2008恒隆數學獎獲獎論文集（簡體書）

《2008恒隆數學獲獎論文集》不僅可供中學生閱讀，亦可供數學教師和數學愛好者閱讀參考。

Preface
by Professor Shing-Tung Yau and Mr.Ronnie C.Chan
Acknowledgement
Hang Lung Mathematics Awards
Organization
Scientific Committee,2008
Steering Committee,2008
Gold,Silver,and Bronze
ISOAREAL AND ISOPERIMETRIC DEFORMATION OF CURVES
A SUFFICIENT CONDITION OF WEIGHT-BALANCED TREE
FERMAT POINT EXTENSION-LOCUS,LOCATION,LOCAL USE
Photos
Honorable Mentions
A CURSORY DISPROOF OF EULER'S CONJECTURE CONCERNING GRAECO-LATIN SQUARES BY MEANS OF CONSTRUCTION
EQUIDECOMPOSITION PROBLEM
COLLATZ CONJECTURE 3n+1 CONJECTURE
GEOMETRIC CONSTRUCTION AREA TRISECTION OF A CIRCLE

5.Conclusions and Reflections
In the previous chapter,we have followed the paths that mathematicianshas lain for us decades ago.Euler has provided the first constructionmethod,while Sade has given us the most recent(along with Parker,Bose,and Shrikhande and their transversal designs).
To summarise their contributions:Euler has proven that Euler squares ofodd order or of an order that is a multiple of four exists(He also provedthe obvious nonexistence of Euler squares of order 2),while Parker,Bose,and Shrikhande constructed Graeco-Latin squares of all orders,includingthose of form4k+2,with the exception ofn = 2 and n = 6.On theother hand,Tarry has shown that Graeco Latin squares of order 6 are notpossible.
Theorem 29.Euler squares exist for every order n except when n = 2or 6.
But the research does not stop here.Recently,more elegant proofs havebrought forward by Stinson,Dougherty,and Zhu Lie.Also,research inthis area has taken on a greater scope.Mathematicians working in thisfield are now researching selforthogonal Latin squares -- squares thatare orthogonal to its transpose.Some error-correcting codes in algebraiccoding theory are also based on MOLS.Speaking of which,perhaps themost exciting developments come from finite projective planes,to whichthe following theorem will link MOLS.
Theorem 30.A complete set of MOLS of order n implies a finite projective plane of order n.
This had all started out as the simple riddle of 36 officers.After leadingto developments in combinatorics,group theory,field theory,transversaldesign,and work done by many mathematicians around the globe,wefinally begin to draw the close to this problem.Yet,the future of Latinsquares is still vast to explore.
Where do we go from here? I list here a few open problems and conjecturesyet to be solved.