HKJSMS Newsletter - Jan 2011
香港聯校數學學會
Hong Kong Joint School Mathematics Society
jsms.hk@gmail.com| http://www.hkjsms.org
Newsletter - Jan 2011
Words from the editor
Au Lawrence Dear readers, this is the second issue of the newsletter. In this issue, there are two articles written by two of our officials. At the end of this newsletter, the problem corner containing a few interesting problems for you to try. If there any contents that you would like us to include in future issues of our newsletter, feel free to tell us through email. I hope you enjoy reading our newsletter.
The Experience
Fung Dalton Yin-Nam In this small world, there is a boy who has an extraordinary passion towards mathematics. Looking back, he has been attending mathematics competitions for about eight years already. Eight years of hard work, and eight years of fun. When he was still studying in primary school, he joined a mathematics competition by chance and unexpectedly won a prize. From then on, most of his time has been given to the world of mathematics. He was quite arrogant in his primary school life on the grounds that he believed his knowledge in mathematics was a lot more superior to his friends. He had started learning trigonometric functions (for instance sine, cosine and tangent) in primary six and he was so proud of himself, not until his experience in secondary one. He had a new life in his secondary school: since no one else really knew he was good at mathematics, he treasured every chance to show his ability and wanted to gain acceptance, appreciation and compliments. Right. One day he met a chance of a lifetime to showcase. His friend challenged him with a question: Expand (x+y)8. For sure, this is rather easy for him now; but then he wrote (x+y)8=x8+y8 on the blackboard, which he knew immediately it was wrong by putting x=y=1 but then he was not able to give a correct answer. The shame and embarrassment came right to him at that very moment in front of his classmates. And he knew his attitude was wrong. Beside from the fact that he started to change his attitude, he also realized many of his friends who used to compete in mathematics competition had already left the path. Though he felt pitiful for them, he had no choice but to continue his mathematical journey on his own. Two years later he was in form three. He attended the Po Leung Kuk invitational mathematics competition, which was considered one of the highlights in form three, and managed to get a gold medal and join the training session. He particularly enjoyed the sessions led by his former tutorial teacher and he wholeheartedly appreciated the efforts paid by him when he was on a wheelchair due to an injury in his leg.
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HKJSMS Newsletter - Jan 2011 Unfortunately, even though he tried his very best, he was not qualified to go to South Africa that year for the international competition; worse still, as an old saying goes, “when it rains, it pours�, his teacher passed away with a stroke in Sep 2009 and he was not with him at the critical moment due to an exam. This had a great impact on him and it took him weeks to recover from the depression and sadness. Last academic year he was in form four and he had been learning so much more: not only mathematics, but also the attitude towards life. He had learnt to accept his incapability to solve some difficult mathematics questions and he knew failure was the stepping stone to success. He had learnt to accept and appreciate others, and discuss unsolved problems with them. He had learnt to explore in the wonderful and astonishing mathematical world as he knew there was always room of improvement for him. His friends and he organized a mathematics core team in his school and started formal training sessions. He spent a day in every week to find training questions and prepare a problem set for the core team members. He insisted despite the frustration of the members not paying attention to training sessions and not handing in problem sets. He was not able to join the Hong Kong Mathematical Olympiad for some reasons; he was both delighted and jealous when he was in the school’s celebration party for the champion at the very end. That was, sadly, his second unforgettable and lifelong regret. Not able to withstand the intense pressure in his academics, he made a tough decision to continue his studies in another school where he skipped year eleven, or equivalently, form five. And with this reason, he was not able to represent Hong Kong and join the China Western Mathematical Olympiad. He would never be able to forget the disappointment when he was informed that the chance slipped away in silence from his hands. Till today, mathematics still gives him a guide of establishing a positive and constructive attitude towards life, events and people. Mathematics is a light tower for him, and leads him to his goals. I bet, you know, that I am the boy. Remember: It is not only the mathematical knowledge you should learn in this mathematical world, but also the right attitude towards people and events in various situation. He gives his utmost appreciation to Henson his classmate, Mr. Ng Chung Chun his teacher, and of course, JSMS for giving him this opportunity to express himself.
Logic and Fallacies
Leung Ka Pui Lyanne Logic is an instrument used for bolstering a prejudice. It is essential in mathematics to have a logical argument. The existence of fallacies always triggers people to come to the wrong conclusion. However, fallacies can be beautiful and grand. One of the first paradoxes was suggested by Zeno. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold the lead. Dear readers, our common sense tells us that this is not true. Yet, where exactly is the flaw? It is, in fact, about the divisibility of time and spaces. Controversies similar to this have lasted throughout history. The refutation of these paradoxes had forced mathematicians to think carefully about their assumptions in dealing with different mathematical concepts.
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HKJSMS Newsletter - Jan 2011 As time passes by, there are still a lot of common fallacies nowadays. One of them is the dissection fallacy. An easy example is shown below. The original figure (A) is a square of 64 square units. It can be rearranged into two other figures (B and C) of 65 and 63 square units respectively. The pieces are cut so that the missing or exceeding area is hidden by tiny, negligible imperfections of shape. However, a closer look at the slanted sides of the trapezoidal and triangular pieces shows that they cannot be aligned in the fallacious illustrations shown.
Apart from the dissection fallacy, there are also fallacies related to algebra. A simple one is to prove that all people have the same age. Let P(1) be the statement “In any group of n people, everyone in that group has the same age�. Consider P(1), in any group that consists of one person, everyone in the group has the same age because there is just one person. Therefore, P(1) is true. The next stage of an induction argument is to prove that whenever P(k) is true, then P(k+1) is true. Assume that P(k) is true for some k. Let there is an arbitrary group of k+1 people. We have to show that if person A and person B are any members of this arbitrary group, then they are of the same age. Consider everyone in this group without person A, these k people are of the same age because of the induction assumption. Similarly, in this group with person B, the remaining people are also of the same age. Let person C be someone else in the group who is not person A or person B. Since person A and person C are of the same age and person B and person C are of the same age, person A and person B must also be of the same age. Since any two people in a group of k+1 people have the same age, it follows that everyone in the group has the same age. Therefore, by induction, everyone has the same age! The above induction proof does sound ridiculous, but which is the step that went wrong? Taking a closer look at the proof and thinking more deeply into the original concept of mathematical induction, you may have noticed that assuming that there is a person R in the group of k+1 people is the major flaw of the whole proof. There might not be anyone else there! Therefore, what we have proved is P(1) is true and P(2) implies P(3) which implies P(4) and so on. However, P(1) does not imply P(2) and this is the reason why the proof is fallacious.
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HKJSMS Newsletter - Jan 2011 Yet, the idea of fallacy does not only limit to mathematics. Paradoxes always lie in simple chats. Your teacher might have told you that there will be a sudden quiz next week. However, is this possible to have a sudden quiz that is announced in advance? If the quiz is not given by Friday, then it is pretty obvious that the quiz will not be “sudden” anymore as everyone would know that it will be given on Friday. Therefore, the quiz cannot be held on Friday. Moreover, if the quiz is not given by Thursday, since the quiz cannot be held on Friday, then the students would know it must be held on Thursday. The same logic can be applied to each day of the week and there would be no “sudden quiz” after all. However, surprisingly, in reality, we are always caught off guard by “sudden quizzes”! Fallacies are fun and it played an important role in the development of mathematics back in time of Aristotle. It is the foundation of the formation of syllogistic reasoning. People often say that by being logical, we would be able to spot out fallacies. Yet, “logical” itself is quite an abstract idea. What does it mean to be logical? In this case, it probably implies knowing how to trace back to the unnecessary assumptions made in every line. All in all, an in-depth look into the study of fallacies is totally worthwhile. Not only can they be used as powerful arguments, they can also be used in jokes! Do pay attention to the things around you, paradoxes are everywhere!
Problem Corner 1.
A triangle is formed by picking three vertices form a regular octagon at random. What is the probability that the triangle is right-angled?
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Find a 9-digit number, A, that satisfies both of the conditions: (i) A contains all of 1,2,3,4,5,6,7,8 and 9 in its digits; (ii) for n = 1,2,3,...,9 , the number made up by A’s first n digits can be divided by n.
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(9th Pui Ching Invitational Mathematics Competition) In a classroom, the teacher said to five students, Alan, Bob, Carl, Dick and Eason, ‘I have written down a five-digit number N which is made up of five different digits. I will let Alan see the ten thousands and thousands digits of N, let Bob see the thousands and hundreds digits, let Carl see the hundreds and tens digits, let Dick see the tens and unit digits and let Eason see the unit and ten thousands digits.’ The teacher then let each student know two digits of N as said, and then everybody sat in a circle and started the following conversation. ‘Raise your hands if you know a prime factor of N,’ said the teacher, and then two students raised their hands. ‘Raise your hands if you know a prime factor of N,’ asked the teacher again, and this time three students raised their hands. ‘Raise your hands if you know a composite factor of N,’ the teacher continued, and then two students raised their hands. ‘Raise your hands if you know two composite factors of N,’ said the teacher, but no student raised their hands. Then the teacher asked, ‘who knows the value of N?’ One student said, ‘I know. N is a multiple of 59.’ Assuming all students to be clever (which means that deductions can be made whenever sufficient information is given), find N.
Answers: 1. 3/7
2. 381654729
3. 50268
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