# Year Book 2012

## Organization

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

## Scientific Committee 2012

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

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

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

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

## Screening Panel 2012

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

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

## Steering Committee 2012

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

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

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

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

## Executive Committee 2012

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

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

## WINNERS of the 2012 Hang Lung Mathematics Awards

### GOLD

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

### SILVER

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

### BRONZE

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

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

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

## 2012 Awards Ceremony Video

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

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

### Presentation of Souvenirs

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

2012 DEFENSE MEETING VIDEO

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

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

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

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