Hong Kong Joint School Mathematics Society HKJSMS Newsletter
Issue 4 12-13
August 2013
From the Editor Dear readers, it’s hot summer and everyone of you are enjoying the summer holidays! Feeling bored at home? Don’t forget to participate our big activity— The Day Camp on 23/8 and 27/8. We will have an enjoyable day in these two days! See you there! Eileen Tam & Jeff Siu
Interview with former HKJSMS president P.2–8 // Reflection on the work of the publication team P.8 // Review on last year’s activities of HKJSMS P.9-10 // Combinatorics from another perspective --- Burnside’s Lemma P.11-P.14// Topology and combinatorics P.14-P.16// Ramsey Theory P.16-18 // Proof without words P.18-19// Fun with Mathematics P.20
Combinatorics
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Interview with former JSMS president Chan Long Tin President of Hong Kong Joint School Mathematics Society (JSMS) 2011-2012 School: Diocesan Boys' School (Graduated) University: The University of Cambridge Major: Mathematics E: L:
Eileen Tam Longtin Chan
J:
Jeff Siu
Interview
Placement examination (AP) when I was F2.
1) Introduction E: Congratulations, Longtin. I am so glad to hear the news that the University of Cambridge has accepted you.
L: Thank you.
J: I understand that you recommended yourself to the president position of JSMS and you were the member of DBS Mathematical Olympiad Team. Why are you interested in Mathematics? Could you tell us about your story?
L: I am interested in Mathematics and I discovered this fact when I was small. When I was small, I gradually joined different classes to learn deeper about Mathematics. The most important thing is that the Education Programme for Gifted Youth programme (EPGY) which I took part in when I was a primary school student. With the acceleratied program I learnt Mathematics in a faster way. For a normal 1-year programme, I only used half a year. After 3 years, I had learnt all the Mathematics of junior high school. I kept studying according the syllabus in the US, for example, Beginning Algebra, until I enrolled Advanced
In the path of learning Mathematics, I found that Mathematics has another side — the Mathematical Olympiad (MO). I encountered MO when I was P5. I once represented my school to participate in the competitions and won some prizes. Since then, I found a sense of success in so I kept on joining MO competitions. As MO built up my confidence, I wanted to improve myself. Everything has different stages. First, I had sense of success and thought myself was smart. Therefore, I kept on joining MO competitions. However, I sensed the fly in the ointment. There was a distance between the peak and my location and I tried to shorten the distance.
The Education Bureau (EDB) had chosen me to represent Hong Kong joining MO competitions for two times. These were excellent experiences for me. These experiences brought me the position of Captain of DBS Mathematical Olympiad Team. You have a lot to do once you are the captain. Besides learning, you have responsibility to teach others. This was a new direction for me. Whenever I gained some knowledge, I passed it to my teammates. This benefits my progress in studying Mathematics.
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I call this as a new chapter.
I recognize myself as someone with leadership talents after years of learning at school. Hence, I wanted to become the president of JSMS. I did not need others to choose me because I recommended myself to the president position. Being confident is very important.
different schools showing up. I really felt happy. I admit that I made mistakes at the early stage. The things were not completed well. I thought it might due to the staff shortage problem. Only 15 people could not do all the things. So, recruiting sub-committee members is crucial. Someone asked me why I not consider recruiting helpers before the events. I answered that our human resource would be unstable if we followed this practice.
2) About JSMS E: I believe you heard of JSMS because of InterSchool Mathematics Contest (ISMC). When did you start thinking to become the president of JSMS?
L: I remember it should be the time when I participated in the ISMC when I was F4. Normally, the posts of joint school societies are taken by F5 students. I wished to have a post and I first came up with JSMS. For your information, the committee is usually formed by people from MO circle. I believed I had the ability to be the president so I recommended myself to the society. Actually, I know some committee members of 23rd cabinet. They told me to form the committee of the next cabinet. There was not much choice as the number of MO people was small.
E: What was your expectation after becoming the president? Did it correspond to the reform brought by you?
L: I planned to recruit sub-committee members. I expected more students joining the events held by JSMS. I expected the participants come from different schools. I expect everyone involved in JSMS was in the MO circle. As expected, there were more guys who come from
Actually, I did not bring any ‘reform’. The significant change was the system of subcommittee.
E: May you give a brief conclusion of the work that the committee had done last year?
L: I can see the improvement of JSMS. I think JSMS was close and isolated from other joint school societies in the past. They have their own regulation and form a circle among themselves. This reveals the phenomenon that JSMS was like an isolated island. There was no subcommittee in JSMS but every societies recruited sub-committee at that time. Innovation was not popular in JSMS because the committee members tended to follow the practice. JSMS almost equals to ISMC. JSMS was therefore lack of features of a joint school society. However, there are more and more different activities are held by JSMS in these years. We can see that JSMS is now widespread compared to the past. For example, the newsletter in the past mainly focused on MO and ISMC. MO is actually something high-class in Hong Kong people’s mind. People might think that the newsletter was just for the MO circle. This lost the meaning of a joint school society. Now, the newsletter is simpler which is suitable for
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secondary school students. Or else, people consider the newsletter as wastepaper. Therefore, I think the biggest contribution made by my cabinet is the popularization of JSMS.
E: What’s your comment on the performance of the sub-committee members in last year? Did they make JSMS difference?
L: With their presence, JSMS became more energetic. The members come from different schools and different levels. In this way, the society can be improved easily as different perspectives produces a clear picture of the matter. This is similar to the society. The government needs to hear different voices, the voice from grassroots, the voice from middleclass and the voice from the rich. Different people have different mind. Back to the topic, there are people in the MO circle, Mathematics lovers and ordinary students. We need to take all their opinions into account so as to make a comprehensive decision. This is our aim: let more students have interest on Mathematics. The most valuable thing contributed by the sub-committee is the ideological impact. The impact becomes the stepping stone of the progress of JSMS.
E: What experience did you gain?
L: I learnt communication skills. As I was the president, I had to give clear instruction and teach them how to get things done. I could not suppress them and decide all the things myself. I also learnt how to persuade others to support me. Besides, I learnt how to influence others to follow my direction in order to unite the team. Any difficulty can be solved with team spirit. My leadership is therefore further
improved. On the other hand, I found out that there are many solutions to a problem. This is due to the evaluation after each event. Which part was not done well? Who need to take the responsibility? How to do better in next time? I need to collect the information and generalize some feasible solution. And this comes up with a challenge. How can I get the information? Some people may suggest this can be done at the meeting. Yet, people always choose not to admit the mistakes or confess the one need to pick up the responsibility before the committee. Other people may suggest using the evaluation form. Still, it is not a nice method because it is too long that only a few committee members will take it serious. And there are many other possible solutions, like having gathering. In this way, committee members can share their view at a relaxing atmosphere and I can get the required information. So, there are many solutions to a problem. Each solution has its own benefits and disadvantages. What I need to do is to find out the optimum case under a specific condition.
The third thing is that you have to observe others’ merit and shortcoming. For instance, there is a guy who is good at thinking up problem only. You would not choose him to be the president. Instead, you should find a person who is good at communication and knows how to present himself. For those whose communication skill is not very well, you would place them at the internal posts. On the other hand, you would find somebody who is adept in jotting notes to be the secretary. That is why observing others’ merit and shortcoming is very important to the president as he needs to allocate the work. In economic terms, the president is the entrepreneur. Making decisions and bearing risks are the tasks of the entrepreneur.
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E: We have talked a lot about JSMS in last year. How do you comment on the performance of the 25th cabinet of JSMS?
L: Average. Nothing is done well or poorly. Everything done in last year is kept this year. The 25th cabinet seemed to have followed the old practice. It seems that everything is done step by step. Besides, there are not much new ideas. As what you have done is standard, my comment to you guys is ‘average’. Of course, this is just my opinion. Your performance is not decided by me only. The participants should be included in judging your performance. Combining all the opinions, the evaluation on you will be pertinent.
JSMS need to develop and equip itself in these few years in order to welcome IMO. At the same time, we should arouse people’s attention to IMO. Although there are still many problems of JSMS, I believe JSMS can improve step by step with our aim.
E: You have just mentioned IMO 2016. Will you come back to Hong Kong to support this event?
L: This depends on my schedule. I really hope I can come as I once was a pioneer of popularizing Mathematics in Hong Kong. Helping Hong Kong to hold IMO is somehow my responsibility. Other alumni should give a helping hand too.
E: What is your expectation on JSMS in the future?
L: I don’t expect the scale of JSMS can match with other joint school societies, having a thousand people joining the events, having much sponsorship or having many committee members. These are too exaggerating as they make the society like a company. We need to be down to earth and perform real deeds. Always remember our aim, let more secondary school students be interested in Mathematics, love Mathematics and contact with Mathematics. Following this direction is the only thing we need to do. I only hope that there will be more participants in the coming ISMC.
Hong Kong will host International Mathematical Olympiad (IMO) in 2016. The aim of the host is to enhance the mathematical level of Hong Kong people. JSMS can contribute to IMO. The alumni and the committee members can be volunteers or participants of IMO. So,
3) Story with Mathematics J: Let’s back to your story with Mathematics. Where did you get training? L: There are different stages, primary school, junior high school and senior high school. The type and nature of competitions are different. I studied at Hong Kong Mathematical Olympiad School (HKMOS) when I was a primary school student. The training of HKMOS is systematic. However, I quit the training before I was F1.
When I was F3, I enrolled the IMO training by the Hong Kong Academy for Gifted Education (HKAGE) and International Mathematical Olympiad Hong Kong Committee (IMOHKC). The training was nice as it was free-of-charge and I gained knowledge. Also, the seniors at my school helped me a lot. It was great because there is a schoolmate who can go to IMO
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or be the reserve every year. I follow their path. Whenever I had questions about Mathematics, I would find them. Besides, they taught me the skills of being a captain. This is another kind of training. It was relaxing and I absorbed knowledge more quickly under this atmosphere. This is different from the traditional training. I built up my foundation of Mathematics during the training by the seniors.
Then, I became a senior. My role was hence changed. I need to self-study. I need to find notes myself, read articles and digest the information on my own. After gaining the knowledge, I had obligation to teach the juniors. Before teaching others, I need to study the knowledge well. And I need to answer their questions. This is another form of training, learning myself. The memory is deeper under this kind of ‘training’. This is similar to the learning method in university.
J: Then how did you prepare for competition?
L: I did many past papers when preparing the competition in primary school. The types of questions are limited. The competitions in junior high school are different, for example, the Pui Ching Invitational Mathematics Contest (PCIMC). I got low marks at this competition. I then realized simply doing the past papers is not enough. I need knowledge to backup and understand the concepts behind the questions. In high school, I stopped doing past papers. There are not many tools available and the skills are quite simple. The main point is the method of solving the problems. Sometimes, you can solve the problems without any difficult theorems, but simply with basic rules. The key is your mental quality. You need to be fit whenever you have competitions or exams. I
will go through all the key points of the knowledge I have learnt to stabilize my knowledge before the competitions. Instead of preparation, it is more important to do practice daily. Mathematics is the core subject because you cannot learn it within a short period of time, like Chinese and English. Your knowledge about Mathematics needs to be built up step by step unless you are a genius. J: How’s your feeling of representing HK to go to competitions [China West Mathematics Olym piad (CWMO) and China Mathematics Olym piad (CMO)]? L: I felt proud of myself. Not only representing Hong Kong, I represent my school DBS at the same time. On the list of winners, the name of DBS is shown. People care the school too instead of only focusing on the region.
The chance of representing Hong Kong is recognition to me. This enhances my confidence. Having the letter from EDB means the approval from the government. It fulfills my satisfaction.
J: What’s your expectation in the university?
L: I don’t have expectation on the future. I go to the university because I want to stud Mathematics. Seeking for knowledge is the only thing I mind. I choose Mathematics due to my interest in Mathematics. I don’t care much about the occupation in the future.
I choose the University of Cambridge not because it is one of the best universities in the world. It is because the Mathematics depart-
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ment in England is respected by others. Mathematics department in Hong Kong is neglected. People only care about the employment condition and whether you can earn money or not. People always ask me why I choose to study Mathematics but not other subjects which can bring me a bright future. Studying Mathematics in Hong Kong is not respected. People do not consider Mathematics when they apply for the universities. This is similar to Engineering. People recognize Engineering as a “blister subject” as engineers cannot be rich in Hong Kong. However, Engineering is a popular subject in India as shown in “3 idiots”. Many Indians would like to study Engineering. As a result, I decide to study mathematics in UK as studying Mathematics cannot gain respect form others in Hong Kong. For Mathematics, the University of Cambridge is the best in UK so I choose it. I hope I can gain knowledge in Cambridge. If I really care about money, I will choose the University of Hong Kong, studying actuarial science instead of studying Mathematics in Cambridge. I believe Cambridge is a suitable place for me to seek for knowledge. J: I understand you don’t care much about your occupation in the future but I really want to know what your dream job is. Could you reveal it to us? Does it relate to Mathematics? L: I want to be a professor in the university. The job vacancy is very few so it is difficult to get the position of professor. Nevertheless, I will work hard because it is my dream. Even though I may not be a professor, I still hope to be a mathematics teacher, such as lecturer and teacher. Teaching the young generation is very meaningful. What I want to do is bringing inspiration to others.
J: I really glad to hear there are still some people would like to study in Mathematics. I wish you can achieve your goal. L: Thank you!
Interview notes - By Eileen Tam I am glad to have an opportunity to meet Longtin again. It is great to have such a conversation with Longtin. After talking with Longtin, I understand JSMS deeper, the problems and the room of improvement. The major thing is that he reminded me the aim of JSMS. Looking back to this year, I find out that I sometimes neglected the aim. I just wanted to finish the event at that time. Therefore, I hope the next cabinet can do better next year and always remember the aim of JSMS. What Longtin said is really helpful.
I am a Mathematics lover. In JSMS, I can find many minded friends. Nevertheless, we are the minority. In fact, how many of us would like to study Mathematics in university? I am not sure. I have struggled on this question many times. The questions from others are somehow shaking my determination on studying Mathematics. Now, Longtin is an example for me. I decide to chase after my dream.
According to the job site CareerCast, mathematician is always on the list of best jobs. However, this is the situation in USA only. The situation in Hong Kong is totally different. Students are not encouraged to study Mathematics, Science or any other subjects that cannot let you be rich. I understand that career is something important in life but does it really important that you need to give up your dream? After graduation, not every student who studies Mathematics can earn a huge profit. However, studying Mathematics can train our thinking and logic. I believe one can be success
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is not due to what subject he studies, is because of his ability.
Reflection on the work of the publication team
Thanks Longtin. It is really a nice interview.
By Jeff Siu & Eileen Tam Time flies, it is near the end for me to be the publication officer in the HKJSMS. In this short article we will reflect the job as publication officer.
First of all, it is a great pleasure to work with all of the sub-committee members in the publication team. They contribute a lot to the newsletters. Moreover, it is a great pleasure for us to interview people who have interests in Mathematics.
Also, what we feel excited is that the content of each newsletter can cater different readers’ interest, such as the corner “Fun with Mathematics” really can arose readers’ interest to know more about Mathematics. Featured articles also cater the needs of readers who want to have a deeper thought on Mathematics. This improvement relies heavily on the hard work of all committee members.
At the same time we also believe our team has things to improve. We suggest the next cabinet can try to divide the content of the 4 newsletters per year to shorter ones, and release each bit of content per week. Therefore it can satisfy the different needs of readers.
To conclude we sincerely hope that our work in this year is up to quality from the others’ eyes. We also hope that the next cabinet will try their best and continue to improve the publication of HKJSMS!
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Review on last year’s activities of HKJSMS
2. Gathering with Singapore Mathematics Rep-
By Jeff Siu and Eileen Tam
lecture with the Singapore representatives at La
1. Annual general meeting
resentatives Our executive committees had a mathematics Salle College on 7th December, 2012. Mr Lim, the teacher from Singapore Team gave an interesting
The annual general meeting was successfully
talk on Knot Theory. All the committees and rep-
held on 23 November 2012 at Hong Kong Acade-
resentatives had a fruitful lesson.
my for Gifted Education (HKAGE). Besides, our executive committees had a dinner We were honorable to invite the Founding Presi-
with the Singapore representatives, one of our
dent of Hong Kong United Commission of Sec-
teacher advisor of HKJSMS, Dr. Li, the trainer of
ondary Students (HKUCSS), Mr. Alfred Lam and
IMO Training 12/13, Dr. Leung, and the Student
the president of 24th HKJSMS (2011-2012), Mr.
Programme Development Officer of HKAGE, Mr.
Chan Long Tin to join the meeting with all the
James Lee on 8th December, 2012 after the China
committee members.
Hong Kong Mathematics Olympiad (CHKMO). In the dinner the representatives and committee
In the meeting the structure of HKJSMS, annual plan was introduced to let the participants and guest to know more about our society. There was also free time to allow communication between
members shared their thoughts on Mathematics and their ways to organize a joint school Mathematics activity. All participants and guests had an enjoyable time.
committee members and the guests in order to build up a sense of belonging to the society.
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3. Inter-school Mathematics Contest 2013 It is the major activity of our society. The contest was successfully held on the 1st May, 2013 at the school hall of Kwun Tong Maryknoll College. There were 130% participation of last year. Both individual event and group event consists of 2 levels, from Junior Group (Form 1,2) and Senior group (Form 3, 4).
The group event results are as follows: Champion
P.L.K. Centenary Li Shiu Chung
4. 25th Anniversary of HKJSMS
Memorial College, Team 1
2012 was the Silver Jubilee Anniversary of our
1st runner up S.K.H. Lam Woo Memorial Secondary School, Team 2 2nd runner up Queen Elizabeth School, Team 1
society. It was generous for the ex- committee members to share their memories in the past HKJSMS by sending photos or even donate heritage such as posters years ago to us. Our commit-
Congratulations for the winning teams and participants!
tee members also made a timeline video to share with the public the good times of HKJSMS. You may also receive the joy of our anniversary by visiting our Facebook page to watch the video!
Moreover, group games were held in the afternoon after all the contests. Participants are divided into groups and cooperate in order to win the prize from the educational games such as transferring small balls . All the participants have fun and enjoyed a great time.
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Combinatorics from another perspective --- Burnside’s Lemma By Tsang Chi Cheuk Readers who are familiar with the topic of combinatorics probably know notations such as . However, these tools seldom help when tackling more difficult problems. Other tools must be found in other topics of mathematics. This time, we shall introduce you to Burnside’s Lemma, a result originally from group theory, but is often useful in combinatorics prob-
In group theory, mathematicians often consider the set and
the set
, which is the
set of permutations acting on . For example, the permutation of switching 1 and 2 is an element of (where ; the permutation of switching 1 with 3, then 3 with 4 is also an But the permutaelement of (where tion of switching 1 and (n+1) is not, since (n+1) is not in
so such permutation cannot
act on . Mathematicians also have a way of denoting permutations conveniently, called the cycle notation:
lems. (1 2) is switching 1 with 2, (3 4)(1 3) is switching Consider the following problem: How many ways can 4 colours be used to paint the faces of a tetrahedron, assuming different surfaces can be painted the same colour, and new ways derived by rotating the tetrahedron only count as one
1 with 3, then 3 with 4, (1 2 3) is shifting 1 to 2, 2 to 3, 3 to 1, (2 1 3 5) is shifting 2 to 1, 1 to 3, 3 to 5, 5 to 2 (For a bit of practice, persuade yourself that (1 2 3 4)=(1 4)(1 3)(1 2)) Readers may have noticed that the number of elements in
is
way (eg. There is only one way that the tetrahedron can be painted with 3 yellow faces and 1
simply
.
red face). This is the type of problem Burnside’s Lemma is extremely useful in solving, hence we shall use this problem to guide your discussion.
Now we denote the set of 4 colours as C, then label the faces of the tetrahedron with numbers 1,2,3,4 and denote the set of the faces as D. Consider mappings of D to C, notice these are precisely the action of painting the faces of the tetrahedron. Let the set of mappings of D to C be , readers should see that the number of elements in is in this case. This is the number of ways that the tetrahedron can be painted if rotations are counted as separate cases (using the previous example, there are 4 ways to paint a tetrahedron with 3 yellow faces and 1 red face
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under this assumption). We call this a labeled colouring. However, the problem we are tackling is an unlabeled colouring problem, so we must find a counting method such that certain elements in
are considered identical.
For each
(remember
is a
way of labeled colouring,
is a way of rotating
the tetrahedron), we can imagine as colouring the tetrahedron with the numbered faces on, then rotating the tetrahedron. Therefore we have the following deduction:
and u lead to indistinguishable colourings when the numbers are removed
Notice rotations of the tetrahedron can actually be represented by permutations in set D= {1,2,3,4}. The rotation on the left is (1 2 3). The rotation on the right is switching (1 2 4) (still remember the cycle notation?) We say are in the same G-orbit for the last line. Therefore the number of G-orbits is precisely the number of unlabelled colourings! (do give the last line some thought, it is an important yet quite difficult step!)
Now that the notational preliminaries are over with (Phew!), we can now state the Burnside’s Lemma:
The set of permutations that represent a rotation of the tetrahedron is denoted G, which is a subset of . (In fact, G is called a subgroup of due to its properties, but we shall not cover that in this article.)
where
is the set of
elements of
‘fixed’ by
is the number of elements in set G is the number of elements in set
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In our case, set G=
Lemma,
where is permutation that does not move the tetrahedron at all (the identity permutation) By our previous discussion,
Therefore Phew! After all this hard work, we finally got the answer! At this point, some readers may have already calculated the answer to this problem using elementary factorial and combination tools, (why?)
even much faster than the method we used here. It may seem that the Burnside’s Lemma is quite
Therefore is the number of colourings that have ‘2 pairs of colours’ (face 1 has
useless then, however, notice we were discussing
the same colour as face 2, face 3 has the same
tion to ask the number of ways of colouring when
the case of 4 colour only. We can extend the questhere are q colours and still get an answer! In-
colour as face 4), hence is
deed, for the case of q colours:
Similarly,
(why?)
Therefore is the number of colourings that have ‘2 pairs of colours’ (face 1 has the same colour as face 2 and face 3), hence is 16. Similarly,
By Burnside’s
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Bet you can’t get such a general result by using elementary tools!
Topology and Combinatorics By Irene Kwok
If you want to try using Burnside’s Lemma yourself, attempt the following questions (the same
I believe every one of you must have played soc-
assumption of repetition of colour and rotation
cer games before and know clearly about how it
of shapes applies, as in the problem discussed):
looks like. It is composed of 12 pentagons and 20 hexagons, arranged that every pentagon is sur-
Question 1 A) Given q colours, how many ways are there to paint the vertices of a tetrahedron? B) Given q colours, how many ways are there to paint the edges of a tetrahedron?
rounded by hexagons. However, have you ever thought of making a new design for the soccer balls while they will still suit the above traditional properties? To achieve this, skill in combinatorics and topology in mathematics will be required.
Question 2 Given q colours and r colours to paint the vertices and edges of an equilateral triangle respectively. How many ways of colouring are there? (Hint: An equilateral triangle is a planar figure, so it cannot be ‘flipped’, it can only be ‘rotated’)
Answers: Question 1
1. Simple transform of soccer ball
A.
A standard soccer ball has precisely 3 faces meeting at each vertex. Nevertheless, here shows some other exceptions:
B.
Question 2
References: http://en.wikipedia.org/wiki/Burnside's_lemma ‘Introduction to Abstract Algebra’ by W. Keith Nicholson
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This can be understood by generating infinite sequences using a topological construction called a branched covering. To make a new soccer ball,
combinatorial or topological—not geometric— objects, so that the polygons can be distorted arbitrarily.
we slice open the ball along this semicircle. Next we shrink the sliced-open coat of the ball in the
3. TOROIDAL SOCCER BALL
east-west direction holding the north and south poles fixed, until the coat covers exactly half the sphere, say the western hemisphere. We can take a second copy of this shrunken coat of the sphere and place it over the eastern hemisphere, so that it is the image of the western hemisphere under half of a full rotation around the north-south axis. Now something remarkable happens: the two pieces can be sewn together, to give the sphere a new structure of a soccer ball with twice as many pentagons and hexagons as we started with!
2. Application of topology and combinatorics in making the new ball It is clear that there are a number of variations on this construction. First we need not start with the standard soccer ball, but we can start with any soccer ball. In particular, we can iterate the passage to branched coverings. Second we can choose the vertices at which to place the branch points arbitrarily, by distorting a given soccer ball in such a way that two chosen vertices are placed at the poles. Third, instead of taking two-fold coverings, we can take D-fold coverings for every positive integer D. This means that after slicing open the coat of the earth, we do not shrink it to fit over a hemisphere, but we shrink it further, so that it fits precisely over one segment of a segmentation of the sphere into D orange segments. Then we spin this around the north-south axis to cover the other D-1 orange segments, and fit everything together along the D seams. For all this it is important that one thinks of soccer balls as
Starting from a sphere, one can construct a torus by removing two small disks from the sphere, and gluing the two boundary circles to each other. This procedure can be applied several times to construct surfaces of arbitrarily large genera. One can arrange the branch points to be vertices of some soccer ball graph on the sphere, and look at the pre-image of this graph on the higher genus surface. This gives the surface the structure of a soccer ball. Here is an easier construction for the torus. Let us take a spherical soccer ball, and cut it open along two disjoint edges. We open up the sphere along each cut, and what we obtain looks rather like a sphere from which two disks have been removed, but now we have a soccer ball pattern on it, and the two boundary circles at which we have opened the sphere each have two vertices
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on them, which are the endpoints of the cut edges. If the cut edges are of the same type, meaning that along both of them two hexagons met in the original spherical soccer ball, or that along both
Ramsey Theory By Brian Lui Ching Yin
of them a pentagon met a hexagon, then we can
Living for only 26 years, Frank Plumpton Ramsey
glue the two boundary circles together so as to
(22 February 1903 – 19 January 1930) was a pre-
match vertices with vertices.
cocious British mathematician, philosopher and economist. He himself did not develop the Ram-
4. Other transforming method
sey theory, but proved a foundational result in combinatorics accidentally as minor lemma in a
Smooth morphing of a trefoil knot into a double-
paper. He might not know at that time that the
covered soccer ball
influence of his effort has triggered even deeper research of other greater minds into graph theory and its applications in life. This leads to the development of the Ramsey theory.
One of the theorems proved by Ramsey in his 1928 paper “On a problem of formal logic” now Folding a soccer ball from paper
bears his name (Ramsey's theorem). While this theorem is the work Ramsey is probably best remembered for, he only proved it in passing, as a minor lemma along the way to his true goal in the paper, solving a special case of the decision problem for first-order logic, namely the decidability of what is now called the Bernays–Schonfinkel– Ramsey class of first-order logic, as well as a characterization of the spectrum of sentences in
A 'breathing' soccer ball
this fragment of logic. It can be seen that he did not mean to focus on the Ramsey’s theorem in the first place. However, a great amount of later work in mathematics was fruitfully developed out of the ostensibly minor lemma, which turned out to be an important early result in combinatorics, supporting the idea that within some sufficiently large systems, however disordered, there must be some order. So fruitful, in fact, was Ramsey's theorem that today there is an entire branch of mathematics, known as Ramsey theory,
1. Biography
which is dedicated to studying similar results.
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can assume at least 3 of these edges, connecting to vertices r, s and t, are blue. (If not, exchange
2. Ramsey’s Theorem Let’s spare some time to digest what Ramsey’s theorem means to us in the following.
red and blue in what follows.) If any of the edges (r, s), (r, t), (s, t) are also blue then we have an entirely blue triangle. If not, then those three edges are all red and we have an entirely red triangle.
In combinatorics, Ramsey's theorem states that in any colouring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For two colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are coloured red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red.
Since this argument works for any colouring, any K6(a complete on 6 vertices with every pair connected) contains a monochromatic K3(a complete graph on 3 vertices with every pair connected), and thereforeR(3,3) ≤ 6. The popular version of this is called the theorem on friends and strangers. In other words, the above simply says that if there are 6 people, then either 3 of them are mutual friends or 3 of them are mutual strangers.
Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds.
b. General Cases However, the main part of the Ramsey theorem does not lie on giving an exact number of what R
a. A Simple Example
(3, 3) is or what R(r, s) is for any given r, s. In-
It is easier to understand this theorem at first by
stead, it gives R(r, s) with inequalities.
starting with some small number, perhaps 3 in this case. In the following example, the formula R (3,3) provides a solution to the question which asks the minimum number of vertices a graph
For the 2-colour case, we have R(r, s) ≤ R(r − 1, s) + R(r, s − 1).
must contain in order to ensure that either (1) at least 3 vertices in the graph are connected Or
For the general c-colour case, we have R(n1, ..., nc) ≤ R(n1, ..., nc−2, R(nc−1, nc)).
(2) at least 3 vertices in the graph are unconnected. Note that owing to the symmetrical nature of the problem space, R(r,s) is equal to R(s,r). Suppose the edges of a complete graph on 6 ver-
These inequalities only give restrictions to the
tices are coloured red and blue. Pick a vertex v.
1. Probability facts on winning a lottery
There are 5 edges incident to v and so (by the
Probability of Death (1):
pigeonhole principle) at least 3 of them must be the same colour. Without loss of generality we
You’re far more likely to suffer a horrible death than to win a lottery. The statistics say that, when
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range of the values that R(r, s) can take. Then you may ask: then what exactly are R(4, 5), R(6, 9) and R(13,14)? The answer is that R(4, 5) equals 18 while the exact values for R(6, 9)and R
Proof without words By Kwan Chun Yin
(13,14) are still unknown. So far, the best result is that R(4, 5) equals 18, with R(5, 5) only known to be between 43 and 49 (inclusive) at this stage. That’s what many mathematicians specialized in combinatorics are concerned about and finding
Consider the following case. A regular hexagon (regardless of its size) is cut out of a triangular grid.
such values for R(r, s), called Ramsey numbers, is the core of the Ramsey theory.
3. Last words: Don’t get puzzled at our Day Camp 2014! Next time when you come to our day camp on 23rd and 27th August and are assigned into a group of 6, please don’t be surprised when the committees say that they know in each group
Then, it is tiled with a special kind of rhombi (pairs of equilateral triangles glued together along an edge). This kind of rhombi has 3 varieties, that is, 3 different orientations when it is used to tile the whole hexagon.
either 3 of you are mutual friends or 3 of you are mutual strangers. The above just told you why:)
It is asked to prove that each type of rhombi must exist in exactly the same number in order to tile the whole hexagon. The following picture demonstrates a particular case.
In fact, the above picture is the cover of the book “Mathematical Puzzles: A connoisseur's collection” by Peter Winkler, who is a research mathematician working in different fields of mathemat-
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ics is produced, giving viewers an impression of
The sum of the numbers in all triangles is 4,
a 3-dimensional object. Therefore, it should be
hence the number of vertical rhombus for filling
noted that, the red and gray rhombi has the same
in this hexagon must be 4. Similarly, by rotating
orientation, while the brown and deeper gray
the hexagon with another line as the base, one of
rhombi shares another orientation, finally, the
the types of horizontal rhombi will become the
black rhombi has a different orientation from the
vertical ones, and it can be shown, using the same
former two, making the hexagon tiled with
method, that there are 4 such type of rhombi.
rhombi of 3 different orientations and ‘proving’
Therefore, each type (orientation) of rhombi
the statement without words.
must exist in the same number in order to tile a regular hexagon. So here is the proof of a simple
However, we are still not sure if the three different types of rhombi exist in the same amount in
mathematics puzzle and there are still more proofs for you to find them out.
every case. Here is a more general approach to tackle the problem “with some words”.
Consider the following hexagon. We can see that it is divided into four rows of equilateral triangles. Then, the first row is labeled 2, the second row is labeled 1, the third row is labeled 0, and the last row is labeled -1. After that, the triangles pointing upwards (△) are labeled with its row number whereas the inverted triangles (△) are labeled with the negative of its row number.
After labeling every triangle, the hexagon is then tiled with the 3 kinds of rhombi (formed by two equilateral triangles), which are:
It can be found that the sum of numbers obtained by each vertical rhombus must be 1, and that obtained by each horizontal rhombus must be 0.
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Fun With Mathematics By Kwan Chun Yin
A woman went to the lottery shop to buy some lottery tickets; she spoke to the boy there, “please, I would like to buy two lottery tickets.” He handed her two and replied, “Thanks, ma’am, come again
compared with winning the lottery, you are 33 next week.” times as likely to be killed by bees, 50 times as likely to be struck by lightning, 8,000 times as
likely to be murdered (that increases after you 3. Comics win the lottery) and 20,000 times as likely to be killed in a car crash. You are even more likely to die in a freak fireworks accident than win millions.
Probability of Death (2): According to some expert mathematical statisticians, the odds of winning a lottery jackpot are so low that actually on your way to buy your ticket gives you a statically higher chance of dying than winning the top prize. But of course, if you died on your way to or from the lottery shop, your chance of winning the money automatically drops to zero.
2. Probability jokes Joke 1:
Acknowledgement Mr Chan Long Tin
Publication Team Andrew Ying // Brian Lui // Eileen Tam // Ian Kwan // Irene Kwok // Jeff Siu // Kwan Chun Hin // Tsang Chi Cheuk
Need your contributions The HKJSMS welcomes contributions to the upcoming issues of our newsletters from teachers and fellow students of our member schools. Please feel free to send your submission, such as event write-ups, articles or jokes related to Mathematics, to jsms.hk@gmail.com. We need your work to make the newsletter more rich!
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