Year Book 2014

ORGANIZATION

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

Scientific Committee 2014

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

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

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

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

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

Screening Panel 2014

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

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

Steering Committee 2014

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

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

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

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

Executive Committee 2014

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

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

Winners of the 2014 Hang Lung Mathematics Awards

GOLD

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

SILVER

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

BRONZE

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

(ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

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

2014 Awards Ceremony Video

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

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

Presentation of Souvenirs

Mr. Ronnie C. ChanChairman, Hang Lung Properties Limited

2014 DEFENSE MEETING VIDEO

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

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

(arranged by school name in alphabetical order)

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