|国家/州||NM，United States of America|
Classifying Magic Squares and Their Associated Symmetries Using a Chord Diagram Approach
Magic squares are defined as matrices whose rows, columns, and main diagonals all add to the same number. They have existed and have been a topic of great interest since as early as 190 BCE, appearing in several different cultures. Some of the greatest minds in mathematics dedicated time to their exploration and construction including Euler, Ramanujan, and Fermat. Even with their prevalence throughout the mathematical world, few general theorems have been proven about magic squares and their properties. Through the course of this project, a novel approach to studying the squares was developed that represents the matrix elements as connected points on a circle with n2 equally spaced points. Furthermore, this visual representation can be used to classify the squares by the symmetries they produce. Unlike previously used visual aids, this method of classification extends to all orders of magic squares. Using this method of circular graphing, generalized theorems about the reflection and rotational symmetries of magic squares were developed as well. It was also proven using these tools that the maximum number of possible axes of symmetry of a given n x n magic square is precisely n/2. Finally, inconsistencies found in prior work by other authors on classification of 4x4 magic squares using permutation matrices were discovered, and further relations to this theory were developed.
高中生科研 英特尔 Intel ISEF
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高中生科研竞赛 英特尔 Intel ISEF 简介
英特尔国际科学与工程大奖赛，简称 "ISEF"，由美国 Society for Science and the Public（科学和公共服务协会）主办，英特尔公司冠名赞助，是全球规模最大、等级最高的中学生的科研科创赛事。ISEF 的竞赛学科包括了所有数学、自然科学、工程的全部领域和部分社会科学。ISEF 素有全球青少年科学竞赛的“世界杯”之美誉，旨在鼓励学生团队协作，开拓创新，长期专一深入地研究自己感兴趣的课题。
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The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. The deductive study of numbers, geometry, and various abstract constructs, or structures.
Algebra (ALB): The study of algebraic operations and/or relations and the structures which arise from them. An example is given by (systems of) equations which involve polynomial functions of one or more variables.
Analysis (ANL): The study of infinitesimal processes in mathematics, typically involving the concept of a limit. This begins with differential and integral calculus, for functions of one or several variables, and includes differential equations.
Combinatorics, Graph Theory, and Game Theory (CGG): The study of combinatorial structures in mathematics, such as finite sets, graphs, and games, often with a view toward classification and/or enumeration.
Geometry and Topology (GEO): The study of the shape, size, and other properties of figures and spaces. Includes such subjects as Euclidean geometry, non-Euclidean geometries (spherical, hyperbolic, Riemannian, Lorentzian), and knot theory (classification of knots in 3-space).
Number Theory (NUM): The study of the arithmetic properties of integers and related topics such as cryptography.
Probability and Statistics (PRO): Mathematical study of random phenomena and the study of statistical tools used to analyze and interpret data.
Other (OTH): Studies that cannot be assigned to one of the above subcategories. If the project involves multiple subcategories, the principal subcategory should be chosen instead of Other.