realised, he said, that nothing in nature be it leaves, trees or mountain scenery - is linear or square. ‘I realised that nature is not made of straight lines … And I realised that all things in the universe … rotate. Rotation became a key concept for me.’ So when he learned that Pi is an endless series of numbers with no pattern or repetition, it made perfect sense to him to take it as a symbol of life, he said. He uses a system he developed, which assigns the syllabic Japanese symbols Kana, to numbers, allowing the memorisation of Pi as a collection of stories. For example 0 can be substituted by o, ra, ri, ru, re, ro, wo, on or oh, and 1 can be substituted by a, i, u, e, hi, bi, pi, an, ah, hy, hyan, bya or byan. The same is done for each number from 2 through 9. Combining these characters, he has created a myriad of stories and poems, including a story about the legendary 12th-century hero Minamoto no Yoshitsune and his sidekick Benkei, who was a Buddhist monk. When he recites digits, he explains that he ‘simultaneously interprets’ his linguistic creations back into numbers.

Hristo Georgiev March 2010 [The, Commutator] 5

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F R A N C I S G A L T O N AND REGRESSION TO THE MEAN Stephen Senn

This year marks the centenary of the death of the great Victorian eccentric and scientist Francis Galton (1822-1911), a cousin of Charles Darwin, who is an important but also a curious ﬁgure in the history of statistics but also in that of many other sciences, in particular psychology and genetics. Galton was born into a wealthy family, the youngest of nine children and appears to have been a precocious child, in support of which, his biographer Forrest cites the following letter to one of his sisters: My dear Adèle, I am four years old and can read any English book. I can say all the Latin Substantives and adjectives and active words besides 52 lines of Latin poetry. I can cast up any sum in addition and multiply by 2,3,4,5,6,7,8,(9),10,(11) I can also say the pence table, I read French a little and I know the clock. Francis Galton, February-15-1827 Apparently Galton was also a truthful child, since, having written the letter, he had realized that what he had claimed about the numbers 9 and 11 was not quite true and had tried to obliterate them. And before you get too impressed, his birthday was 16 February so he was very nearly ﬁve! Galton’s later progress in education was not quite so smooth. He dabbled in medicine and

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Feature Francis Galton

then read mathematics at Cambridge but eventually had to take a pass degree. In fact he subsequently freely acknowledged his weakness in formal mathematics but this weakness was compensated by an exceptional ability to understand the meaning of data. Galton was a brilliant natural statistician. Many words in our statistical lexicon were coined by Galton. For example, correlation and deviate are due to him, as is regression and he was the originator of terms and concepts like quartile, decile and percentile, and of the use of median as the mid point of a distribution. Of course, words have a way of developing a life of their own, so that, unfortunately decile is increasingly being applied to mean tenth. There are, pretty obviously, ten tenths of a distribution but there are, slightly less obviously, only nine deciles, since the deciles are the boundaries between the tenths. To use decile to mean tenth, as for example speaking of students ‘in the top decile’ (according to their examination marks) is not only pompous but wrong and means that yet another word will eventually have to be invented to perform the function that Galton created decile to fulfil. In fact, I am tempted to say that to use decile to mean tenth is the mark of an imbedecile but I digress… To take another example, we no longer use the term regression in quite the way Galton did, now usually reserving it for the fitting of linear relationships. In Galton’s usage regression was a phenomenon of bivariate distributions and something he discovered through his studies of heritability. However the use of regression in Galton’s sense does survive in the phrase regression to the mean – a powerful phenomenon it is the purpose of this article to explain. Galton first noticed it in connection with his genetic study of the size of seeds but it is perhaps his 1886 study of human height that really caught the Victorian imagination. Galton had compared the height of adult children to the

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heights of their parents. For this purpose he had multiplied the heights of female children by 1.08. For as he put it In every case I transmuted the female statures to their corresponding male equivalents and used them in their transmuted form, so that no objection grounded on the sexual difference of stature need be raised when I speak of averages. It is interesting to note, by the by, that he also considered whether 1.07 or 1.09 might not be a better factor to use but remarked The final result is not of a kind to be affected by these minute details, for it happened that, owing to a mistaken direction, the computer to whom I first entrusted the figures used a somewhat different factor, yet the result came out closely the same. The year being 1886 the computer in question was, of course, a human and not an electronic assistant! The more interesting point, however, is that Galton provides an early example of what is now recognised as a general scientific phenomenon. Scientists never seem to fail the robustness checks they report. It is interesting to speculate why. Galton’s data consisted of 928 adult children and 205 ‘parentages’ that is to say father and wife couples. (The mean number of children per couple was thus just over 4.5.) He represented the height of parents using a single statistic ‘the mid-parent,’ this being the mean of the height of the father and of his wife’s height multiplied by 1.08. Of course, as previously noted, for the female children the heights were also multiplied by 1.08. For the male children they were unadjusted. Figure 1 is a more modern graphical representation of Galton’s data. (The data have been taken from the very useful website provided by the University of Alabama in Huntsville.) Galton had grouped his results by intervals of one inch and in consequence, if a given

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Figure 1: Galton’s height data. Two scatter plots showing the regression phenomenon child’s recorded height were plotted amount of ‘jitter’ in either dimension against its recorded ‘mid-parent’ to separate the points, which are height, many points would be over- shown in blue. The data are plotted plotted. I have added a small amount two ways: child against mid-parent on of ‘jitter’ in either dimension to sepa- the left and mid-parent against child rate the points, which are shown in on the right. The thin solid black diagblue. The data are plotted two ways: onal line in each case is the line of child against mid-parent on the left equality. If a point lies on this line then and mid-parent against child on the child and mid-parent were identical in right. The thin solid black diagonal height. Also shown in red in each case line in each case is the line of equality. are two different approaches one If a point lies on this line then child might use to predicting ‘output’ from and mid-parent were identical in ‘input’. The dashed line is the least height. Also shown in red in each case squares fit, what we now (thanks to are two different approaches one Galton) call a regression line. The thick might use to predicting ‘output’ from black line is a more local fit, in fact a ‘input’. The dashed line is the least so-called LOWESS (or locally squares fit, what we now (thanks to weighted scatterplot smoothing) line. Galton) call a regression line. The thick The point about either of these two apblack line is a more local fit, in fact a proaches irrespective of whether we preso-called LOWESS (or locally dict child from mid-parent or vice versa is weighted scatterplot smoothing) line. that the line that is produced is less The point about either of these two ap- steep than the line of exact equality. proaches irrespective of whether we The consequence is that we may expredict child from mid-parent or vice pect that an adult child is closer to avversa is that the line that is produced erage heights than its parents but also is less steep than the line of exact paradoxically, that parents are closer equality. The consequence is that we to average height than is their child. may expect that an adult child is closer This particular point is both deep and to average heights than its parents but trivial. It is deep because the first time also paradoxically, that parents are that students encounter it (I can still recloser to average height than is their member my own reaction) they aschild.Figure 1 is a more modern graph- sume that it is wrong: its truth is well ical representation of Galton’s data. hidden. Once understood, however, it (The data have been taken from the becomes so obvious that one is very useful website provided by the amazed at how regularly it is overUniversity of Alabama in Huntsville.) looked. So, let’s leave Francis Galton for the Galton had grouped his results by intervals of one inch and in consequence, moment and consider another examif a given child’s recorded height were ple, this time a simulated one. Figure plotted against its recorded ‘mid-par- 2 shows simulated values in diastolic ent’ height, many points would be blood pressure (DBP) for a group of over-plotted. I have added a small 1000 individuals measured on two oc-

8 [The, Commutator] March 2011

casions: at baseline and at outcome(blue circles) or inconsistent, tentatively tensive perhaps(orange stars). The distributions at outcome and baseline are very similar with, means close to 90mmHg and a spread that can be defined by that Galtonian statistic the inter-quartile range as being close to 11mmHg on either occasion. In other words, what the picture shows is a population that all in all has not changed over time although, since the correlation (to use another Galtonian term) is just under 0.8 and therefore less than one, there is, of course, variability over time for many individuals. However, in the setting of many clinical trials, Figure 2 is not a figure we would see for the simple reason that we would not follow up individuals who were observed to be normotensive at baseline. Instead what we would see is the picture given in Figure 3. Of the 1000 subjects seen at baseline 285 had observed DBP values in excess of 95mmHg. We did not bother to call the other 715 back and concentrated instead on those we deemed to have a medical problem. If now, however, we compare the outcome values of the 285 subjects we have left to the values they showed at baseline we will find that mean DBP at outcome is more than 2mmHg lower than it was at baseline. What we have just observed is what Francis Galton called regression to the mean. It is a consequence of the observation that on average extremes do not survive: extremely tall parents tend to have children who are taller than average and extremely small parents tend to

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Feature Francis Galton

Figure 2: Simulated diastolic blood pressure for 1000 patients measured on two occasions. Blue circles, ‘Normotensive,’ red diamonds, ‘hypertensive’ on both occasions, orange stars, ‘inconsietent.’ have children who are smaller than average but in both cases the children tend to be closer to the average than were their parents. If that were not the case the distribution of height would have to get wider over time. Of course there can be changes in such distributions over time and it is the case that people are taller now than in Galton’s day but this is separate phenomenon in addition to the regression to the mean phenomenon. However, regression to the mean is not restricted to height nor even to genetics. It can occur anywhere where repeated measurements are taken. In our blood pressure example there has been an apparent spontaneous improvement in blood pressure. Apparently many patients who were hypertensive at baseline became normotensive. It is important to understand here that this observed ‘improvement’ is a consequence of this stupid (but very common) way of looking at the data. It arises because of the way we select the values. What is missing because of our selection method is bad news. We can only see patients who remain hypertensive or who become normotensive. The patients who were normotensive but became hypertensive are shown in Figure 4. If we had their data they would correct the misleading picture in Figure 3 but the way we have gone about our study means that we will not see their outcome values. Does it happen that scientists get

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Figure 3: Diastolic blood pressure on two occasions for patients observed to be hypertensive at baseline.

fooled by Galton’s regression to the mean? All the time! Right this moment all over the world in dozens of disciplines, scientists are fooling themselves, either by not having a control group that would also show the regression effect or if they do have a control group by concentrating on the differences within groups between outcome and baseline rather than the differences between groups at outcome. It is regression to the mean that is a very plausible explanation for the placebo effect since entry into clinical trials is usually only by virtue of an extreme baseline value. This does not matter as long as you compare the treated group to the placebo group since both groups will regress to the mean. It does mean however, that you have to be very careful before claiming that any improvement in the placebo group is due to the healing hands of the physician or psychological expectancy. To prove that would require a three arm trial: an active group, a placebo group and a group given nothing at all. Then all three groups would have the same regression to the mean improvement and differences between the placebo and the open arm could be judged to be due to a true placebo effect. Not surprisingly very few such trials have been run. However, analysis of those that have been run suggest that only in the area of pain control do we have reliable evidence of a placebo effect. But regression to

Figure 4: Patients from Figure 2 who were normotensive at baseline but hypertensive at outcome.

the mean is not just limited to clinical trials. Did you choose dangerous intersections in your region for corrective engineering work based on their record of traffic accidents? Did you fail to have a control group of similar black spots that went untreated? Are you going to judge efficacy of your intervention by comparing before and after? Then you should know that Francis Galton’s regression to the mean predicts that sacrificing a chicken on such black spots can be shown to be effective by the methods you have chosen. Did you give failing students a remedial class and did they improve again when tested? Are you sure that subsequence means consequence? What have you overlooked? A Victorian eccentric who died 100 years ago, although no great shakes as a mathematician, made an important discovery of a phenomenon that is so trivial that all should be capable of learning it and so deep that many scientists spend their whole career being fooled by it.

Stephen Senn is a professor at the . University of Glasgow and the author of ‘Dicing with Death: Chance, Risk and Health’, published by Cambridge University Press.

March 2010 [The, Commutator]

Photograph courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach

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PAUL ERDÖS The Mathematician Who Saw Proofs Under Every Roof 10 [The, Commutor] March 2010

Paul Erdős, unarguably one of the most eccentric and deﬁnitely the most proliﬁc mathematician of the 20th century wrote and co-authored 1,475 academic papers. Many of which monumental and all of them substantial. Moreover, this enormously large number of mathematical works, in fact more than any other author has ever written, is yet outshined by their amazing quality: ‘There is an old saying,’ said Erdős ‘Non numerantur, sed ponderantur’ They are not counted but weighed. Born in Budapest, Hungary on March 26, 1913 to Jewish parents, Anna and Lajos, Erdős’s love for

numbers started at a very young age - due mainly to both parents being high-school mathematics teachers. At three he could multiply three-digit numbers in his head and calculate how many seconds a person had lived. A year later he discovered negative numbers. ‘I told my mother,’ he said,‘ that if you take two hundred and fifty from a hundred you get minus a hundred and fifty.’ Paul had two sisters, aged three and five, who died of scarlet fever just days before he was born. This naturally had the effect of an over protective upbringing and is why Erdős was kept home from school until the

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Feature Biography

Basic Facts Born: 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

“God may not play dice with the universe, but something strange is going on with the prime numbers” Paul Erdös 1913 - 1996

Notable Awards: Wolf Prize (1983/84), AMS Cole Prize (1951) age of 10. Perhaps it was due to this that Paul tied his shoes himself for the ﬁrst time at 11. He never cooked nor drove a car, although he was familiar enough with the theoretical part of driving. He didn't have a license and depended on a network of friends known as “Uncle Paul sitters”- to give him lifts from one place to another. He never boiled water for tea and he admitted having buttered his ﬁrst piece of toast at the age of 21. As a high school student the young mathematician became a keen solver of the problems proposed monthly in the Hungarian Mathematical and Physical Journal for Secondary Schools, KöMaL, Középiskolai Matematikai és Fizikai Lapok. Eventually he published several articles in it about the problems in elementary plane geometry. Despite the restrictions on Jews entering universities in Hungary, seventeen-year-old Erdős was allowed to enter in 1930 as a winner of a national examination. He studied for his doctorate at the University Pázmány Péter in Budapest where as a university fresher he became popular in the mathematical circles with an amazingly simple proof of Chebyshev’s theorem (which says that a prime can always be found between any positive integer and its double). In four years he completed his undergraduate work

case with plenty of room left for his radio. The only possessions that mattered to him were his mathematical notebooks. He completed ten of them by the time he died. He proved theorems and solved problems in England, United States, Israel, China, Australia, and 22 other countries. Inspired, Erdős moved from one university or

At three he could multiply three digit numbers in his head. research centre to the next. His motto was ‘Another roof, another proof’. Very often he would show up unannounced on the doorstep of a fellow mathematician declaring ‘My brain is open!’ and stay as long as his colleague served up interesting mathematical challenges. In many cases he would ask the current host about whom he should visit next. His working style has been humorously compared to traversing a linked list. Erdős had his own peculiar mathematical vocabulary describing the surrounding world. He believed that God, whom he aﬀectively called the S.F. or “Supreme Fascist,” had a transﬁnite book containing the shortest but most beautiful and elegant proofs for every mathematical problem. The S.F.

Erdös structured his life to maximize the amount of time he had for mathematics. and earned a Ph.D. in mathematics. was often accused of hiding Paul’s Then it was time for Erdős’s legendary socks and Hungarian passports and journey in the world of numbers to ﬁ- sometimes keeping the best mathematnally begin. Erdős structured his life ical proofs for himself. Children were to maximize the amount of time he Erdős’s “epsilons”, music was just had for mathematics. He had no wife “noise”, married people were “capor children, no job, no hobbies, not tured” and giving a mathematical leceven a home to tie him down. His ture was “preaching”. When he said whole wardrobe ﬁt into a small suit- someone had “died”, Erdős meant

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that the person had stopped doing mathematics. When he said someone had “left”, the person had actually died. The highest compliment he could pay to a colleague’s work was to say, ‘That’s straight from The Book’. After the death of Paul’s mother, Ronald Graham, another one of the world's best-known mathematicians, computer theorists and technology visionaries, took the responsibility of looking after the Hungarian mathematician. Graham, at the time a director of the Mathematical Sciences Research Centre at AT&T Bell Laboratories, was in charge of Erdős’s money and mathematical papers - previously the work of his mother. The articles, more than a thousand in number, were kept in a closet full of ﬁling cabinets. Near the closet there was a sign in capital letters saying: ‘ANYONE WHO CANNOT COPE WITH MATHEMATICS IS NOT FULLY HUMAN. AT BEST HE IS A TOLERABLE SUBHUMAN WHO HAS LEARNED TO WEAR SHOES, BATHE, AND NOT MAKE MESSES IN THE HOUSE.’ Graham also handled Erdős’s mail correspondence which involved sending out 1,500 letters a year, only few of them non-mathematics related. ‘I am in Australia’ a typical letter would began. ‘Tomorrow I leave for Hungary. Let k be the largest integer …’ A fascinating fact about Ron Graham is that he holds a world record. He is in the Guinness Book of World Records for using (in 1977) the largest number ever in a mathematical proof. So large there isn't even a notation for it and now known as "Graham's number." A great testimony for Erdős’s love for maths came when he had to have surgery on one of his eyes to treat cataracts. He wanted, during the operation, to read a maths book with the other eye. The physicians objected for

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practical and scientific reasons both. As a compromise the doctors arranged for a local mathematician to stay with Paul during the surgery and talk maths. In 1949 Erdős had his most satisfying victory over the prime numbers when he and Atle Selberg gave “The Book” proof of the prime number theorem (which is a statement about the frequency of primes at larger and larger numbers). In 1951 John von Neumann presented the Cole Prize to Erdős for his work in prime number theory. The Hungarian mathematician helped the founding of the Graph Theory field and attended the first International Conference on the subject in 1959. During the next three decades Erdős continued to do important work in combinatorics, partition theory, set theory, number theory, and geometry - the diversity of the fields he worked in was unusual. What little money Erdős received in stipends or lecture fees he gave away to relatives, colleagues, students, and strangers. He could not pass a homeless person without giving him money. In 1984 he won the prestigious Wolf prize, the most lucrative award in mathematics. He contributed most of the $50,000 he received to a scholarship he established in Israel in the name of his parents. ‘I kept only seven hundred and twenty dollars,’ Erdős said, ‘and I remember someone commenting that for me even that was a lot of money to keep.’ Whenever Erdős learned of a good cause he promptly made a small donation. In the late 1980s Erdős heard of a promising high school student named Glen Whitney who wanted to study mathematics at Harvard but was a little short the tuition. Erdős arranged to see him and, convinced of the young man's talent, lent him $1,000. He asked Whitney to pay him back only when it would not cause financial strain. A decade later Ronald Graham heard from Whitney who at last had the money to repay Paul. ‘Did Erdős expect me to pay interest?’ Whitney wondered. ‘What should I do?’ he asked Graham. Graham talked to Erdős. ‘Tell him,’ Erdős said, ‘to do with the $1,000 what I did.’ In 2009, Glen Whitney quit his job as an Algorithm Manager at the giant quantitative hedge fund Renaissance Technologies on Long Island to devote himself to one of his biggest dreams: creating a world-class interactive Mu-

12 [The, Commutor] March 2010

seum of Mathematics. The $20-millioncapital project (abbreviated MoMath) is now within $1 million of reaching its goal. Negotiations concerning a location near Madison Square Park are in their ﬁnal stages and Whitney expects to cut the ribbon in the spring of 2012. The mathematics of Paul Erdős is the mathematics of beauty and insight. He is the consummate problem solver, his hallmark is the succinct and clever argument often leading to a solution from “The Book”. He loves areas of mathematics which do not require an excessive amount of technical knowledge but give scope for ingenuity and

also asked basic questions outside mathematics but he never remembered the answers and asked the same questions again and again. For decades Erdős vigorously sought out new, young collaborators and ended many working sessions with the remark, ‘We'll continue tomorrow if I live.’ With 485 co-authors, Erdős collaborated with more people than any other mathematician in history. Because of his proliﬁc output friends created the Erdős number as a humorous tribute. Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators (those lucky 485) could

What little money Erdös received in stipends or lecture fees he gave away to relatives, colleagues and strangers. surprise. To Erdős, the proof had to provide insight into why the result was true and not just a complicated sequence of steps which would constitute a formal proof yet somehow fail to provide any understanding. He did not just want to solve problems, however, he wanted to solve them in an elegant and elementary way. Part of his mathematical success stemmed from his willingness to ask fundamental questions, to ponder critically things that others had taken for granted. He

claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and the most famous contemporary “computer personality” with a small Erdős number is Bill Gates, who has an Erdős number of 4. The great unwashed who have never written a mathematical paper have an Erdős number of inﬁnity. The University of Glasgow’s representatives with small Erdős number are

Paul Erdös with Ronald Graham and his wife Fan Chung Graham www.the-commutator.com

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PAUL ERDÖS: PRIZES 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. It was established and originally funded by Paul Erdős. The Paul Erdős Award is established to recognise contributions of mathematicians which have played a significant role in the development of mathematical challenges at the national or international level and which have been a stimulus for the enrichment of mathematics learning. Each recipient of the award is selected by the Executive and Advisory Committee of the World Federation of National Mathematics Competitions on the recommendation of the WFNMC Awards Subcommittee. The Anna and Lajos Erdős Prize in Mathematics is a prize given by the Israel Mathematical Union to an Israeli mathematician (in any field of mathematics and computer science), “with preference to candidates up to the age of 40.” The prize was established by Paul Erdős in 1977 in honour of his parents, and is awarded annually or biannually. The name was changed from Erdős Prize in 1996, after Erdős's death, to reflect his original wishes. Dr. Robert Irving, who has an Erdős number of 2, through Michael E. Saks. Dr. David Manlove, Dr. Lorna Love, Dr. Patrick Prosser, Dr. John D. McClure, Dr. Martin W. McBride all having an Erdős number of 3, through Dr. Irving. Jerrold Grossman at Oakland University in Rochester, Michigan, runs an Internet site called the Erdős Number Project which tracks the coveted numbers. It reveals one very interesting observation: take the 485 mathematicians with Erdős number 1 and represent them by 485 points on a sheet of paper. Draw an edge between

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any two points whenever the corresponding mathematicians published together. The resulting graph, which at last count had 1,381 edges, is the Collaboration Graph. Some of Erdős's colleagues have published papers about the properties of the Collaboration Graph, treating it as if it were a real mathematical object. One of these papers made the observation that the graph would have a certain very interesting property if two particular points had an edge between them. To make the Collaboration Graph have that property, the two disconnected mathematicians immediately got to-

gether, proved something trivial, and wrote up a joint paper. Defying the conventional wisdom that mathematics was a young man’s game, Erdős went on proving and conjecturing until the age of 83 succumbing to a heart attack only hours after disposing of a nettlesome problem in geometry at a conference in Warsaw. The epitaph Paul Erdős wrote for himself was ‘Vegre nem butulok tovabb’; Finally I am becoming stupider no more.

Hristo Georgiev

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THE Futurama is an animated sci-fi sitcom created by Matt Groening and developed by Groening and David X. Cohen. The series follows the adventures of a pizza delivery boy, Philip J. Fry, from New York City who after being cryogenically frozen by mistake just after midnight on January 1, 2000, finds himself woken up 1,000 years later and starts working for his only living relative, Professor Hubert J. Farnsworth. The distant brilliant-minded grand-son is the owner of Planet Express, a delivery company bearing the slogan “Our crew is replaceable, your package isn’t!” started to help fund his research and Fry’s job is not too different from his last one – he works as a cargo delivery boy. In the 98th episode (the 10th episode of series 6) of Futurama called ‘The Prisoner of Benda,’ Professor Farnsworth and Amy, a long-term intern from Mars University, in order for each of them to be able to live their biggest dreams, invent a machine (Figure 5) which allows two users to switch minds. It is, as they find out later, due to the brain’s natural immune response, unable to do so when these same two people already switched their minds with each other before, in other words, the machine cannot be used twice in a row on the same pairing of bodies. To try to return to their actual bodies, the rest of the crew of Planet Express, and a few others are involved into

18 [The, Commutor] March 2010

THEOREM the mind switching game. Later on in the episode, an equation for enough switches is invented and proved, so that everything goes back to normal. One of the participants discovers that by including no more than two new people in this mind swap game, everyone can always return to their original body. A theorem, later called the Futurama Theorem, based on group theory was speciﬁcally written and proven by Ken Keeler. Problem: Suppose that k people have swapped bodies. Label these people so that the 1st person is in the 2nd person’s body, the 2nd person is in 3rd person’s body, and so on, so that the last person (the kth person) must be in the 1st person’s body: 1 2 f k k+1 f n r=d n 2 3 f 1 k+1 f n Figure 1. Solution: With two additional people x and y we can get everyone back to normal using the following procedure: Let <a,b> represent the transposition that switches the contents of a and b. By hypothesis π is generated by distinct switches on [n]. Introduce two ‘new bodies’ <x,y> and write:

1 2 f k k+1 f n x y * r =d n 2 3 f 1 k+1 f n x y Figure 2. For any i=1 ... k-1 let σ be the series of switches: v = (G x,1 H G x,2 H f G x, i H (G y,1 + 1 H G y,1 + 2 H f G y, k H)( G x,1 + 1 H)( G y,1 H)

Figure 3. Note that each switch exchanges an element of [n] with one of <x,y> so they are all distinct from the switches within [n] that generated π and also from <x,y>. By routine veriﬁcation: 1 2 f n y x * r v=d n 1 2 f n y x Figure 4. i. e. σ reverts the k-cycle and leaves x and y switched (without performing <x,y>). Now let π be an arbitrary permutation on [n]. It consists of disjoint cycles and each can be inverted as above in sequence after which x and y can be switched if necessary via <x,y>, as was desired.

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Feature Futurama Theorem

FUTURAMA WRITERS AND STAFF He soon left Bell Labs to write for David Letterman and subsequently for various sitcoms, including The Simpsons and Futurama. Keeler has won Writer's Guild and Emmy Awards.

David S. Cohen has a Bachelor’s Degree in physics from Harvard University and a Master’s Degree in theoretical computer science from University of California, Berkeley. His most notable academic publication (with Manuel Blum) concerned the computer science problem of pancake sorting, which was also the subject of an academic paper by Bill Gates. In addition, Cohen is credited as a co-author on several papers by the computer vision researcher Alan Yuille. Before starting professional scriptwriting Cohen used to write the humour column for his high school newspaper and later was a writer for the Harvard Lampoon Magazine. He has won Emmy Awards for his work on both The Simpsons and Futurama.

Jeﬀ Westbrook majored in physics and the history of science at Harvard University and he received his Ph.D. in computer science from Princeton University in 1989. The title of his doctoral thesis was Algorithms and Data Structures for Dynamic Graph Algorithms. He was an Associate Professor in the Department of Computer Science at Yale University and also worked at AT&T Labs before writing for Futurama. He published an article entitled Short Encodings of Planar Graphs and Maps with Ken Keeler.

Ken Keeler has a Ph.D. in applied mathematics from Harvard University. Hs doctoral thesis was on Map Representations and Optimal Encoding for Image Segmentation. He also published a paper on Short Encodings of Planar Graphs and Maps with Jeﬀ Westbrook. After earning his doctorate in 1990, Keeler joined the Performance Analysis Department at AT&T Bell Laboratories.

J. Stewart Burns graduated magna cum laude with a bachelor's degree in mathematics from Harvard University in 1992. His senior thesis was on The Structure of Group Algebras. He received his master's degree in mathematics from University of California, Berkeley in 1993. He has worked on dynamic graph problems, in which the goal is to maintain connectivity and planarity informa-

tion about graphs that are growing online, and various problems that can be attacked with competitive analysis. He has studied the problem of robot navigation in an unfamiliar environment, with co-authors Dana Angluin and Wenhong Zhu, both of Yale University. Sarah J. Greenwald has B.S. in mathematics from the Union College in Schenectady, NY and a Ph.D. in mathematics from the University of Pennsylvania. She is an associate professor at Appalachian State University and a 2005 Mathematical Association of America Alder Award winner for distinguished teaching, including her use of popular culture in the classroom, and the 2010 Appalachian State University Wayne D. Duncan Award for Excellence in Teaching in General Education. She is a moderator for both The Simpsons and Futurama. Bill Odenkirk has a PhD in in organic chemistry from the University of Chicago in 1995.

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) Suppose that Professor Farnsworth

x (mind) ← x (body)

Figure 5. Prof. Farnsworth, Amy and their mind-switching machine and Amy knew of this theorem at the beginning: then they would have known they could switch back with the aid of two other people x and y, which is described below (move and mind←body correspondence at each stage). Initial arrangement: Prof. (mind)  Amy (mind) Prof. (mind) ← Amy (body) Amy (mind) ← Prof. (body)

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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)

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 is an original work inspired by the plot of the episode is a great example of how critical thinking can be used to solve all types of problems.

Hristo Georgiev March 2010 [The, Commutator] 19

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Feature Futurama Theorem

Here is the formal mathematical explanation of what happens in the episode along with a proposed more eďŹ&#x192;cient solution.

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Mathematics Lagrange’s Theorem

A COUNTEREXAMPLE TO THE FULL CONVERSE OF LAGRANGE’S THEOREM Lagrange's Theorem is a fundamental result in ﬁnite group theory and states that the order of a subgroup divides the order of the group. Although there are partial converses which hold in certain circumstances (for example if we have a ﬁnite cyclic group) the full converse is false. The usual counterexample given is the alternating group A4 which is the subgroup of

S4 containing all even permutations on 4 letters (geometrically, the rotation group of a regular tetrahedron). Although 6 divides ; A4 ;= 12 , the group A4 has no subgroup of order 6. Below we give two proofs of this fact. The ﬁrst proof makes use of cosets and the second requires the additional concepts of normal subgroups, quotient groups and isomorphisms. Second proof:

First proof:

order 6, and

H is a subgroup of A4 of order 6. Since H has index 2 in A4 , it follows that H 1 A4 and A4 /H , Z2 , i.e. H is a normal subgroup of A4 and the quotient group A4 /H is isomorphic to Z2 . Since the quotient group has

We must consider three cases:

order 2, both its elements are self inverse and therefore square to give the identity element. Therefore, for all we have a ! A4

Assume that

H is a subgroup of A4 of order 6. Since

Once again we assume that

A4 = {e,( 12)( 34),( 13)( 24),( 14)( 23),( 123),( 132), (124),( 142),( 134),( 143),( 234),( 243) contains eight elements of order 3, it follows that any subgroup of

A4 of

H in particular, must contain at least two elements of order 3. Let a be a general element of order 3. Then since ; A4 : H ;= 2 i.e. H has index 2 in A4 , 2 3 at most two of the cosets aH , a H , and a a H = eH = H are distinct. H = aH & a ! H 2 2 2 1 (ii) H = a H & a ! H & (a ) - = a ! H 2 2 2 (iii) aH = a H & [aa = e ! aH] / [e ! a H] & H = aH & a ! H (i)

(aH) 2 a2 H = eH = H , i.e. for all a ! A4 we have a2 ! H . If a ! A4 is a general element of order 3, then

a = a4 = (a2) 2 ! H . Hence H must

In all three cases we have

contain all eight elements of order 3, contradicting our assumption that H

has no subgroup of order 6.

has order 6. We conclude that no subgroup of order 6.

a ! H so H must contain all eight elements of order 3, contradicting our assumption that H has order 6. Therefore A4

A4 has

Colin Pratt www.the-commutator.com

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What Weâ&#x20AC;&#x2122;ve Been Up To!

Cheese

and Wine Night

Our first event, was a great success, we managed 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!

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Geek Chic Pub Crawl 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.

MOVIE NIGHTS 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!

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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 Macsoc@hotmail.com no later than the 1st of April! www.the-commutator.com

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BUROKKU ブロック A

BxC

BxC E

A BxC

A E

A - header number B - sum of all four surrounding cell numbers C - number of connections D - cell number E - connections

BxC

Figure 1. A 1x1 Burokku puzzle with a legend The meaning of the word ’burokku’ in Japanese (ブロック) is ’block’, thus the name of the puzzle suggests its main goal: to find all defined blocks, using the initially given clues and then fill them up with appropriate numbers. There are a few simple rules which need 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 the zeros placed outside a block.

x x x Here are some useful hints you might Figure 2.1. A sample 2x2 puzzle want to consider when working on a Burokku puzzle solution: • The ﬁrst multiplier in each corner is the actual number occupying the corner 1x4 4x4 3x2 cell, because it turns out to be the only cell taken into account when making the sum of all adjacent to the circle cells. • There are no circles with only one connection, so all header numbers are 9x3 4x2 5x4 formed by multiplying the adjacent cells sum by a number of connections ≥ 2. • If the header number in one of the corners is 0, the cells sum of the same circle is always 0, while the number of connec4x2 5x3 1x2 tions can be either 0 or a positive integer.

In addition, the following list of Figure 2.2. Solution to puzzle 2.1 slightly more sophisticated rules defines www.burokku.org the general flavour of the puzzle: • Each circle contains a number that is Burokku is a newly invented puzzle inspired by other going to be referred to as a header numpopular formats like Sudoku, KenKen and Slitherlink. ber. You have the unique opportunity to test your skills on • The header number represents a prodthe first officially published in print 3x3 Burokku puzuct of the sum of all adjacent cells and zle on page 27. If you find it too easy, there are a couple the number of connections between the of dozen puzzle sets on the official website. circle and its neighbour circles. • The connections between circles define and divide the grid into several blocks. • Each cell may contain either a posix x x 6x2 11 x 3 15 x 3 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 this x x x 25 x 3 15 x 4 10 x 2 block’s cells is doubled as well; if there are two double connections, then the sum is doubled twice etc. Basically, the main initial approach of solving each Burokku puzzle should x x x 14 x 3 10 x 2 4x2 consist of guessing each circle's sum and its number of connections with Figure 3.1. Another sample 2x2 puzzle Figure 3.2. Solution to puzzle 3.1 neighbours by logical deduction based Hristo Georgiev on the given header number.

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Games & Puzzles

PUZZLES FROM MARTIN GARDNER’S COLLECTION Martin Gardner was an American mathematics and science writer born on Oct 21, 1914 in Tulsa, OK, USA and died on May 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 also wrote a ‘puzzle’ story column for Asimov's Science Fiction magazine in the late 1970s and early 1980s. Problem 1: The amazing code Dr. Zeta is a scientist from Helix, a galaxy in another spacetime dimension. One day Dr. Zeta visited the Earth to gather information about humans. His host was an American scientist named Herman. Herman: Why don't you take back a set of the Encyclopaedia Britannica? It's a great summary of all our knowledge. Dr. Zeta: Splendid idea, Herman. Unfortunately, I can't carry anything with that much mass. However, I can encode the entire encyclopaedia on this metal rod. One mark on the rod will do the trick. Herman: Are you joking? How can one little mark carry so much information? Dr. Zeta: Elementary, my dear Herman. There are less than a thousand diﬀerent letters and symbols in your encyclopaedia. I will assign a number from 1 through 999 to each letter or symbol, adding zeros on the left if needed so that each number used will have three digits. Herman: I don't understand. How would you code the word cat? Dr. Zeta: It's simple. We use the sort of code I just showed you. Cat might be coded 003001020. Using his powerful pocket computer, Dr. Zeta scanned the encyclopaedia quickly, translating its entire content into one gigantic number. By putting a decimal point in front of the number, he made it a decimal fraction. Dr. Zeta then placed a mark on his rod, dividing it accurately into lengths a and b so that the fraction a/b was equivalent to the decimal fraction of his code. Dr. Zeta: When I get back to my planet, one of our computers will measure a and b exactly, then compute the fraction a/b. This decimal fraction will be decoded, and the computer will print your encyclopaedia. Would this approach work in practice?

le if I add that x and y are the two positive integers that begin the longest possible generalised Fibonacci chain ending in a term of 1,000,000. Problem 3: The blank column A secretary, eager to try out a new typewriter, thought of a sentence shorter than one typed line, set the controls for the two margins and then, starting at the left and near the top of a sheet of paper, proceeded to type the sentence repeatedly. 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 a line: When she saw that the next word (including whatever punctuation marks may have followed it) would not fit the remaining space on a line, she shifted to the next line. Each line, therefore, started flush at the left with a word of her sentence. She finished the page after typing 50 single-spaced lines. Is there sure to be at least one perfectly straight column of blank spaces on the sheet, between the margins, running all the way from top to bottom? Problem 4: A chess problem

Figure 1.

Your king is on a corner cell of a chessboard and your opponent’s knight is on the corner cell diagonally opposite, as shown in Figure 1. No other pieces are on the board. The knight moves ﬁrst. For how many moves can you avoid 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 2: The twenty bank deposits Problem 6: Arrange the superqueen A Texas oilman who as an amateur number theorist opened a new bank account by depositing a certain integral number of dollars, which we shall call x. His second deposit, y, also was an integral number of dollars. Thereafter each deposit was 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 of x and y, his ﬁrst two deposits? The problem reduces to a Diophantine equation that is somewhat tedious to solve, but a delightful shortcut using the golden ratio becomes availab-

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Figure 2.

A ‘superqueen’ is a chess queen that also moves like a knight. Place four superqueens on a five-by-five board (Figure 2) so that no piece attacks another. If you solve this, try arranging 10 superqueens on a 10-by-10 board so that no piece attacks another. Both solutions are unique if rotations and reflections

are ignored.

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Problem 13: The three pennies

Problem 7: Guess the four digits ABCD DCBA ? ? ?? 12 30 0 ABCD are four consecutive digits in increasing order. DCBA are the same four in decreasing order. The four dots represent the same four digits in an unknown order. If the sum is 12,300, what number is represented by the four question marks Problem 8: The coloured socks Ten red socks and ten blue socks are all mixed up in a dresser drawer. The twenty socks are exacty alike except for their colour. The room is in pitch darkness and you want two matching socks. What is the smallest number of socks you must take out of the drawer in order to be certain that you 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 searching through the cash register,‘but I can't do it with the coins I have here.’ ‘Can you change a half dollar then?’ Miss Jones shook her head. In fact, she said, she couldn't even 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 directly toward each other. The instant they started, a fly on the handle bar of one bicycle started flying straight toward the other cyclist. As soon as it reached the other handle bar it turned and 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, and the fly flew at a constant speed of 15 miles an hour, how far did 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’s diagonal that runs from comer A to corner B (Figure 3)?

26 [The, Commutator] March 2011

Joe: ‘I’m going to toss three pennies in the air. If they all fall heads, I’ll give you a dime. If they all fall tails, I’ll give you a dime. But if they fall any other way, you have to give me a nickel.’ Figure 3. Find AB. Jim: ‘Let me think about this 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 if two are alike, then the third penny will either match the other two or not match them. The chances are even that the third penny will or won't match. Therefore the chances must be even that the three pennies will be all alike or not all alike. But Joe is betting a dime against my nickel that they won’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 tribe always tell the truth, members of the other always lie. A missionary met two of these natives, one tall, the other short. ‘Are you a truth-teller?’ he asked the taller one. ‘Oopf,’ the tall native answered. The missionary recognised this as a native word meaning either yes or no, but he couldn't recall which. The short native spoke English, so the missionary asked him what his 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 digits have been arranged: 8-5-4-9-1-7-6-3-2-0

You can ﬁnd the solutions to all problems and puzzles in the Solutions section of our website or through scanning the QR code on the right with your mobile device which will lead you directly to the URL.

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[The, Commutator] Vol 2 Ed 1 (1)_Commutator Volume 2 Edition 1 20/03/2011 13:40 Page 27

Games & Puzzles

Chess Puzzle

Microsums

Choose one of the white pieces. A valid move is one that captures any black piece acquiring the properties of whatever it has just captured. White pieces cannot be captured. Only the ﬁrst white piece you choose is removed from the chessboard. The goal is to capture the black king in as few moves as possible.

Fill in the numbers 1 through 16, so that the numbers on the right and on the bottom correspond exactly to each of the column, row and diagonal sums.

25 34 38 39 26 32 34 44 46

7 One Hundred is a digit in each of the cells. 6 There You may choose to either add an-

other digit to the front or to the back and make a two-digit number or leave the one-digit number as it is. The goal is for the sum of each row and column to be exactly 100.

2 Skyscrapers

4 1

Fill in each square with an interger from 1 to 6 indicating the height of each building. No number may appear Write down your solution in the spaces provided on the bottom right of the page. Note that your solution does not have to be of exactly 16 twice in any row or column. The nummoves as the number of the spaces. bers along the edge of the puzzle indicate the number of buildings which you would see from Fill in all missing numbers and lines, refering to the rules on page that direction if 24. The sums of all deﬁned regions must be equal. there was a series of skyscrapers with heights equal the entries in that row or column.

3 5

2 3

Burokku

2 1

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Chess Puzzle Solution Moves: 1.___________________________________________________ 2.___________________________________________________ 3.___________________________________________________ 4.____________________________________________________ 5.____________________________________________________ 6.___________________________________________________ 7.___________________________________________________ 8.___________________________________________________ 9.___________________________________________________ 10.__________________________________________________ 11.__________________________________________________ 12.__________________________________________________ 13.__________________________________________________ 14.__________________________________________________ 15.__________________________________________________ 16.__________________________________________________

March 2010 [The, Commutator]

[The, Commutator] Vol 2 Ed 1 (1)_Commutator Volume 2 Edition 1 20/03/2011 13:40 Page 28

Games & Puzzles

Planetarium

Arithmetic Tree

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 up to the sum. 3. The four weights along the four spirals (the dashed lines) must add up to the sum. 4. You may use only positive integers for weights and each of them may not be used more than once, in other words, repetition of numbers is not allowed.

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

× ×

^ × 3

1024

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

Hex Sudoku 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 the digits 0,1,2,3,4,5,6,7,8,9A,B,C,D,E,F.

D 2

C D 4 F 0

0 3 8 1

6 9

5 A 4 0 3 B F 6 2 B F A 1 6 0 8 9 D 7 1

2 C D B 1 6 8 2 9 4 0 E 3 A 0 3 6 7 1 A 3 2 D B 7 8 C B A 8 6 3 D A B 6 D 4 2 8 F 4

28 [The, Commutator] March 2011

5 3 A

D 7

B F D C 6 1 7 2 4 B 5 C 0 A C 3 2 6 E B 2 8 E 6 9 C

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