# Year Book

## Organization

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2018 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Zhouping Xin 2004 Morningside Medal of Mathematics Gold Medalist The Chinese University of Hong Kong Professor Tony F. Chan, JP King Abdullah University of Science and Technology Professor Shiu Yuen Cheng Tsinghua University Professor Lawrence C. Evans University of California, Berkeley Professor David Gabai Princeton University Professor Brendan Hasset Brown University Professor Bong Lian Brandeis University Professor Rafe Mazzeo Stanford University Professor Ngaiming Mok The University of Hong Kong Professor Viet Trung Ngo Vietnam Academy of Science and Technology Professor Raman Parimala Emory University Professor Hyam Rubinstein University of Melbourne Professor Tom Yau Heng Wan The Chinese University of Hong Kong Professor Po Lam Yung The Chinese University of Hong Kong Professor Jun Zou The Chinese University of Hong Kong

## Screening Panel

The members of the Screening Panel of the 2018 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau Heng Wan The Chinese University of Hong Kong Dr. Ping Shun Chan The Chinese University of Hong Kong Dr. Man Chuen Cheng The Chinese University of Hong Kong Dr. Charles Chun Che Li The Chinese University of Hong Kong Dr. Mark Jingjing Xiao The Chinese University of Hong Kong Professor Po Lam Yung The Chinese University of Hong Kong

## Steering Committee

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2018 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shiu Yuen Cheng 2007 Chern Prize Recipient Tsinghua University Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Professor Kwok Wai Chan The Chinese University of Hong Kong Professor Raymond Chan The Chinese University of Hong Kong Professor Tony F. Chan, JP King Abdullah University of Science and Technology Professor Wing Sum Cheung The University of Hong Kong Mr. Siu Leung Ma, BBS, MH Fung Kai Public School Professor Ngaiming Mok The University of Hong Kong Professor Tai Kai Ng Hong Kong Academy of Gifted Education Professor Yang Wang The Hong Kong University of Science and Technology Ms. Susan Wong Hang Lung Properties Limited Professor Zhouping Xin The Chinese University of Hong Kong Dr. Chee Tim Yip Princeton (Shenzhen) International School

## Executive Committee

The members of the Executive Committee of the 2018 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Dr. Kai Leung Chan The Chinese University of Hong Kong Professor Kwok Wai Chan The Chinese University of Hong Kong Professor Ka Luen Cheung The Education University of Hong Kong Dr. Chi Hin Lau The Chinese University of Hong Kong Secretariat: Ms. Aggie So Ching Law* Ms. Judy Wing Lam Chu Ms. Serena Wing Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong

*Note: Ms. Law participated up to October 2017.

## Winners of the 2018 Hang Lung Mathematics Awards

### GOLD

 Topic On the Trapezoidal Peg Problem among Convex Curves Team Member Zhiyuan Bai Teacher Mr. Pui Keung Law School La Salle College Abstract The Trapezoidal Peg Problem, as one of the generalizations of the famous Square Peg Problem, asks when a prescribed trapezoid can be inscribed in a given Jordan curve. We investigated a possible approach towards the problem by first weakening the similarity condition, in which we have shown that for any trapezoid, some classes of convex curves can actually inscribe, up to two kinds of weaker forms of similarity, infinitely many trapezoids. Our main theorem further analyzed the properties of one of these infinite family of trapezoids, and showed that any given trapezoid can be uniquely inscribed in any strictly convex C1 curve, which we named ‘oval’, up to only translation and a kind of transformation, which we called ‘stretching’, but without rotation, and the resulting trapezoid moves continuously when the given trapezoid rotates. Through this, we consequently obtained a necessary and sufficient condition for an oval to inscribe an arbitrary trapezoid up to similarity, which could be taken as an answer to the problem among ovals. Some other variations are also discussed.

### SILVER

 Topic Containing Geometric Objects with Random Inscribed Triangles in a Circle Team Member Tsz Hin Chan Teacher Mr. Ho Fung Lee School Pui Ching Middle School Abstract In this paper, we aim to investigate the probability of an inscribed triangle in a given circle containing certain geometric objects. Our paper is motivated by a Putnam problem in 1992. We study three generalizations in containing: (i) an arbitrary point, (ii) an arbitrary line segment which lies on a diameter, and (iii) a concentric circle. For the case of an arbitrary point, a closed form expressed by the Spence’s function is obtained. For the case of an arbitrary line segment, we use numerical approximations to calculate the probability, namely the trapezoidal rule and the Monte Carlo integration. For the case of a concentric circle, we successfully find an explicit formula that depends on the radius of the concentric circle.

### BRONZE

 Topic On the Divisibility of Catalan Numbers Team Member Tsz Chung Li Teacher Dr. Kit Wing Yu School United Christian College Abstract In this paper, we propound an efficacious method to derive the p-adic valuation of the Catalan number by analyzing the properties of the coefficients in the base p-expansion of n. We unearth a new connection between those coefficients and the p-adic valuation of the Catalan number. In fact, we have discovered that the highest power of p dividing the Catalan number is relevant to the number of digits greater than or equal to half of p + 1, the nature of distribution of digits equal to half of p – 1, and the frequency of carries when 1 is added to n. Meanwhile, we remark that the method we apply is more natural than the current way used by Alter and Kubota, which is quite artificial. Applications, examples of our new formula, and details about Catalan numbers are also included in this paper.

### (Arranged by school name and research topic in alphabetical order)

 Topic Doing Indefinite Integrals without Integration Team Member Chun Szeto Teacher Mr. Alexander Kin Chit O School G.T. (Ellen Yeung) College Abstract Residue Theorem has been frequently used to tackle certain complicated definite integrals. However, it is never applied to indefinite integrals. Therefore, in this report, Residue Theorem and some small tricks are applied to find antiderivatives. There are mainly three interesting results: Antiderivatives can be found without integration: antiderivatives can be represented by residues, while calculation of residues requires no knowledge of integration. A universal functional form of antiderivatives can be obtained: the antiderivative obtained by this method has a functional form that converges wherever it should converge. The functional form has the largest possible region of convergence on the complex plane. As a weak tool for analytic continuation: since the universal functional form of an antiderivative is obtained, differentiating yields a universal functional form of the integrand. Topic A Generalization of the Gauss Sum Team Member Ho Leung Fong Teacher Mr. Hing Pan Fong School Hoi Ping Chamber of Commerce Secondary School Abstract This essay will analyze a function that is a generalization of the Gauss sum. The function happens to be closely related to the cycle index of the symmetric group, which will also be analyzed. Some properties of the Gauss sum will be generalized. A number-theoretic inequality is also obtained. Topic A Markov Model of the Busy Footbridge Problem Team Members Lok Kan Yuen, Omega Nok To Tong, Ethan Lok Kan Tsang Teacher Mr. Ho Fung Lee School Pui Ching Middle School Abstract The central problem we are investigating is based on a problem from the 2018 Singapore International Mathematics Challenge. It is about a mathematical model of the probabilities that the people on a footbridge from two sides meet. In our paper, we generalize the contest problem in various cases. We develop a Markov model, and then formulate a transition matrix to solve the generalized problem. Also, we define an expansion rule of the transition matrices to reduce the time complexity to compute. Furthermore, we propose a new topic on the expected number of collisions. We tackle the problem by performing Jordan decomposition. Lastly, we optimize the method of finding eigenvalues by observing the recursive relationships in transition matrices. Topic Investigation on Mordell’s Equation Team Member Tin Wai Lau Teacher Mr. Ho Fung Lee School Pui Ching Middle School Abstract This paper aims to investigate the integral solutions of the Mordell’s Equation y2 = x3 + k for a particular class of integers k. We employ some classical approaches, i.e. factorization in number fields and quadratic reciprocity. When k = p2 for certain primes p, we can determine the set of solutions. The equation for two other classes of integers k are also solved in this paper. Topic Old and New Generalizations of Classical Triangle Centres to Tetrahedra Team Members Trevor Kai Hei Cheung, Hon Ching Ko Teacher Mr. Pak Leong Cheung School St. Paul’s Co-educational College Abstract The classical triangle centres, namely centroid, circumcentre, incentre, excentre, or thocentre, and Monge point, will be generalized to tetrahedra in a unified approach as points of concurrence of special lines. Our line characterization approach will also enable us to create new tetrahedron centres lying on the Euler lines, which will be a family with nice geometry including Monge point and the twelve-point centre. Two tetrahedron centres generalizing orthocentre of triangles from new perspectives will be constructed through introducing antimedial tetrahedra, tangential tetrahedra, and a new kind of orthic tetrahedra. The first one, defined as the circumcentre of the antimedial tetrahedron of a tetrahedron, will be proven to lie on the Euler line. The second one, defined as the incentre or a suitable excentre of the new orthic tetrahedron of a tetrahedron, will be discovered to be collinear with its circumcentre and twenty-fifth Kimberling centre X25. Surprisingly, these two differently motivated geometric generalizations turn out to have analogous algebraic representations. A clear definition of tetrahedron cent res, as a generalization of triangle centres to tetrahedra, will be coined to set up a framework for studying the analogies between geometries of triangles and tetrahedra. The fundamental properties of tetrahedron centres will be studied.

### 2018 Award Ceremony Video

Young talents were recognized by a Gold, a Silver, a Bronze and five Honorable Mentions.
They received the trophies from world class scholars.
Many guests shared their joy and honor.

## ORGANIZATION

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2004

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2004 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing Tung Yau The Chinese University of Hong Kong and Harvard University Professor John Coates Cambridge University Professor Tony Chan University of California, Los Angeles Professor Shiu Yuen Cheng The Hong Kong University of Science and Techology Professor Kulkarni Harish-Chandra Research Institute Professor Ka Sing Lau The Chinese University of Hong Kong Professor Jun Li Stanford University Professor Chang-Shou Lin National Chung Cheng University Professor Ngai Ming Mok The University of Hong Kong Professor John Morgan Columbia University Professor Duong Phong Columbia University Professor Dan Stroock Massachusetts Institute of Technology Professor Tom Wan The Chinese University of Hong Kong Professor Lo Yang Institute of Mathematical Sciences, Academic Sinica Professor Andrew Yao Center for Advanced Study, Tsing Hua University

## Screening Panel 2004

The members of the Screening Panel of the 2004 Hang Lung Mathematics Award are:

 Chair: Professor Tom Yau-heng Wan The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong

## Steering Committee 2004

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educators and heads of mathematics departments  in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards  committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2004 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing Tung Yau Director, The Institute of Mathematical Sciences, CUHK and Professor, Harvard University Ms Susan Wong Representative from Hang Lung Properties Mr. Bankee Kwan Chairman, Celestial Asia Securities Holdings Ltd Professor Hung Hsi Wu Professor, Mathematics Department, UC Berkeley Professor Lo Yang Institute of Mathematical Sciences, Academic Sinica Mr. Siu Leung Ma Mr. Chun Kau Poon Former principal, St. Paul Co-educational College Mr. Chee Tim Yip Principal, Pui Ching Middle School Professor Shiu Yuen Cheng Chairman, Mathematics Department, HKUST Professor Ka Sing Lau Chairman, Mathematics Department, CUHK Professor Man Keung Siu Chairman, Mathematics Department, HKU Professor Thomas Au Program Chairman, EPYMT, CUHK

## Executive Committee 2004

The members of the Executive Committee of the 2004 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-keung Au The Chinese University of Hong Kong Academic Resource Center: Dr. Leung-fu Cheung The Chinese University of Hong Kong Academic Resource Center: Dr. Ka-luen Cheung The Chinese University of Hong Kong Secretariat: Ms Serena Yip The Chinese University of Hong Kong

2018 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## WINNERS of the 2004 Hang Lung Mathematics Awards

### GOLD

 Topic: Marked Ruler as a Tool for Geometric Constructions – from angle trisection to n-sided polygon Team Members: Edward Sin Tsun Fan School: Sha Tin Government School

### SILVER

 Topic: Further Investigation on Buffon’s needle problem Team Members: Fan Fan Lam, Ho Fung Tang, Ho Yin Poon, Lok Hin Yim, Yiu Tak Wong School: Munsang College (Hong Kong Island)

### BRONZE

 Topic: 畫鬼腳 Team Members: Ting Fai Man, Hoi Kwan Lau, Shek Yeung, Man Kit Ho School: Sha Tin Government Secondary School

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic: Deconstructing Kafka Team Members: Yuk Hing Chiu, Man Wai Ho, Hoi Yin Hui, Yu Hang Lee School: SKH Lam Woo Memorial Secondary School Topic: 連成積不能成方的探究 Team Members: Chit Ma, Ho Man Lam, Wing Hei Kwok, Kai Chung Wan, Shun Yip School: Shatin Pui Ying College Topic: Illumination Problem Team Members: Wang Hei Ku, Hou Chen Lee, Wai Hung Chan, Tak San Fong, Hoi Cheung Cheung School: Shatin Tsung Tsin Secondary School Topic: 商場中電梯分佈的最優化 Team Members: Hing Ming Chan, Wilson Sze Ngo Chan, Kok Leung Fung, Ho Ko, Man Kin Leung School: Hong Kong Chinese Women’s Club College

### (NON-MONETARY CERTIFICATE)

 Topic: The Construction of South-Western HK Island Line Team Members: Ming Chung Lau, Tak Man Lee, Tsz Yeung Suen, Chak Kwan Tam, To Man Wong School: Sing Yin Secondary School Topic: Perimeter of Ellipse and Generalization in n-Dimensional Case Team Members: Ching King Chan, Ka Leung Chu School: Yuen Long Merchants Association Secondary School

## Finalist Teams Selected for the Oral Defense at the 2018 Hang Lung Mathematics Awards

### (Arranged by school name and research topic in alphabetical order)

 Solving Cubic Pell’s Equation by Bifurcating Continued Fraction   Team Member: Ka Lam Wong Teacher: Mr. Yiu Chung Leung School: Bishop Hall Jubilee School Heine-Cantor Theorem, Lebesgue’s Number and Compactness Team Member: Kin Fung Wong Teacher: Dr. Chi Kwan Leung School: Cognitio College (Kowloon) Doing Indefinite Integrals without Integration Team Member: Chun Szeto Teacher: Mr. Alexander Kin Chit O School: G.T. (Ellen Yeung) College Finding the Expected Number of Random Reversals to Sort a Permutation Using Matrix Equation for Application in Genetics  Team Members: Man Hon Fan, Kwok Yan Lo Teacher: Mr. Ho Cheung Lai School: HKUGA College A Generalization of the Gauss Sum Team Member: Ho Leung Fong Teacher: Mr. Hing Pan Fong School: Hoi Ping Chamber of Commerce Secondary School On the Trapezoidal Peg Problem among Convex Curves Team Member: Zhiyuan Bai Teacher: Mr. Pui Keung Law School: La Salle College A Markov Model of the Busy Footbridge Problem Team Members: Lok Kan Yuen, Omega Nok To Tong, Ethan Lok Kan Tsang Teacher: Mr. Ho Fung Lee School: Pui Ching Middle School Containing Geometric Objects with Random Inscribed Triangles in a Circle Team Member: Tsz Hin Chan Teacher: Mr. Ho Fung Lee School: Pui Ching Middle School From Close Match Problem to the Generation of Identities of Binomial Coefficients and Trigonometric Terms Team Members: Chi Hang Tam, Joshua Pui Sang Cheung, Chi Ki Ngai Teacher: Mr. Ho Fung Lee School: Pui Ching Middle School Investigation on Mordell’s Equation Team Member: Tin Wai Lau Teacher: Mr. Ho Fung Lee School: Pui Ching Middle School Graphical Approach to the Lonely Runner Conjecture Team Members: Ka Ho Mok, Sum Kiu Law, Ho Lam Wan, Cheuk Yin Lee Teacher: Mr. Pun Sin Chan School: S.K.H. Tsang Shiu Tim Secondary School Inradius Numbers and the Investigations on the Inradius Number Diophantine Equation Team Members: Long Kiu O, Hei Long Young Teacher: Ms. Jasmine Sze Lai Ku School: St. Joseph’s College Old and New Generalizations of Classical Triangle Centres to Tetrahedra Team Members: Trevor Kai Hei Cheung, Hon Ching Ko Teacher: Mr. Pak Leong Cheung School: St. Paul’s Co-educational College On the Divisibility of Catalan Numbers Team Member: Tsz Chung Li Teacher: Dr. Kit Wing Yu School: United Christian College On Length Preserving Curve Flow to Isoperimetric Inequality Team Member: Man Hei Law Teacher: Mr. Ching Ping Lam School: Wah Yan College, Hong Kong

## 2004 頒獎禮實錄

「今天真正的明星，不是誰，正是恒隆數學獎的各位得獎者。」

「恒隆數學獎的成績超出想像中的卓越。」

「天空海闊，未來是年青人的世界。」

「祝願各參賽同學能成長為香港、中國以至全球的傑出社會棟樑，成為出色的數學家、科學家、大商人，一起貢獻這個大同世界。」

「本人在此恭喜各位得獎者，並祝願香港的數學和不同學科的發展能更上一層樓。」

> 參賽者︰范善臻同學
> 領隊老師︰王徽女士
> 校長︰周金祥先生

2004 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## FINALIST TEAMS SELECTED FOR THE ORAL DEFENSE AT THE 2004 HANG LUNG MATHEMATICS AWARDS

### (arranged by school name in alphabetical order)

 Reconstruction of 3-Dimensional Model of Blood Vessels: Simple Idea with Great Impact! Buddhist Mau Fung Memorial College An interesting journey involving Chebyshev Polynomials with applications Buddhist Mau Fung Memorial College Predictable Model of SARS in Hong Kong Buddhist Mau Fung Memorial College Flow Cheung Sha Wan Catholic Secondary School 商場中電梯分佈的最優化 Hong Kong Chinese Women’s Club College Invigilators Allocation by Linear Programming Immanuel Lutheran College “Lift it up” – Mathematics project on comparing different lift system La Salle College Further investigation on Buffon’s needle problem Munsang College (Hong Kong Island) Marked Ruler as a Tool for Geometric Constructions – from angle trisection to n-sided polygon Sha Tin Government Secondary School 畫鬼腳 Sha Tin Government Secondary School 連乘積不能成方的探究 Shatin Pui Ying College Illumination Problem Shatin Tsung Tsin Secondary School The construction of South-Western HK Island Line Sing Yin Secondary School Deconstructing Kafka SKH Lam Woo Memorial Secondary School Perimeter of Ellipse and Generalization in n-Dimensional Case Yuen Long Merchants Association Secondary School

## Organization

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2016

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2016 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing Tung Yau 1982 Fields Medalist Harvard University The Chinese University of Hong Kong Professor Raymond Hon Fu Chan* The Chinese University of Hong Kong Professor Shiu Yuen Cheng Tsinghua University Professor Dejun Feng The Chinese University of Hong Kong Professor Wee Teck Gan National University of Singapore Professor Wei Ping Li The Hong Kong University of Science and Technology Professor Bong Lian Brandeis University Professor Chang Shou Lin Taiwan University Professor Ngai Ming Mok The University of Hong Kong Professor Ye Tian Chinese Academy of Sciences Professor Tom Yau Heng Wan The Chinese University of Hong Kong Professor Michael Zieve University of Michigan

*Note: Professor Chan was unable to join the Oral Defense and will be represented by Professor Dejun Feng.

## Screening Panel 2016

The members of the Screening Panel of the 2016 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau Heng Wan The Chinese University of Hong Kong Dr. Man Chuen Cheng The Chinese University of Hong Kong Dr. Chi Hin Lau The Chinese University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Charles Chun Che Li The Chinese University of Hong Kong

## Steering Committee 2016

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2016 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics The Chinese University of Hong Kong Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Professor Tony F. Chan The Hong Kong University of Science and Technology Professor Shiu Yuen Cheng Tsinghua University Professor Wing Sum Cheung The University of Hong Kong Professor Ka Sing Lau The Chinese University of Hong Kong Mr. Siu Leung Ma Fung Kai Public School Ms. Michelle Sau Man Mak Hang Lung Properties Limited Professor Tai Kai Ng The Hong Kong University of Science and Technology Professor Zhouping Xin The Chinese University of Hong Kong Dr. Chee Tim Yip Pui Ching Middle School

## Executive Committee 2016

The members of the Executive Committee of the 2016 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Dr. Kai Leung Chan The Chinese University of Hong Kong Dr. Ka Luen Cheung The Education University of Hong Kong Dr. Leung Fu Cheung The Chinese University of Hong Kong Secretariat: Ms. Aggie So Ching Law Ms. Konnie Wan Yu Pak* Ms. Serena Wing Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong

*Note: Ms. Pak participated up to August 2015

## Winners of the 2016 Hang Lung Mathematics Awards

### GOLD

 Topic On the Summation of Fractional Parts and its Application Team Members Sun Kai Leung Teacher Mr. Yiu Chung Leung School Bishop Hall Jubilee School Abstract The summation of fractional parts is an old topic in number theory since the time of G.H.Hardy and J.E.Littlewood (see [3]). Throughout the years, many mathematicians have contributed to the estimation of the sum $\sum_{n \leq N} \left\{\alpha n\right\}$ , where α is an irrational number. In Section 2, we estimate the fractional part sum of certain non-linear functions, which can be applied to refine an existing bound of the discrepancy. In Section 3, we continue to make use of the sum in order to study the distribution of quadratic residues and ‘relatively prime numbers’ modulo integers.

### SILVER

 Topic On the Iterated Circumcentres Conjecture and its Variants Team Members Tsz Fung Yu, Tsz Chun Wong, Janice Ling Teacher Mr. Ho Fung Lee School Pui Ching Middle School Abstract We study the Iterated Circumcentres Conjecture proposed by Goddyn in 2007: Let $P_1,P_2,P_3,\dotsc$ be a sequence of points in $R^d$ such that for every $i \geq d + 2$ the points $P_i-1,P_i-2,P_i-3,\dotsc,P_i-d-1$ are distinct, lie on a unique sphere, and further, Pi is the center of this sphere. If this sequence is periodic, then its period must be $2d + 4$. We focus on cases of $d = 2$ and $d = 3$ and obtain partial results on the conjecture. We also study the sequence and prove its geometrical properties. Furthermore, we propose and look into several variants of the conjecture, namely the Skipped Iterated Circumcentres Conjecture and the Spherical Iterated Circumcentres conjecture.

### BRONZE

 Topic A Geometric Approach to the Second Non-trivial Case of the Erdös-Szekeres Conjecture Team Members Wai Chung Cheng Teacher Ms. Dora Po Ki Yeung School Diocesan Girls’ School Abstract The Erdös-Szekeres conjecture, developed from the famous Happy-Ending Problem, hypothesizes on the number of points in general position needed on a plane to guarantee the existence of a convex n-gon. The research conducted aims to examine geometric characteristics of different constructions of points in general position, organized by number of points forming the convex hull of the set. This paper has explored the case of pentagons, reestablishing the previously proven result of the case using a geometrical approach in contrast to the combinatorial approaches generally adopted when exploring this problem. This paper also proves that the lower bound to the conjecture is not sharp under certain circumstances, an aspect never explored in the past.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic Congruences of Solutions of the Pell’s Equation Team Members Man Yi Kwok Teacher Mr. Kim Fung Lee School Baptist Lui Ming Choi Secondary School Abstract In this research, we are interested in how the solutions of the famous Pell’s equation look like. It is well known that the solutions of the Pell’s equation are generated by the fundamental solution of the equation, which could be represented by a set of recursive equations. Therefore, we would like to explore the characteristics of such recurrence sequences and tell the relationship between the cycle length of the congruence modulo a number and divisibility of the terms. Topic A Synthetic Approach on Studying the Mysterious Right Kite and its Applications on Cryptography in related to Poincaré Disk Model in the Views of Euclid Geometry Team Members Chit Yuen Lam, Christy Sze Wai Kok, King Chun Chan, Hin Tung Chung Teacher Mr. Tat Cheong Wong School G.T. (Ellen Yeung) College Abstract In this study, it gives a synthetic approach to the quadrilateral “Kite” and right kite. It mainly based on the definitions, postulates (axioms), propositions (theorem and constructions) from the Euclid’s Elements, which is known as one of the most successful and influential mathematical textbook attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt, c. 300 BC. Linked with the definitions of “Right Kite” and the lines which are that meet the boundary of a said circle orthogonally described in the Poincaré Disk Model, we attempt to combine it in a mathematical task namely “Cryptography”. The application of Poincaré Disk Model will be acted as a bridge to form a single key for encryption and decryption. Even the single common trick we use, it leads to infinite possibilities by experiencing various and distinct mathematical skills in cryptography. Last but not least, we would like to dedicate to the publish of Euclid’s Element and the discovery of Euclid’s Geometry so that we can admire the Beauty of Mathematics. Our ultimate goal is to lay the new insight into some of the most enjoyable and fascinating aspects of geometry regarding to the most unaware quadrilateral, Kite. Topic The Generalized Tower of Hanoi Problem Team Members Hoi Wai Yu Teacher Ms. Mee Lin Luk School La Salle College Abstract In this paper, we look into a generalized version of the well-known Tower of Hanoi problem. We will investigate the shortest methods of traversing between any two valid configurations of discs in the standard problem, as well as in some variants. Topic On Hilbert Functions and Positive-definite Quadratic Forms Team Members Chak Him Au Teacher Mr. Yan Ching Chan School P.L.K. Centenary Li Shiu Chung Memorial College Abstract In this project, we give an explicit construction of positive definite quadratic forms of arbitrary dimension by using a family of real analytic functions whose coefficients in their Taylor expansions are strictly positive. We also prove a variant result that allows the construction if the number of positive coefficients has a positive upper density. Topic Triples of Sums of Two Squares Team Members Kin Ip Mong, Chun Ming Lai, Siu Hong Mak Teacher Mr. Chun Yu Kwong School Wong Shiu Chi Secondary School Abstract In 1903, an anonymous reader submitted a question to Mathematical Questions in The Educational Times: Find all consecutive triples of sums of two squares. J.E. Littlewood later posed a question on whether in general there exist infinitely many triples $n, n + h, n + k$ that are simultaneously sums of two squares? By solving the equation a $a^2 + 2 = (a – l)^2 + b^2$, we give all consecutive triples of sums of two squares such that the first number is a perfect square. This method is generalised to solve Littlewoods problem for the case when $h$ is a perfect square. We also prove that there are infinitely many pairs of consecutive triples of sums of two squares such that the first numbers of the two triples differ by 8.

## ORGANIZATION

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2006

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2006 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Tony F. Chan University of California, Los Angeles Professor Shiu Yuen Cheng The Hong Kong University of Science and Technology Professor John Coates Cambridge University Professor Jean Pierre Kahane Universite Paris-Sud Orsay Professor Ka-Sing Lau Chinese University of Hong Kong Professor Peter Lax Courant Institute, New York University Professor Chris Lennard University of Pittsburgh Professor Kenneth Millett University of California, Santa Barbara Professor Ngaiming Mok University of Hong Kong Professor Cathleen Morawetz Courant Institute, New York University Professor Gilbert Strang Massachusetts Institute of Technology Professor Robert Strichartz Cornell University Professor Tom Yau-heng Wan Chinese University of Hong Kong Professor Lo Yang Institute of Mathematical Sciences, Academic Sinica Professor Andrew Chi-Chih Yao Centre for Advanced Study, Tsing Hua University

## Screening Panel 2006

The members of the Screening Panel of the 2006 Hang Lung Mathematics Award are:

 Chair: Professor Tom Yau-heng Wan The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong

## Steering Committee 2006

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2006 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James Mirrlees 1996 Nobel Laureate in Economics Professor Thomas Kwok-keung Au The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Ka-Sing Lau Chairman, Mathematics Department, CUHK Professor Jian Shu Li The Hong Kong University of Science and Technology Mr. Siu Leung Ma CEO, Fung Kai Public Schools Mr. Chun Kau Poon St. Paul Co-educational College (retired) Ms Susan Wong Hang Lung Properties Professor Lo Yang Deputy Director, Morningside Center of Mathematics, Chinese Academy of Sciences Mr. Chee Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2006

The members of the Executive Committee of the 2006 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-keung Au The Chinese University of Hong Kong Dr. Leung-Fu Cheung The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Secretariat: Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong

### 2016 Award Ceremony Video

Young talents were recognized by a Gold, a Silver, a Bronze and five Honorable Mentions.
They received the trophies and certificates from world class scholars.
Many guests shared their joy and honor.

### Presentation of Souvenirs

##### Mr. Ronnie C. ChanChairman, Hang Lung Properties Limited

2016 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## Finalist Teams Selected for the Oral Defense at the 2016 Hang Lung Mathematics Awards

### (arranged by school name in alphabetical order)

 Congruences of Solutions of the Pell’s Equation Baptist Lui Ming Choi Secondary School On the Summation of Fractional Parts and its Application Bishop Hall Jubilee School A Geometric Approach to the Second Non-trivial Case of the Erdös-Szekeres Conjecture Diocesan Girls’ School A Synthetic Approach on Studying the Mysterious Right Kite and its Applications on Cryptography in related to Poincaré Disk Model in the Views of Euclid Geometry G.T. (Ellen Yeung) College On Family of Triangles – from Medians to Concurrent Lines and Angle Bisectors Hong Kong Chinese Women’s Club College The Generalized Tower of Hanoi Problem La Salle College Are Gray Code and Gros Sequence the Solution of Chinese Ring? Maryknoll Fathers’ School On Hilbert Functions and Positive-definite Quadratic Forms P.L.K. Centenary Li Shiu Chung Memorial College On the Iterated Circumcentres Conjecture and its Variants Pui Ching Middle School Rational Distance in Rational-sided Triangles Pui Ching Middle School Voting Power Queen Elizabeth School Triples of Sums of Two Squares Wong Shiu Chi Secondary School

## ORGANIZATION

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2014

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2014 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing-Tung Yau Harvard University and The Chinese University of Hong Kong Professor Lars Andersson Albert Einstein Institute, Germany Professor Raymond Hon-Fu Chan* The Chinese University of Hong Kong Professor Tony F. Chan The Hong Kong University of Science and Technology Professor Shiu-Yuen Cheng Mathematical Sciences Center, Tsinghua University Professor Jaigyoung Choe Korea Institute for Advanced Study Professor Ingrid Daubechies Duke University Professor Ka-Sing Lau The Chinese University of Hong Kong Professor John M. Lee University of Washington Professor Ngai-Ming Mok The University of Hong Kong Professor Duong H. Phong Columbia University Professor Mark A. Stern Duke University Professor Ye Tian Chinese Academy of Sciences Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Michael Wolf Rice University

*Note: Professor Chan was unable to join the Oral Defense and will be represented by Professor Jun Zou.

## Screening Panel 2014

The members of the Screening Panel of the 2014 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Charles Chun-Che Li The Chinese University of Hong Kong Dr. Chi-Hin Lau The Chinese University of Hong Kong

## Steering Committee 2014

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society including leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2014 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics Master, Morningside College, CUHK Professor Thomas Kwok-Keung Au EPYMT and Mathematics, The Chinese University of Hong Kong Professor Tony F. Chan Vice-Chancellor, The Hong Kong University of Science and Technology Professor Shiu-yuen Cheng Professor, Tsinghua University Professor Wing-Sum Cheung Professor, Mathematics, The University of Hong Kong Professor Ka-Sing Lau Professor, Mathematics, The Chinese University of Hong Kong Mr. Siu-Leung Ma CEO, Fung Kai Public School Ms. Michelle Sau-Man Mak Chairman’s Office, Hang Lung Properties Limited Professor Tai-Kai Ng Associate Dean of Science, The Hong Kong University of Science and Technology Professor Zhouping Xin Professor, Mathematics, The Chinese University of Hong Kong Dr. Chee-Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2014

The members of the Executive Committee of the 2014 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Dr. Leung-Fu Cheung The Chinese University of Hong Kong Secretariat: Ms. Mandy Ka-Man Leung Ms. Konnie Wan Yu Pak Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong

## Winners of the 2014 Hang Lung Mathematics Awards

### GOLD

 Topic Investigation of the Erdös-Straus Conjecture Team Members Yuk Lun Fong Teacher Mr. Kwok Kei Chang School Buddhist Sin Tak College Abstract In this paper, we are going to investigate the ${\bf{\textit{Erdős-Straus Conjecture }}}$: For any positive $n \geq 2$, there exists positive integers $k,k_1,k_2$ such that $$\dfrac{4}{n} = \dfrac{1}{k}+\dfrac{1}{k_1}+\dfrac{1}{k_2}$$ Firstly, we will solve a simpler form $\dfrac{3}{n} = \dfrac{1}{x} + \dfrac{1}{y}$ as a starting point. Next we will investigate the Erdos-Straus Conjecture in the following dimensions: the related geometric representation of the Erdos-Straus Conjecture, the properties of solutions of the Erdos-Straus Conjecture, further investigation of some paper of the Erdos-Straus Conjecture, existence of special forms of solutions of the Erdos-Straus Conjecture, and the investigation of the Erdos-Straus Conjecture in algebraic dimension. The aim of this report is to find evidence that shows the ${\bf{\textit{Erdős-Straus Conjecture}}}$ is true. If evidence is not strong enough, we still hope that this report can make an improvement to the researched result at present.

### SILVER

 Topic Pseudo Pythagorean Triples Generator for Perpendicular Median Triangles Team Members Yan Lam Fan, Wai Pan Yik Teacher Mr. Ho Fung Lee School Pui Ching Middle School Abstract The problem of finding all integral sides and lengths of a right-angled triangle is famous and the solution set is called the Pythagorean Triple. Now, instead of the sides of a triangle, we concern ourselves with the orthogonality of lines from vertices to their opposite sides. We want to generalize the problem to the arbitrary rational ratio on the sides.

### BRONZE

 Topic Probability, Matrices, Colouring and Hypergraphs Team Members Hok Kan Yu, Dave Lei, Ka Chun Wong, Sin Cheung Tang Teacher Mr. Yan Ching Chan School P.L.K. Centenary Li Shiu Chung Memorial College Abstract In this project, we achieved various results using Probabilistic Methods. By exploiting the concept of probability and expected value, we managed to achieve three results: distribution of entries on a cube, colouring of vertices of a hypergraph and a lower bound of a maximal independent set on a hypergraph.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic Classification of Prime Numbers by Prime Number Trees Team Members Man Him Ho, Chun Lai Yip, Yat Wong, Yin Kei Tam Teacher Mr. Alexander Kin Chit O School G.T. (Ellen Yeung) College Abstract The traditional sieve of Eratosthenes gives a simple algorithm for finding all prime numbers. However, prime numbers seem to appear unpredictably but with regular population ratio in the ranges of integers, as Gauss had found a density function of prime numbers within a range of x. On the other hand, there are few methods of classification of prime numbers. We developed a new classification of prime numbers by prime number trees. In the prime number trees, the following number is generated by attaching a digit either 1, 3, 7, or 9 to the right hand side of the preceding prime number. If the number generated remains a prime, then the process continues, otherwise it stops. The prime number trees group prime numbers with similar digits together and show the elegance of a shorthand of prime numbers. This method also shows a regular classification of prime numbers. Topic Two Methods for Investigating the Generalized Tic-Tac-Toe Team Members Kam Chuen Tung, Luke Lut Yin Lau Teacher Ms. Mee Lin Luk School La Salle College Abstract In this paper, we look into the (m,n,k,p,q) game, one of the generalizations of the well-known Tic-Tac-Toe game. The objective of the game is to achieve ‘k-in-a-row’ with one’s pieces before one’s opponent does. We use two methods – exhaustion and pairing strategy – to investigate the results of the (m,n,k,p,q) game for several different values of the five parameters. Topic A General Formula to Check the Divisibility by All Odd Divisors and its Extensions Team Members Chun Kit Du, Tung Him Lam, Hok Leung Chan, Chi Ming Ng, Kai Yin Ng Teacher Mr. Kwok Tai Wong School S.K.H. Lam Woo Memorial Secondary School Abstract The paper places much emphasis on the method of checking, without using division, the divisibility of an integer by an odd divisor. In part A, it mainly focuses on getting the general way to perform the divisibility test by an algorithm using the unit digit and the rest of the truncated digits of the dividend. Parts B and C are extensions of part A. In part B, it attaches importance to using the last two or more digits of the dividend and so the divisibility test is not just restricted to the ones digit. While parts A and B focus on the method of verifying the divisibility of a number, part C mainly concentrates on finding the quotient without performing division algorithm. This unique method of division is discovered in the process of investigation in part A. Topic On the Geometric Construction of Triangles and the Algebraic Interpretation of the Notion Team Members Chi Cheuk Tsang, Ho Lung Tsui, Justin Chi Ho Chan Tang, Ian Yu Young Kwan Teacher Mr. Perrick King Bor Ching School St. Joseph’s College Abstract This study centers on the Euclidean construction of triangles under several given pre-conditions, and carries out several major objectives surrounding this aim: 1. to devise a scheme to primarily distinguish cases in which Euclidean construction is impossible; 2. to seek the simplest agenda in the construction of possible cases; 3. to give a strict definition of Euclidean constructability; and 4. to determine the methods and rigorous proofs of inconstructability. Topic The Application of Graph Theory to Sudoku Team Members Josephine Yik Chong Leung, Wai Shan Lui Teacher Mr. Tad Ming Lee School Ying Wa Girls’ School Abstract In this project, we establish the Sudoku graph by studying the relationship between Sudoku and graphs with the help of NEPS (Non-complete Extended P-Sum). The approach is to look for the chromatic polynomial of the Sudoku graph, so that we can determine the total number of possible solved Sudoku puzzles. Although the chromatic polynomial of the Sudoku graph is not presented in this research, we have found some properties of the polynomial that may provide inspirations for further research.

### 2014 Awards Ceremony Video

The 2014 Hang Lung Mathematics Awards winners were announced and recognized on December 11, 2014.  Eight awards were announced: a Gold Award, a Silver Award, a Bronze Award and five Honorable Mentions.

Winning Students, teachers, and schools were recognized on stage, and received crystal trophies and certificates from world renowned scholars.

### Presentation of Souvenirs

##### Mr. Ronnie C. ChanChairman, Hang Lung Properties Limited

2014 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the video to watch.

## FINALIST TEAMS SELECTED FOR THE ORAL DEFENSE AT THE 2014 HANG LUNG MATHEMATICS AWARDS

### (arranged by school name in alphabetical order)

 Probability Study using Matrix to Analyze Football Tournaments Baptist Lui Ming Choi Secondary School Investigation of the Erdös-Straus Conjecture Buddhist Sin Tak College Snake Chiu Lut Sau Memorial Secondary School Classification of Prime Numbers by Prime Number Trees G.T. (Ellen Yeung) College On the Investigation of Fundamental Solutions to the Pell Equation G.T. (Ellen Yeung) College Two Methods for Investigating the Generalized Tic-Tac-Toe La Salle College Probability, Matrices, Colouring and Hypergraphs P.L.K. Centenary Li Shiu Chung Memorial College Pseudo Pythagorean Triples Generator for Perpendicular Median Triangles Pui Ching Middle School A General Formula to Check the Divisibility by All Odd Divisors and its Extensions S.K.H. Lam Woo Memorial Secondary School Passing Through the Surface Shatin Pui Ying College More about a Finger-Counting Trick Sir Ellis Kadoorie Secondary School (West Kowloon) On the Geometric Construction of Triangles and the Algebraic Interpretation of the Notion of Constructability St. Joseph’s College How to Keep Water Cold II – A Study about the Wet Contact Surface Area in a Cube St. Stephen’s Girls’ College Rock-Paper-Scissors Tsuen Wan Public Ho Chuen Yiu Memorial College The Application of Graph Theory to Sudoku Ying Wa Girls’ School

## Organization

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2012

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2012 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing-Tung Yau Harvard University and The Chinese University of Hong Kong Professor Shiu-Yuen Cheng The Hong Kong University of Science and Technology Professsor Rafe Mazzeo Stanford University Professor Duong H. Phong Columbia University Professor Raymond H. Chan The Chinese University of Hong Kong Professor Reyer Sjamaar Cornell University Professor Ngai-Ming Mok The University of Hong Kong Professor Seiki Nishikawa Tohoku University Professor Ye Tian University of Science and Technology of China Professor Michael Zieve University of Michigan Professor Tom Yau-Heng Wan The Chinese University of Hong Kong

## Screening Panel 2012

The members of the Screening Panel of the 2012 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Charles Chun-Che Li The Chinese University of Hong Kong Dr. Chi-Hin Lau The Chinese University of Hong Kong

## Steering Committee 2012

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society including leading educators and  heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration ofthe  competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2012 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics Master, Morningside College, CUHK Professor Thomas Kwok-Keung Au EPYMT and Mathematics, The Chinese University of Hong Kong Professor Tony F. Chan Vice-Chancellor, The Hong Kong University of Science and Technology Professor Shiu-yuen Cheng Professor, Mathematics, The Hong Kong University of Science and Technology Professor Wing-Sum Cheung Professor, Mathematics, The University of Hong Kong Professor Ka-Sing Lau Professor, Mathematics, The Chinese University of Hong Kong Professor Zhou-Ping Xin Professor, Mathematics, The Chinese University of Hong Kong Professor Lo Yang Academician, Chinese Academy of Sciences Dr. Stephen Tommis Executive Director, HK Academy for Gifted Education Mr. Siu-Leung Ma CEO, Fung Kai Public Schools Ms. Carolina Yip Chairman’s Office, Hang Lung Properties Limited Mr. Chee-Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2012

The members of the Executive Committee of the 2012 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Dr. Leung-Fu Cheung The Chinese University of Hong Kong Secretariat: Ms. Mavis Kit-Ying Chan Ms. Mandy Ka-Man Leung Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong

## WINNERS of the 2012 Hang Lung Mathematics Awards

### GOLD

 Topic: Towards Catalan’s Conjecture Team Members: Chung Hang KWAN Teacher: Mr. Yat Ting TONG School: Sir Ellis Kadoorie Secondary School (West Kowloon) Abstract: The presented project aims at having an insight on one of the most famous, hard but beautiful problems in number theory─ Catalan’s Conjecture (This conjecture has become a theorem in 2002). Throughout this project, very advanced techniques and results established are avoided. Most of the results established in this report only require the concepts in elementary number theory, for example: divisibility and congruence. Yet, these techniques can be used delicately to establish a number of particular cases.

### SILVER

 Topic: Cutting Twisted Solid Tori (TSTs) Team Members: Yiu Shing WONG, Ho Yin LAU, Kai Lai CHAN, Kai Shing MOK, Tsz Nam CHAN Teacher: Mr. Sai Hung CHAN School: Sha Tin Government Secondary School Abstract: In the paper, we generalize cutting a Möbius strip and similar strips to a larger extent than Möbius, Listing, Ball-Coxeter and Fatehi’s papers. We generalize the object from a strip to a “twisted solid torus” (tst) and consider the result after cutting it. The Argand diagram has been used to describe lines in the cross section of tst. We have used a technique of checking the concurrence of lines defined by parametric equations by applying the concept of pole-polar duality. Euler’s celebrated formula on graphs has also been employed. Then we study the resultant objects formed from the cutting process and call them “knotted tst”. We then deduce a general formula for the number of different knotted tsts. After that, we consider the links that are formed from cutting tsts, which we call “tst links”. General forms of their braid words, Seifert matrices and Alexander polynomials are then deduced. Then we consider cutting a tst in the form of a non-trivial knot and study the resultant links. Finally, we study the cutting of combinations of more than one tsts in the form of virtual knots, which we call “tst products”, and derive a general formula for the result.

### BRONZE

 Topic: Complexity Reduction of Graphs Team Members: Tsam Kiu PUN Teacher: Mr. Cyril LEE School: St. Mary’s Canossian College Abstract: Many real-world problems can be modeled mathematically as graphs. Some of these graphs are complex because of their large number of vertices and edges. To develop applications over any of these graphs, a graph which is less complex but having characteristics similar to the original graph will always be very useful. We propose in this report a new graph reduction method by performing a singular value decomposition on the adjacency matrix of a complex graph. We also propose a notion of loop decomposition which is a generalization of graph triangulation, from which we also derive a measure of graph complexity.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic: Trajectories in Regular Pentagon Team Members: Yuk Kei LEUNG, Chun Shing WONG, Tsz Hei LAM, Ho Wai CHAN, Hiu Ying MAN Teacher: Mr. Kim Fung LEE School: Baptist Lui Ming Choi Secondary School Abstract: It is known that a light ray must obey the law of reflection when it is reflected by a plane mirror. In this report, we are going to find out whether a light ray in a regular pentagon1 formed by 5 congruent plane mirrors can go back to the starting position and what the possible emitting angles are. Also, we will investigate the looping of the light trajectory after finite reflection. First, we make an observation on some special cases. Then, we will consider the general cases and try to classify the looping trajectories. Properties of looping trajectories will be studied. Lastly, another approach, vectors, will be used to investigate this problem. Topic: A Study on Polyhedron with All Triangular Faces: Nine-point Circle Co-sphere Team Members: Yat Long LEE, Kwok Chung TAM Teacher: Mr. Lai Shun Nelson CHUNG School: Carmel Holy Word Secondary School Abstract: Previous articles have discussed about the properties of orthocentric tetrahedrons: nine-point circles on each face cospherical and the 3D Euler line. This paper aims at finding the sufficient and necessary conditions for the nine-point circles to be cospherical in the triangular polyhedrons. First, we discussed the conditions for the nine-point circles to be cospherical in a tetrahedron, in a hexahedron and in an octahedron. Next, we found that the 3Dorthocenter $H_C$ , the center of the 24-point sphere (48-point sphere) $N_C$ and the 3D circumcenter $O_C$ of a tetrahedron (an octahedron), if they exist, must be collinear and the ratio of the distance between them is $H_C N_C:N_C O_C = 1 : 1$. After studying the properties of triangular polyhedrons, we have found that the existence of the 3D orthocenter and the 3D circumcenter is the necessary condition for the nine-point circles to be cospherical. Topic: Manipulating the Fermat’s Equation Team Members: Long Hin SIN, Ka Kit KU, Wing Man CHIK, Ming Hong LUI Teacher: Mr. Yan Ching CHAN School: Po Leung Kuk Centenary Li Shiu Chung Memorial College Abstract: In our report, we will manipulate the Fermat’s Equation by allowing one of the exponents to be arbitrary. It turns out that if a prime base is restricted, there are either no solutions or a unique primitive solution, depending on the residue class that the prime belonging to modulo 4. Topic: From ‘Chopsticks’ to Periodicity of Generalized Fibonacci Sequence Team Members: Hui Hon Ka HUI, Kwun Hang LAI, Tin Chuen TSANG, Kin Lam TSOI, Cheong Tai YEUNG Teacher: Mr. Chi Keung LAI School: Shatin Pui Ying College Abstract: The ultimate objective of this paper is to examine the periodicity of the Generalized Fibonacci Sequence (GFS) modulo $j$ with different starting numbers. In this paper, we introduce a brand new method to study the period of the sequence inspired by the hand game ‘Chopsticks’ usually played in primary schools. We first prove that the period of GFS modulo a prime $p$ other than 5 is either half of the $p$-th Pisano Period or exactly equal to it in Theorem 16. We then investigate the decomposition from the period of the game modulo j to the least common multiple of the periods of the game modulo the primepower factors of $j$ in Theorem 23. We continue our investigation on the periodicity of GFS modulo $p$ other than 5 and prime powers $p^k$ in Corollary 18-20, Lemma 7 and Theorem 26. Finally, we use Theorem 27 to give a general expression for the period of GFS modulo $j$ in terms of the $p_i$-th Pisano period, where $p_i$’s are the prime factors of $j$. Topic: How to Cut a Piece of Paper – Making Paper Cones with the Greatest Total Capacity Team Members: Him Shek KWAN, Chin Ching WAN, Ka Chun LO Teacher: Mr. Chun Yu KWONG School: Wong Shiu Chi Secondary School Abstract: Given a regular polygonal paper inscribed in a unit circle, the paper is cut along its radii and each division (consisting of one or more sub-divisions) is made into a cone. These cones are allowed to be slanted to obtain a greater capacity. The purpose of this study is to maximize the total capacity of cones made from the paper over all ways of divisions. The methodology in this report is streamed into two parts – minimax strategy and bounds by inequalities. For triangular paper, the rims of cones are parameterized before their water depths are expressed explicitly. The capacities of cones are maximized over angles of slant. Different ways of division are compared to find out the optimal solution. Probing into general cases, various inequalities are set up analytically and exhaustively to bound the total capacities for comparisons. To obtain the greatest capacities, cones made from one sub-division should be slanted but those from multiple sub-divisions should be held vertically. For a polygonal paper of six or more sides, it should be divided into two divisions, each comprising two or more sub-divisions with a central angle ratio of 0.648:1.352, approaching the way of division in circular paper.

## 2012 Awards Ceremony Video

The 2012 Hang Lung Mathematics Awards winners were announced and recognized on December 18, 2012. Eight awards were announced: a Gold Award, a Silver Award, a Bronze Award, and five Honorable Mentions.

Winning Students, teachers, and schools were recognized on stage, and received crystal trophies and certificates from world renowed scholars.

### Presentation of Souvenirs

##### Mr. Ronnie Chi-Chung ChanChairman, Hang Lung Properties

2012 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## Finalist Teams Selected for the Oral Defense at the 2012 Hang Lung Mathematics Awards

### (arranged by school name in alphabetical order)

 Trajectories in Regular Pentagon Baptist Lui Ming Choi Secondary School Construction of Uni-transversal Path Maryknoll Convent School (Secondary Section) From ‘Chopsticks’ to Periodicity of Generalized Fibonacci Sequence Shatin Pui Ying College Continued Fraction and Related Topics In A Geometric Perspective Buddhist Sin Tak College Manipulating the Fermat’s Equation Po Leung Kuk Centenary Li Shiu Chung Memorial College Cutting Twisted Solid Tori(TSTs) Sha Tin Government Secondary School An Investigation into Gamma Function St. Francis’ Canossian College Complexity Reduction of Graphs St. Mary’s Canossian College Application of Generalized Fibonacci Sequence -The Probabilistic Behaviors of Propagation of SARS G.T. (Ellen Yeung) College A Study on Polyhedron with All Triangular Faces: Nine-point Circle Co-sphere Carmel Holy Word Secondary School An approach to non-numerical method of finding the non-diagonalizable solution of quadratic matrix equation Queen Elizabeth School Predictions on Usain Bolt in London Olympics 2012 G.T. (Ellen Yeung) College How to cut a piece of paper – making paper cones with greatest total capacity Wong Shiu Chi Secondary School Towards Catalan’s Conjecture Sir Ellis Kadoorie Secondary School (West Kowloon)

## ORGANIZATION

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2010

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2010 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing-Tung Yau Harvard University Professor Chong-Qing Cheng Nanjing University Professor Shiu-yuen Cheng The Hong Kong University of Science and Technology Professor John H. Coates University of Cambridge Professor Jean-Marc Fontaine Paris-Sud 11 University Professor Ka-Sing Lau The Chinese University of Hong Kong Professor Eduard Looijenga Universiteit Utrecht Netherland Professor Ngai-Ming Mok The University of Hong Kong Professor Duong H. Phong Columbia University Professor Hyam Rubinstein University of Melbourne Professor Gilbert Strang Massachusetts Institute of Technology Professor Ngo Viet Trung Institute of Mathematics, Vietnam Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Chin-Lung Wang National University of Taiwan

## Screening Panel 2010

The members of the Screening Panel of the 2010 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Chi-Hin Lau The Chinese University of Hong Kong

## Steering Committee 2010

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society including  leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2010 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics Master, Morningside College, CUHK Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Professor Tony Chan The Hong Kong University of Science and Technology Professor Shiu-yuen Cheng The Hong Kong University of Science and Technology Professor Wing-Sum Cheung The University of Hong Kong Professor Ka-Sing Lau Chairman, Mathematics Department, CUHK Professor Lo Yang Chinese Academy of Sciences Dr. Stephen Tommis Executive Director, HK Academy for Gifted Education Mr. Siu-Leung Ma CEO, Fung Kai Public Schools Ms Carolina Yip Hang Lung Properties Limited Mr. Chee-Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2010

The members of the Executive Committee of the 2010 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Dr. Leung-Fu Cheung The Chinese University of Hong Kong Dr. Charles Chun-Che Li The Chinese University of Hong Kong Secretariat: Ms. Mavis Kit-Ying Chan Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong

## WINNERS of the 2010 Hang Lung Mathematics Awards

### GOLD

 Topic: Expressibility of Cosines as Sum of Basis Team Members: Kwok Wing TSOI, Ching WONG Teacher: Mr. Yan Ching CHAN School: Po Leung Kuk Centenary Li Shiu Chung Memorial College Abstract: The central issue we are investigating is based on a problem from The Hong Kong (China) Mathematical Olympiad. It is basically about whether a cosine ratio is expressible as sum of rational numbers to powers of reciprocals of primes. In our project, we give the generalization of this problem by using some tricks in Elementary Number Theory and Galois Theory.

### SILVER

 Topic: Curve Optimization Problem Team Members: Ping Ngai CHUNG Teacher: Ms. Mee Lin LUK School: La Salle College Abstract: In this project, we shall introduce a new quantity associated with any given shape on the plane: “optimal curve”, which is defined as the shortest curve such that its convex hull fully covers a given shape S. Here curve can involve straight lines or union of straight lines. We shall investigate on some properties of this kind of curve and also prove a theorem that among shapes with a given fixed length of perimeter, the circle has the maximal optimal curve. Moreover, we will introduce an algorithm to find the shortest curve with convex hull equals a given shape in polynomial time.

### BRONZE

 Topic: Orchard Visibility Problem Team Members: Trevor Chak Yin CHEUNG, Yin To CHUI Teacher: Mr. Kwok Kei CHANG School: Buddhist Sin Tak College Abstract: In this paper, we discuss the generalization of the orchard visibility problem – from that of grid shapes to that of the shapes of the trees. We will even take a look at the problem of the visibility problem on a sphere surface and 3-D space.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic: A Study of Infectious Diseases by Mathematical Models Team Members: On Ping CHUNG, Winson Che Shing LI, Hon Kei LAI, Wing Yan SHIAO, Sung Him WONG Teacher: Mr. Wing Kwong WONG School: Hong Kong Chinese Women’s Club College Abstract: Diseases are devastating. The SARS in 2003 and the swine influenza in 2009 sparked myriad of questions in our minds. Our major concern is the spread of germs. Throughout the entire project, we investigate disease-related issues and try to study the impacts of a disease by mathematical modeling. We first start with the simplest model followed by more complicated ones. We focus on different factors that affect the spread of diseases. Diagrams are included in each chapter to see how the values of different groups vary. Then we come up with possible ways to prevent epidemics. Altering the models by adding more conditions, we find one that fits the real life situation – the SEIRS model. The situation in Hong Kong (Swine Influenza from April 2009 to April 2010 in Hong Kong) is simulated by putting the data into the model and our goal is fulfilled. Topic: Dividing a Circle with the Least Curve Team Members: Chung Yin CHAN, Rennie LEE Teacher: Mr. Ka Wo LEUNG School: Hong Kong True Light College Abstract: In this project we planned to study the division of a circle with the shortest curve. In a party, we often divide a circular cake into equal and unequal parts. Suppose that bacteria grow on the exposed surface area of a cake. In order to keep the cake hygienic, we should divide the cake with the shortest cut. We investigated this problem by using a simple mathematical model: dividing a circle into equal or unequal areas with the shortest curve. The first possible solution was the radius method. It meant that we used radii to divide a circle into parts. But, were there any ways to divide a circle with a curve shorter than that of the radius method? The results included: Radius method is the solution of the problem for $n$ = 2, 3 and equal division. Radius method is not a solution of the problem for $n$ = 4 and equal division. Orthogonal circular arc is the solution of the problem for $n$ = 2 and unequal division. We found a necessary condition of the problem for $n$ = 3 and unequal division by a “Y-shaped” curve. Topic: Magic Squares of Squares Team Members: Pak Hin LI Teacher: Mr. Chi Ming CHAN School: P.L.K. Vicwood K.T. Chong Sixth Form College Abstract: In this report, we want to know whether there is a magic square whose entries are distinct perfect squares. Firstly, we analyze the basic properties of a magic square and find that the magic sum of a magic square is equal to 3 times of the central entry and the 9 entries of a magic square contain 8 arithmetic progressions. Secondly, we focus on our main target, magic square of squares. Investigating the properties of the prime factors of those 9 entries, we find that if the greatest common divisor of all entries is equal to 1, the prime factors of central entry are of the form $p \equiv 1$ (mod 4), the central entry must not be a square of a prime number and the common prime factors of any two adjacent entries (if exist) are not of the form $p \equiv 3$ (mod 4). Thirdly, we find that this problem is equivalent to a system of Diophantine equations with ten variables. We provide a construction method of the solution to these partial equations: $$a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2 = 2M^2$$ , where these nine perfect squares are distinct. Finally, based on the theorems obtained, we find that given a positive integer $N$, there exists a positive integer $M$ such that it has $N$ essentially different representations of a sum of two perfect squares. Topic: Spherical Fagnano’s Problem and its Extensions Team Members: Ho Kwan SUEN, Tin Yau CHAN, Tsz Shan MA, Pak Hay CHAN Teacher: Mr. Cheuk Yin AU School: Pui Ching Middle School Abstract: In a given acute triangle, the inscribed triangle with minimum perimeter is the orthic triangle. This problem was proposed and solved using calculus by Fagnano in 1775. Now we wonder, will the result remain unchanged when the problem is discussed on a sphere? In this paper, we will first try to find the answer of the “spherical Fagnano’s problem”. Based on our results in spherical triangle cases, we will go further to generalize the problem to quadrilateral and n-sided spherical polygon in spherical geometry. Topic: The Erdős-Szekeres Conjecture (“Happy End Problem”) Team Members: Ho Ming WONG, Man Han LEUNG, Wing Yee WONG, Hon Ka HUI, Tin Chak MAK Teacher: Mr. Chi Keung LAI School: Shatin Pui Ying College Abstract: The survey [1] conducted by W. Morris and V. Soltan mentioned that in 1935 Erd˝os-Szekeres proved that for any integer $n \geq 3$, there exists a smallest positive integer $g(n)$ points in general position in the plane containing n points that are the vertices of a convex n-gon. [See reviewer’s comment (3)] They also conjectured that $g(n) = 2n−2 + 1$ for any integer $n \geq 3$. The conjecture is far from being solved for decades though many mathematicians had tried their very best on it. This paper is to investigate the Erd˝os-Szekeres conjecture by studying the greatest positive integer $f(n)$ points in general position in the plane which contains no convex n-gons. We successfully prove the cases when $n = 4$, 5 i.e. $f(4) = 4$ and $f(5) = 8$. For $n = 6$, we arrive at the conclusion that $f(6) \geq 16$ by creating an example of 16 points containing no convex hexagons. Moreover, we excitedly find an elegant proof for this example that one more point added to it will certainly give birth to a convex hexagon.

### 2010 Awards Ceremony Video

The 2010 Hang Lung Mathematics Awards winners were announced and recognized on December 18, 2010. Eight awards were announced: a Gold Award, a Silver Award, a Bronze Award, and five Honorable Mentions.

Winning students, teachers, and schools were recognized on stage, and received crystal trophies and certificates from world renowned scholars.

### Closing Remarks

##### Dr. Gerald L. ChanNon-Executive Director, Hang Lung Group

2010 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## Finalist Teams Selected for the Oral Defense at the 2010 Hang Lung Mathematics Awards

### (arranged by school name in alphabetical order)

 The Erdos-Szekeres Conjecture Shatin Pui Ying College Dividing a circle with the least curve Hong Kong True Light College Orchard Visibility Problem Buddhist Sin Tak College A Study of Infectious Diseases by Mathematical Models Hong Kong Chinese Women’s Club College The Centres of Tetrahedra Baptist Lui Ming Choi Secondary School Investigation on Mastermind and its generalization Munsang College (HK Island) Expressibility of Cosines as Sum of Basis Po Leung Kuk Centenary Li Shiu Chung Memorial College Introduction and Applications of Fuzzy Homotopy and Fuzzy Deformation Retraction Tang Shiu Kin Victoria Government Secondary School Zero-knowledge mutual authenticaion in the two-party password-authenticated key exchange setting: future goal or mission impossible? Carmel Secondary School Spherical Fagnano’s Problem and Its Extensions Pui Ching Middle School Starting from Combinatorial Geometry St. Joseph’s College Curve Optimization Problem La Salle College A new method to improve the ranking system for students studying the NSS: Using the calculation of eigenvector to find the weights of different subjects in NSS Carmel Holy Word Secondary School Magic squares of squares P.L.K. Vicwood K.T.Chong Sixth Form College n-Puzzle: An Innovation Sha Tin Government Secondary School

## ORGANIZATION

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2008

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2008 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing-Tung Yau Harvard University Professor Tony F. Chan University of California, Los Angeles Professor David C. Chang Polytechnic Institute of New York University Professor Chong-Qing Cheng Nanjing University Professor John H. Coates University of Cambridge Professor Benedict H. Gross Harvard University Professor Ka-Sing Lau The Chinese University of Hong Kong Professor Jian-Shu Li The Hong Kong University of Science and Technology Professor Chang-Shou Lin National University of Taiwan Professor Jill P. Mesirov Broad Institute of MIT and Harvard Professor Kenneth C. Millett University of California, Santa Barbara Professor Ngai-Ming Mok The University of Hong Kong Professor Stanley J. Osher University of California, Los Angeles Professor Duong H. Phong Columbia University Professor Wilfried Schmid Harvard University Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Hung-Hsi Wu University of California, Berkeley

## Screening Panel 2008

The members of the Screening Panel of the 2008 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong

## Steering Committee 2008

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educator and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2008 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Professor Wing-Sum Cheung The University of Hong Kong Professor Ka-Sing Lau Chairman, Mathematics Department, CUHK Professor Jian-Shu Li The Hong Kong University of Science and Technology Professor Lo Yang Chinese Academy of Sciences Mr. Siu-Leung Ma CEO, Fung Kai Public Schools Mr. Chun-Kau Poon Principal, The Hong Kong Federation of Youth Group Lee Shau Kee College St. Paul Co-educational College (retired) Ms Susan Wong Hang Lung Properties Limited Mr. Chee-Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2008

The members of the Executive Committee of the 2008 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Dr. Leung-Fu Cheung The Chinese University of Hong Kong Dr. Charles Chun-Che Li The Chinese University of Hong Kong Secretariat: Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong

## WINNERS of the 2008 Hang Lung Mathematics Awards

### GOLD

 Topic: Isoareal and Isoperimetric Deformation of Curves Team Members: Kwok Chung Li, Chi Fai Ng Teacher: Mr. Wing Kay Chang School: Shatin Tsung Tsin Secondary School Abstract: In the report, we want to answer the following question: how to deform a curve such that the rate of change of perimeter is maximum while the area and the total kinetic energy are fixed? First we work on isosceles triangle as a trial. Then we study smooth simple closed curve and obtain information about the velocity of each point of the curve and its relation to the curvature. We also consider the applications of the results and the velocity for the dual isoperimetric problem.

### SILVER

 Topic: Sufficient Condition of Weight-Balance Tree Team Members: Chi Yeung Lam, Yin Tat Lee Teacher: Mr. Chun Kit Ho School: The Methodist Church Hong Kong Wesley College Abstract: Huffman’s coding provides a method to generate a weight-balance tree, but it is not generating progressively. In other words, we cannot have meaningful output if we terminate the algorithm halfway in order to save time. For this purpose, we want to design an alternative algorithm, therefore this paper aims at finding out a sufficient condition of being a weight-balance tree. In this paper, we have found out the sufficient condition. Besides, as the solution of building a weight-balance tree can be applied to solving other problems, we abstract the problem and discuss it in the manner of graph theory. The applications are also covered.

### BRONZE

 Topic: Fermat Point Extension – Locus, Location, and Local Use Team Members: Fung Ming Ng, Chi Chung Wan, Wai Kwun Kung, Ka Chun Hong Teacher: Mr. Yiu Kwong Lau School: Sheng Kung Hui Tsang Shiu Tim Secondary School Abstract: Published in 1659, the solutions of Fermat Point problem help people find out the point at which the sum of distances to 3 fixed points in the plane is minimized. In this paper, we are going to further discuss when the number of fixed points is greater than 3, the relationship between the fixed points and the point minimizing the sum of distances to more than three given points. Also, we would like to find out if there exists a way such that the location of point minimizing the sum of distances to more than three given points can be determined just by compass and ruler, or approximated by mathematical methods.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic: A Cursory Disproof of Euler’s Conjecture Concerning Graeco-Latin Squares by means of Construction Team Members: Jun Hou Fung Teacher: Mr. Jonathan Hamilton School: Canadian International School of Hong Kong Abstract: In this report, our team has explored a mathematical structure commonly known as Graeco-Latin squares. Although we do give a broad scope of this field, we are particularly focused on one aspect: Euler’s Conjecture. According to this conjecture, there are certain types of Graeco-Latin squares that do not exist. In this report, we disprove this conjecture by demonstrating a means to construct an infinite number of these so-called non-existent squares. This branch of mathematics is highly related to group theory, combinatorics, and transversal design; therefore, we will also provide a brief overview of these topics throughout this report. Topic: Equidecomposition Problem Team Members: Cheuk Ting Li Teacher: Miss Mee Lin Luk School: La Salle College Abstract: The equidecomposition problem is to divide a shape into pieces, and then use the pieces to form another shape. In this project, we are going to investigate the conditions under which a given shape can be broken down and combined into another specified shape. The classical problem on the equidecomposability of polygons has already been solved by mathematicians. We start by presenting the proof of the classical problem, which is the keystone of this research. Then the problem is generalized to weighted shapes, shape with curves, etc. Some interesting new results are obtained. Topic: 3n+1 Conjecture Team Members: Shun Yip Teacher: Mr. Chi Keung Lai School: Shatin Pui Ying College Abstract: The aim of our project is to investigate the 3n + 1 conjecture. It is very hard to give a general path for each natural number to arrive at 1. So we investigate its negation i.e. there exists a natural number k with no path to 1. There are two possibilities: either ktakes a path which is a cycle to itself after n steps or its path is increasing indefinitely. These two possibilities lead us to study pre-numbers of any odd natural number and the number of peaks of paths. In the project, several interesting results were obtained by studying backward paths, number of peaks and cycles or forward paths. Topic: Geometric Construction – Area Trisection of a Circle Team Members: Shun On Hui, Kin Ho Lo, Kai Ming To, Maureen Tsz Yan Ho , Wai Hang Ng Teacher: Mr. Wai Hung Ho School: Tsuen Wan Public Ho Chuen Yiu Memorial College Abstract: When dividing a cake of circle shape into equal parts, it is quite easy to divide it from the centre. However, if we need to divide it from its edge, how can we accomplish this task accurately? This report aims to find a method to divide the area of a circle into 3 equal parts with two straight lines by Euclidean construction, i.e. the construction with compass and straightedge only. However, we were aware that it is difficult, if not impossible, to find the exact method of construction. Therefore, we try to find some methods to divide the area of circle approximately into three equal parts. In this report, we have three analytic approaches: by Lagrange Interpolating Polynomial, by infinite series of sine function and by method of bisection. Then, we will discuss three methods of construction, which are: inscribing a regular polygon with a large number of sides, inscribing a regular polygon with a small number of sides and bisecting the slope. At last, we will give a comparison of these three methods

### 2008 Awards Ceremony Video

The 2008 Hang Lung Mathematics Awards winners were announced and recognized on December 21, 2008. Seven awards were announced: a Gold Award, a Silver Award, a Bronze Award, and four Honorable Mentions.

Winning Students, teachers, and schools were recognize on stage, and received crystal trophies and certificates from world renowed scholars.

### Closing Remarks

##### Dr. Gerald L. ChanNon-Executive Director, Hang Lung Group

2008 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the video to watch:

## Finalist Teams Selected for the Oral Defense at the 2008 Hang Lung Mathematics Awards

### (arranged by school name in alphabetical order)

 Sufficient Condition of Weight-Balance Tree The Methodist Church Hong Kong Wesley College Interesting Findings in Rational Triangles and its relation to an Elliptic Curve Buddhist Sin Tak College 3n + 1 Conjecture Shatin Pui Ying College Least Wet Under Rain Kwok Tak Seng Catholic Secondary School The tactics of winning “RISK” S.D.B. NG SIU MUI SECONDARY SCHOOL “Fermat Point” – Locus, Location, and Local Use Sheng Kung Hui Tsang Shiu Tim Secondary School Analysis of Mastermind Immanuel Lutheran College Geometric Construction – Area Trisection of A Circle Tsuen Wan Public Ho Chuen Yiu Memorial College Wire In Magnetic Field SKH Lam Woo Memorial Secondary School Equidecomposition Problem La Salle College Isoareal and Isoperimetric Deformation of Curves Shatin Tsung Tsin Secondary School A Cursory Disproof of Euler’s Conjecture Concerning Graeco-Latin Squares By Means of Construction Canadian International School of Hong Kong Be a Smart Banker－ A research on 1 vs 1 gambling Shatin Tsung Tsin Secondary School Capture King – a Mathematics game Munsang College (HK Island) Is there a function for the absolute percentage error in the period of a pendulum when using the standard formula Sha Tin College

## WINNERS of the 2006 Hang Lung Mathematics Awards

### GOLD

 Topic: How to Keep Water Cold – A Study about the Wet Contact Surface Area in Cylinder Team Members: Cheuk Hin Cheng Teacher: Mr. Kwok Tai Wong School: S.K.H. Lam Woo Memorial Secondary School

### SILVER

 Topic: On the Prime Number Theorem Team Members: Yun Pui Tsoi Teacher: Mr. Wai Man Ko School: Shatin Government Secondary School

### BRONZE

 Topic: Construction of Tangents to Circles in Poincare Model Team Members: Fai Li, Chung Yam Li, Daniel Chung Sing Poon, King Ching Li Teacher: Mr. Chun Yu Kwong School: Wong Shiu Chi Secondary School

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 Topic: Circle Packing Members: Wa Yip Lau, Hon Yiu So Teacher: Mr. Kwok Kei Chang School: Buddhist Sin Tak College Topic: An Investigation in Secret Sharing Members: Kin Shing Lo, Ho Kwan Lee, Chi Wong Teacher: Mr. Ka Wo Leung School: Fung Kai Liu Man Shek Tong Secondary School Topic: Two Interesting Mathematical Games Team Members: Kin Ying Chan, Man Chung Cheung, Kwok Chun Li, Chi Fai Ng Teacher: Mr. Pik Yee Lam School: Shatin Tsung Tsin Secondary School Topic: Rolling without Sliding Team Members: Yin Hong Tin, Cheuk Ying Tsim, Yat Au Yeung Teacher: Mr. Hon-wai Yung School: South Tuen Mun Government Secondary School Topic: Decrypting Fibonacci and Lucas Sequences Team Members: Yin Kwan Wong, David Tak Wai Lui, Theodore Heung Shan Hui Teacher: Ms. Yau Man Sum School: St. Paul’s Co-educational College Topic: Developing 3D Human Model by Using Mathematical Tools Team Members: Ka Leung Chu, Ka Hei Wong Teacher: Mr. Chi Keung Chan School: Yuen Long Merchants Association Secondary School

### (NON-MONETARY CERTIFICATE)

 Topic: Exploration on the Odd Perfect Number Team Members: Kong Fard Man, Tsz Ching Wong, Ling Chit Li, Ka Wai Leung, Shun Yip Teacher: Mr. Chi Keung Lai School: Shatin Pui Ying College

### 2006 Awards Ceremony Video

The 2006 Hang Lung Mathematics Awards winners were announced and recognized on December 19, 2006.  Ten awards were announced: a Gold Award, a Silver Award, a Bronze Award, 6 Honorable Mentions, and one Special Commendation.

Winning students, teachers, and schools were recognized on stage, and received crystal trophies and certificates from world renowned scholars.

### Token Presentation

2006 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the video to watch.

## FINALIST TEAMS SELECTED FOR THE ORAL DEFENSE AT THE 2006 HANG LUNG MATHEMATICS AWARDS

### (arranged by school name in alphabetical order)

 Circle Packing Buddhist Sin Tak College A New Direction Measurement for Analysing Stroke Sequences Immanuel Lutheran College Decrypting Fibonacci and Lucas Sequence St Paul’s Co-educational College On the Prime Number Theorem Sha Tin Government School Speedy Tunnel Shatin Tsung Tsin Secondary School Two Interesting Mathematical Games Shatin Tsung Tsin Secondary School Knight Tour Shatin Tsung Tsin Secondary School An Investigation in Secret Sharing Fung Kai Liu Man Shek Tong Secondary School Exploration on the odd perfect numbers Shatin Pui Ying College A Theory of Blocking Geometry: From Viewpoint of Light and Shadow Tang Shiu Kin Victoria Government Secondary School Construction of tangents to circles in Poincare model Wong Shiu Chi Secondary School Graphical Functionalization Kiangsu-Chekiang College (Shatin) How to Keep Water Cold – A study about the wet contact surface area in cylinder SKH Lam Woo Memorical Secondary School Rolling without Sliding South Tuen Mun Government Secondary School Developing 3D human model by using mathematical tools Yuen Long Merchants Association Secondary School