The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 
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 
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 
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.
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 C^{1} 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. 
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. 
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 padic valuation of the Catalan number by analyzing the properties of the coefficients in the base pexpansion of n. We unearth a new connection between those coefficients and the padic 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. 
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:

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 numbertheoretic 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 y^{2} = x^{3} + k for a particular class of integers k. We employ some classical approaches, i.e. factorization in number fields and quadratic reciprocity. When k = p^{2} 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 Coeducational 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 twelvepoint 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 twentyfifth Kimberling centre X_{25}. 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. 
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.
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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  HarishChandra Research Institute 
Professor Ka Sing Lau  The Chinese University of Hong Kong 
Professor Jun Li  Stanford University 
Professor ChangShou 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 
The members of the Screening Panel of the 2004 Hang Lung Mathematics Award are:
Chair: Professor Tom Yauheng 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 
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 Coeducational 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 
The members of the Executive Committee of the 2004 Hang Lung Mathematics Awards are:
Chair: Professor Thomas Kwokkeung Au  The Chinese University of Hong Kong 
Academic Resource Center: Dr. Leungfu Cheung 
The Chinese University of Hong Kong 
Academic Resource Center: Dr. Kaluen 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.
Topic:  Marked Ruler as a Tool for Geometric Constructions – from angle trisection to nsided polygon 
Team Members:  Edward Sin Tsun Fan 
School:  Sha Tin Government School 
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) 
Topic:  畫鬼腳 
Team Members:  Ting Fai Man, Hoi Kwan Lau, Shek Yeung, Man Kit Ho 
School:  Sha Tin Government Secondary School 
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 
Topic:  The Construction of SouthWestern 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 nDimensional Case 
Team Members:  Ching King Chan, Ka Leung Chu 
School:  Yuen Long Merchants Association Secondary School 
Solving Cubic Pell’s Equation by Bifurcating Continued Fraction Team Member: Ka Lam Wong Teacher: Mr. Yiu Chung Leung School: Bishop Hall Jubilee School 
HeineCantor 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 Coeducational 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 
恒隆數學獎2004頒獎禮終於在同年的12月17日假香港禮賓府完滿結束。大家請按下圖片以收看當日精彩片段。
恒隆數學獎 2004 頒獎禮
恒隆集團主席
陳啟宗先生
致辭
頒獎典禮開首，由恒隆集團主席陳啟宗先生致歡迎辭，介紹是次典禮的榮譽嘉賓出場︰丘成桐教授、路甬祥教授，李國章教授和羅范椒芬女士。同時，陳先生亦公開答謝了各位著名評審教授和工作人員們為是次數學獎作出的貢獻和努力。
「今天真正的明星，不是誰，正是恒隆數學獎的各位得獎者。」
鼓掌聲和觀笑聲不絕於耳。
恒隆數學獎 2004 頒獎禮
香港中文大學數學科學研究所所長
哈佛大學數學系教授
丘成桐教授
致辭
「恒隆數學獎的成績超出想像中的卓越。」
丘成桐教授認為是次比賽的水準可與 Intel Science Talent Search 和 Siemens Westinghouse 等等國際級大型獎項媲美。丘教授回想當年，少年時期已經十分醉心數學研究，他寄望恒隆數學獎的各位獲獎同學能夠領略到數學的美，並為自己的作品感到驕傲。
「天空海闊，未來是年青人的世界。」
恒隆數學獎 2004 頒獎禮
中國全國人民代表大會常務委員會副委員長
中國科學院院長
路甬祥教授
致辭
路教授非常高興能夠參加是次隆重的頒典禮，並對首屆恒隆數學獎得主表示祝福。數學是一項嚴謹的科學，無論什麼科學研究、經濟社會或是人的思維，都離不開數學。「學好數理化，走遍天下都不怕。」從中學時代開始培養數學研究的精神，學習到的經驗可以終身受用。對於恒隆數學獎這類鼓勵創作思維發展的活動，路教授表示希望可以繼續將其發揚光大，帶領潮流，一屆比一屆好。
「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。」
恒隆數學獎 2004 頒獎禮
香港教育統籌局局長
李國章教授
致辭
數學是一門被廣泛應用的學科，數學教育不但能夠培養同學學會學習和勇於接受挑戰的潛能，還可以加強大家的自信。在現今高科技和高競爭的世代，學生的多元技能，例如邏輯思維和解難能力等等，成為了最有力的爭勝條件。數學教育的重要性可見一斑。恒隆數學獎致力推廣數學教育，對香港數學教育有傑出的貢獻，為老師同學建立了一個強大和充滿前景的數學學習環境。
「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。」
李局長演辭全文輯錄
恒隆數學獎 2004 頒獎禮
第一部份– 頒獎花絮
第二部份– 冠軍隊伍致辭︰沙田官立中學
> 參賽者︰范善臻同學
> 領隊老師︰王徽女士
> 校長︰周金祥先生
王徽老師︰「能夠教導范同學是我的光榮。其實范同學的啟蒙老師不是本人，學校許多老師亦有為范同學灌溉施肥。我十分幸運地可以成為收成的農夫。恒隆數學獎可以提升同學對數學的興趣和肯定同學的數學潛能。」
周金祥校長︰「十分榮幸能夠站在台上致辭。本校非常幸運地可以獲得銅獎和金獎，這些獎項不但可以發揮到同學們的數學潛力，還可以增加他們對數學的興趣。多謝各合辦機構成功舉辦此比賽，祝各得獎同學將來可以成為像丘成桐教授般的傑出數學家。」
范善臻同學︰「我沒準備講辭，多謝恒隆數學獎，多謝我媽媽和多謝各位老師一直以來的支持。能夠得到金獎，除了自己的努力外，亦有賴其他夥佯的幫忙。再次多謝鄭紹遠教授和中文大學數學系給予的指導。」
2004 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the team’s video you want to watch.
Reconstruction of 3Dimensional 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 nsided 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 SouthWestern HK Island Line Sing Yin Secondary School 
Deconstructing Kafka SKH Lam Woo Memorial Secondary School 
Perimeter of Ellipse and Generalization in nDimensional Case Yuen Long Merchants Association Secondary School 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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.
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 
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 
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
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 nonlinear 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. 
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 \(P_i1,P_i2,P_i3,\dotsc,P_id1\) 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. 
Topic  A Geometric Approach to the Second Nontrivial Case of the ErdösSzekeres Conjecture 
Team Members  Wai Chung Cheng 
Teacher  Ms. Dora Po Ki Yeung 
School  Diocesan Girls’ School 
Abstract  The ErdösSzekeres conjecture, developed from the famous HappyEnding Problem, hypothesizes on the number of points in general position needed on a plane to guarantee the existence of a convex ngon. 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. 
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 wellknown 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 Positivedefinite 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. 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 ParisSud Orsay 
Professor KaSing 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 Yauheng Wan  Chinese University of Hong Kong 
Professor Lo Yang  Institute of Mathematical Sciences, Academic Sinica 
Professor Andrew ChiChih Yao  Centre for Advanced Study, Tsing Hua University 
The members of the Screening Panel of the 2006 Hang Lung Mathematics Award are:
Chair: Professor Tom Yauheng 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 
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 Kwokkeung Au  The Chinese University of Hong Kong 
Professor Wing Sum Cheung  The University of Hong Kong 
Professor KaSing 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 Coeducational 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 
The members of the Executive Committee of the 2006 Hang Lung Mathematics Awards are:
Chair: Professor Thomas Kwokkeung Au  The Chinese University of Hong Kong 
Dr. LeungFu Cheung  The Chinese University of Hong Kong 
Dr. KaLuen Cheung  The Hong Kong Institute of Education 
Secretariat: Ms. Serena WingHang Yip  The Chinese University of Hong Kong 
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.
2016 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the team’s video you want to watch.
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 Nontrivial Case of the ErdösSzekeres 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 Positivedefinite 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 Rationalsided Triangles Pui Ching Middle School 
Voting Power Queen Elizabeth School 
Triples of Sums of Two Squares Wong Shiu Chi Secondary School 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 ShingTung Yau  Harvard University and The Chinese University of Hong Kong 
Professor Lars Andersson  Albert Einstein Institute, Germany 
Professor Raymond HonFu Chan*  The Chinese University of Hong Kong 
Professor Tony F. Chan  The Hong Kong University of Science and Technology 
Professor ShiuYuen Cheng  Mathematical Sciences Center, Tsinghua University 
Professor Jaigyoung Choe  Korea Institute for Advanced Study 
Professor Ingrid Daubechies  Duke University 
Professor KaSing Lau  The Chinese University of Hong Kong 
Professor John M. Lee  University of Washington 
Professor NgaiMing 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 YauHeng 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.
The members of the Screening Panel of the 2014 Hang Lung Mathematics Awards are:
Chair: Professor Tom YauHeng Wan  The Chinese University of Hong Kong 
Professor Conan Nai Chung Leung  The Chinese University of Hong Kong 
Dr. Charles ChunChe Li  The Chinese University of Hong Kong 
Dr. ChiHin Lau  The Chinese University of Hong Kong 
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 KwokKeung Au  EPYMT and Mathematics, The Chinese University of Hong Kong 
Professor Tony F. Chan  ViceChancellor, The Hong Kong University of Science and Technology 
Professor Shiuyuen Cheng  Professor, Tsinghua University 
Professor WingSum Cheung  Professor, Mathematics, The University of Hong Kong 
Professor KaSing Lau  Professor, Mathematics, The Chinese University of Hong Kong 
Mr. SiuLeung Ma  CEO, Fung Kai Public School 
Ms. Michelle SauMan Mak  Chairman’s Office, Hang Lung Properties Limited 
Professor TaiKai 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. CheeTim Yip  Principal, Pui Ching Middle School 
The members of the Executive Committee of the 2014 Hang Lung Mathematics Awards are:
Chair: Professor Thomas KwokKeung Au  The Chinese University of Hong Kong 
Dr. KaLuen Cheung  The Hong Kong Institute of Education 
Dr. LeungFu Cheung  The Chinese University of Hong Kong 
Secretariat: Ms. Mandy KaMan Leung Ms. Konnie Wan Yu Pak Ms. Serena WingHang Yip 
The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong 
Topic  Investigation of the ErdösStraus 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 ${\bf{\textit{ErdősStraus Conjecture }}}$: For any positive $n \geq 2$, there exists positive integers $k,k_1,k_2$ such that 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 ErdosStraus Conjecture in the following dimensions: the related geometric representation of the ErdosStraus Conjecture, the properties of solutions of the ErdosStraus Conjecture, further investigation of some paper of the ErdosStraus Conjecture, existence of special forms of solutions of the ErdosStraus Conjecture, and the investigation of the ErdosStraus Conjecture in algebraic dimension. The aim of this report is to find evidence that shows the ${\bf{\textit{ErdősStraus Conjecture}}}$ is true. If evidence is not strong enough, we still hope that this report can make an improvement to the researched result at present. 
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 rightangled 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. 
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. 
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 TicTacToe 
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 wellknown TicTacToe game. The objective of the game is to achieve ‘kinarow’ 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 preconditions, 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 (Noncomplete Extended PSum). 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. 
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.
2014 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the video to watch.
Probability Study using Matrix to Analyze Football Tournaments Baptist Lui Ming Choi Secondary School 
Investigation of the ErdösStraus 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 TicTacToe 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 FingerCounting 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 
RockPaperScissors Tsuen Wan Public Ho Chuen Yiu Memorial College 
The Application of Graph Theory to Sudoku Ying Wa Girls’ School 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 ShingTung Yau  Harvard University and The Chinese University of Hong Kong 
Professor ShiuYuen 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 NgaiMing 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 YauHeng Wan  The Chinese University of Hong Kong 
The members of the Screening Panel of the 2012 Hang Lung Mathematics Awards are:
Chair: Professor Tom YauHeng Wan  The Chinese University of Hong Kong 
Professor Conan Nai Chung Leung  The Chinese University of Hong Kong 
Dr. Charles ChunChe Li  The Chinese University of Hong Kong 
Dr. ChiHin Lau  The Chinese University of Hong Kong 
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 KwokKeung Au  EPYMT and Mathematics, The Chinese University of Hong Kong 
Professor Tony F. Chan  ViceChancellor, The Hong Kong University of Science and Technology 
Professor Shiuyuen Cheng  Professor, Mathematics, The Hong Kong University of Science and Technology 
Professor WingSum Cheung  Professor, Mathematics, The University of Hong Kong 
Professor KaSing Lau  Professor, Mathematics, The Chinese University of Hong Kong 
Professor ZhouPing 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. SiuLeung Ma  CEO, Fung Kai Public Schools 
Ms. Carolina Yip  Chairman’s Office, Hang Lung Properties Limited 
Mr. CheeTim Yip  Principal, Pui Ching Middle School 
The members of the Executive Committee of the 2012 Hang Lung Mathematics Awards are:
Chair: Professor Thomas KwokKeung Au  The Chinese University of Hong Kong 
Dr. KaLuen Cheung  The Hong Kong Institute of Education 
Dr. LeungFu Cheung  The Chinese University of Hong Kong 
Secretariat: Ms. Mavis KitYing Chan Ms. Mandy KaMan Leung Ms. Serena WingHang Yip 
The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong 
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. 
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, BallCoxeter 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 polepolar 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 nontrivial 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. 
Topic:  Complexity Reduction of Graphs 
Team Members:  Tsam Kiu PUN 
Teacher:  Mr. Cyril LEE 
School:  St. Mary’s Canossian College 
Abstract:  Many realworld 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. 
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: Ninepoint Circle Cosphere 
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: ninepoint circles on each face cospherical and the 3D Euler line. This paper aims at finding the sufficient and necessary conditions for the ninepoint circles to be cospherical in the triangular polyhedrons. First, we discussed the conditions for the ninepoint 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 24point sphere (48point 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 ninepoint 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 1820, 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 subdivisions) 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 subdivision should be slanted but those from multiple subdivisions 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 subdivisions with a central angle ratio of 0.648:1.352, approaching the way of division in circular paper. 
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.
2012 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the team’s video you want to watch.
Trajectories in Regular Pentagon Baptist Lui Ming Choi Secondary School 
Construction of Unitransversal 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: Ninepoint Circle Cosphere Carmel Holy Word Secondary School 
An approach to nonnumerical method of finding the nondiagonalizable 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) 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 ShingTung Yau  Harvard University 
Professor ChongQing Cheng  Nanjing University 
Professor Shiuyuen Cheng  The Hong Kong University of Science and Technology 
Professor John H. Coates  University of Cambridge 
Professor JeanMarc Fontaine  ParisSud 11 University 
Professor KaSing Lau  The Chinese University of Hong Kong 
Professor Eduard Looijenga  Universiteit Utrecht Netherland 
Professor NgaiMing 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 YauHeng Wan  The Chinese University of Hong Kong 
Professor ChinLung Wang  National University of Taiwan 
The members of the Screening Panel of the 2010 Hang Lung Mathematics Awards are:
Chair: Professor Tom YauHeng 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. ChiHin Lau  The Chinese University of Hong Kong 
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 KwokKeung Au  The Chinese University of Hong Kong 
Professor Tony Chan  The Hong Kong University of Science and Technology 
Professor Shiuyuen Cheng  The Hong Kong University of Science and Technology 
Professor WingSum Cheung  The University of Hong Kong 
Professor KaSing Lau  Chairman, Mathematics Department, CUHK 
Professor Lo Yang  Chinese Academy of Sciences 
Dr. Stephen Tommis  Executive Director, HK Academy for Gifted Education 
Mr. SiuLeung Ma  CEO, Fung Kai Public Schools 
Ms Carolina Yip  Hang Lung Properties Limited 
Mr. CheeTim Yip  Principal, Pui Ching Middle School 
The members of the Executive Committee of the 2010 Hang Lung Mathematics Awards are:
Chair: Professor Thomas KwokKeung Au  The Chinese University of Hong Kong 
Dr. KaLuen Cheung  The Hong Kong Institute of Education 
Dr. LeungFu Cheung  The Chinese University of Hong Kong 
Dr. Charles ChunChe Li  The Chinese University of Hong Kong 
Secretariat: Ms. Mavis KitYing Chan Ms. Serena WingHang Yip 
The Chinese University of Hong Kong The Chinese University of Hong Kong 
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. 
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. 
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 3D space. 
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 diseaserelated 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:

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 nsided spherical polygon in spherical geometry. 
Topic:  The ErdősSzekeres 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˝osSzekeres 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 ngon. [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˝osSzekeres conjecture by studying the greatest positive integer \(f(n)\) points in general position in the plane which contains no convex ngons. 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. 
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.
2010 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the team’s video you want to watch.
The ErdosSzekeres 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 
Zeroknowledge mutual authenticaion in the twoparty passwordauthenticated 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 
nPuzzle: An Innovation Sha Tin Government Secondary School 
The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising worldrenowned 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.
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 ShingTung Yau  Harvard University 
Professor Tony F. Chan  University of California, Los Angeles 
Professor David C. Chang  Polytechnic Institute of New York University 
Professor ChongQing Cheng  Nanjing University 
Professor John H. Coates  University of Cambridge 
Professor Benedict H. Gross  Harvard University 
Professor KaSing Lau  The Chinese University of Hong Kong 
Professor JianShu Li  The Hong Kong University of Science and Technology 
Professor ChangShou 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 NgaiMing 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 YauHeng Wan  The Chinese University of Hong Kong 
Professor HungHsi Wu  University of California, Berkeley 
The members of the Screening Panel of the 2008 Hang Lung Mathematics Awards are:
Chair: Professor Tom YauHeng 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 
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 KwokKeung Au  The Chinese University of Hong Kong 
Professor WingSum Cheung  The University of Hong Kong 
Professor KaSing Lau  Chairman, Mathematics Department, CUHK 
Professor JianShu Li  The Hong Kong University of Science and Technology 
Professor Lo Yang  Chinese Academy of Sciences 
Mr. SiuLeung Ma  CEO, Fung Kai Public Schools 
Mr. ChunKau Poon  Principal, The Hong Kong Federation of Youth Group Lee Shau Kee College St. Paul Coeducational College (retired) 
Ms Susan Wong  Hang Lung Properties Limited 
Mr. CheeTim Yip  Principal, Pui Ching Middle School 
The members of the Executive Committee of the 2008 Hang Lung Mathematics Awards are:
Chair: Professor Thomas KwokKeung Au  The Chinese University of Hong Kong 
Dr. KaLuen Cheung  The Hong Kong Institute of Education 
Dr. LeungFu Cheung  The Chinese University of Hong Kong 
Dr. Charles ChunChe Li  The Chinese University of Hong Kong 
Secretariat: Ms. Serena WingHang Yip  The Chinese University of Hong Kong 
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. 
Topic:  Sufficient Condition of WeightBalance 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 weightbalance 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 weightbalance tree. In this paper, we have found out the sufficient condition. Besides, as the solution of building a weightbalance 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. 
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. 
Topic:  A Cursory Disproof of Euler’s Conjecture Concerning GraecoLatin 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 GraecoLatin 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 GraecoLatin squares that do not exist. In this report, we disprove this conjecture by demonstrating a means to construct an infinite number of these socalled nonexistent 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 prenumbers 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 
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.
2008 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the video to watch:
Sufficient Condition of WeightBalance 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 GraecoLatin 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 
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 
Topic:  On the Prime Number Theorem 
Team Members:  Yun Pui Tsoi 
Teacher:  Mr. Wai Man Ko 
School:  Shatin Government Secondary School 
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 
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. Honwai 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 Coeducational 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 
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 
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.
2006 DEFENSE MEETING VIDEO
Please click on the “Playlist” menu below to select the video to watch.
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 Coeducational 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 KiangsuChekiang 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 
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test2
Speech by Professor Arthur KC Li,
Secretary for Education and Manpower
at the Presentation Ceremony of the Hang Lung Mathematics Awards
17 December 2004
Prof. Lu, Prof. Yau, Mr Chan, Distinguished Guests, Ladies and Gentlemen,
It is indeed my great pleasure to be here this evening to take part in Hang Lung Mathematics Awards Presentation Ceremony – a very prestigious occasion which marks a milestone in the mathematics education of Hong Kong. I see among us renowned mathematicians and academics who have made significant contribution in the advancement of knowledge, as well as promising young people who would one day join the rank of distinguished mathematicians and scientists.
Mathematics is not only a specialised subject (and perhaps a difficult subject to many people) but also an essential tool to enable us to understand the world around us. It provides means to create, acquire, organize and apply knowledge. Many amazing advances in science and technology have their roots in mathematics and they would never have been possible without mathematical and computational models. The use of mathematics is commonplace in daily life, and its application is found in all aspects of our life and society, including economies, finance, medicine, science, technology, environment, and even in policy decisions. Advanced mathematics is even more widely used, but often in an unseen and unadvertised way. It is often said that advances are made with supercomputers. However, there has to be a mathematical theory, which guides the computer in terms of speed and accuracy in every step of its operation. Mathematics is involved in matters related to life and death too. For example, statistics is essential in analyzing the causes of illnesses and the effectiveness of new drugs. The importance of mathematics is universally recognised and mathematical sciences are in a period of unprecedented growth.
In this technologically complex and highly competitive era, we need to equip our students with a powerful set of tools, which include logical reasoning, problem solving skills, and the ability to think in abstract ways. As one of the key learning areas in our school curriculum, mathematics education aims not only to provide students with mathematical knowledge, but also to equip them with the necessary skills so that they can develop the capabilities to learn how to learn and the confidence to face the challenges of the knowledgebased society.
As you are aware, we are proposing a reform of our academic system. Under the new “3+3+4” academic structure, all students will have the opportunity to study the 3year senior secondary course, with enhanced language and mathematical abilities and a broadened knowledge base. The 4year undergraduate programme that follows will allow more time and space for broader and diversified learning experiences. This will widen students’ horizons and expose them to both specialized and broad knowledge for a more balanced wholeperson development. The direction of our education reform will focus more on learning skills, generic abilities, attitudes and values. We hope children coming through our education system will not only be effective learners at school, but will remain as critical, reflective and independent thinkers after they leave school. We also hope that the solid foundation we build for our children can help some of them proceed confidently beyond first degrees to engage in more challenging research work.
We will continue to pay great attention to mathematics education in the new senior secondary curriculum. In addition to English, Chinese and Liberal Studies, Mathematics is also one of the four core and compulsory subjects. A distinctive feature of the new curriculum is its flexibility. The proposed senior secondary mathematics curriculum will consist of a Compulsory Part and an Elective Part. The Compulsory Part aims to provide a broad and balanced curriculum for all students. The Elective Part is designed for students who want more mathematical knowledge and skills for their future endeavours.
Apart from the formal curriculum, we also try to develop students’ interests and talents in mathematics through competitions and other activities. Since 2001, the Education and Manpower Bureau has launched the “Support Measures for the Exceptionally Gifted Students Scheme” to help nurture and develop the potential and talents of exceptionally gifted children. Parallel support for their parents and teachers is also organized to help strengthen their competence in nurturing and supporting talented students. Around 3000 secondary students with outstanding performance/potential in leadership, mathematics or sciences have joined the Scheme to participate in various enhancement programmes.
It is gratifying to know that our students have been making remarkable achievements in international studies and competitions, and we should really be proud of them. Hong Kong remains the top of the league in Mathematics in the Programme for International Student Assessment (PISA) which was announced last week. Our 15yearold students also rank 2nd in problem solving and 3rd in science, which is the envy of many developed economies around the world. In the Trends in International Mathematics and Science Study (TIMSS) which was released just two days ago, our students also performed substantially better than their counterparts in other places. In mathematics, Hong Kong rank 3rd among the 46 participating economies at the 8th grade (ie our secondary 2), and 2nd among the 24 participating economies at the 4th grade (ie our primary 4). In science, we rank 4th at both levels, representing a big leap forward from the 1995 and 1999 results. It is interesting to note that the percentage of total instructional time intended for mathematics in Hong Kong was among the lowest in the participating economies, showing that our education system has been a very effective one.
Our students also fared well in a number of international and national mathematics competitions. At the 45th International Mathematical Olympiad, our team has won two silvers, two bronzes and two honourable mentions. Our students also won a silver award and 7 bronze awards in the National Mathematical Olympiad this year.
Traditional mathematics competitions can be one of the most enjoyable and valuable experiences in mathematics for many students as they can change students’ attitudes about what they are learning and motivate students to do more challenging mathematics. However, it still requires a big leap from successes in school mathematics competitions to productive work in mathematics. A student can excel in school and in competitions by becoming adept at solving problems to which a standard answer or solution is already known to exist. To become a research mathematician, however, a person has to be able to identify and make progress on interesting problems that may yet have no solutions. Studies have, in fact, shown that research projects have the potential to stimulate students’ learning and foster classroom engagement by stimulating their interest with a variety of challenging and authentic problemsolving tasks.
Unlike many speed competitions with questions that expect standard answers or solutions, the Hang Lung Mathematics Award emphasizes mathematical insight, creativity and originality. In addition to those traditional mathematics areas such as algebra, geometry, probability or analysis, other topics could be chosen from a very broad range of fields, such as applications in science, engineering, medical research, finance, logistics, transport, etc. In this regard, I am sure that the Hang Lung Mathematics Awards will further motivate secondary school students to develop research skills, to boost up their confidence and to promote success in their future pursuits.
Thanks to Mr Ronnie Chan whose vision and generous donation have made the Hang Lung Mathematics Awards a reality. Our gratitude should also go to Prof Shingtung Yau for all his efforts in making the scheme a success. I would also like to thank the Institute of Mathematical Sciences of the Chinese University of Hong Kong, the Hong Kong Education City Limited and other donors for their generous contributions to the Awards. Thanks should also be given to all teachers and school heads for their encouraging support to students and their efforts in promoting mathematics education.
On this special occasion, I would like to pay a tribute to the late Professor Chern ShingShen, who was widely recognized as one of the greatest mathematicians of the 20th century. It is indeed a great pity that Prof Chern cannot be with us today. Prof Chern was one of the creators of modern differential geometry and also winner of numerous prestigious honors in mathematics. His research led to the development of the laternamed Chern characteristic classes in fiber spaces that play a role in a wide area of mathematics and mathematical physics. Throughout his career, he has been known for his kindness, generosity, and contributions towards mathematics. Prof Chern is certainly a role model for all our students and also everyone here. His passing away is indeed our great loss and he will no doubt be missed. However, I am sure that he would be delighted to see our advancement and efforts towards building a strong and prosperous environment in the study of mathematical sciences.
Finally, I would like to congratulate all winners of the Hang Lung Mathematics Awards for their outstanding achievements and wish them every success in the future. I am sure, with our joint efforts, Hong Kong will continue to excel in Mathematics and other disciplines as well.
Thank you.
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