PrInformation and Summary.
Summary:
In this article we are going to study mathematical games. By Mathematical game we mean games that can be studied by mathematics.
In this article we are goint to study 3 games.
Combinatorial games. In this theme we are going to study the theory of Nim and variants of Nim.
The Josephus game. In this theme we are going to study the theory of the Josephus Problem. In 2008 we have presented our research on the Josephus Problem, but this time we are going to present new results here. We have studied self-similarity in the Josephus Problem in both directions, the linear Josephus Problem and linear Josephus Problem in both directions. We have made recursive relations and we proved the self-similarity of the linear Josephus Problem in both directions mathematically. We have also studied the Josephus Problem Mod n. The Josephus Problem can produce many interesting sequences when divided by natural number.
The Knight's tour game. In this theme we are going to present variants of Knight's tour problem. Here we can show beautiful graphs made by the problem. The graphs made by us are presented in Mathart of Visual Mathematics in Serbia and Archimedes-laboratory in Italy.
We have presented our result at the Research Institute of Mathematical Science of Kyoto University in 2007 and 2008.
The result of our research is going to be published by the Research Institute of Mathematical Science of Kyoto University, and the Mathematical Gazette.
We used a computer algebra system called Mathematica to discover formulas and theorems, and we occasionally used C and C++ language when we needed speedy calculations.
Project Types: Descriptive
Grade Division: 10-12
Team Size: 5
Subject Area: Mathematics
Software Tools:
Website:
LateX
WinShell for Latex
LaTeX2HTML
Mathematica 6.0 for Windows
Mathematica 6.0 for Mac OS X
Adobe Photoshop CS for Mac OS X
Adobe GoLive CS for Mac OS X
Project:
Mathematica 6
Turbo J Builder 2007
Special Skills:
Website:
Knowledge of Latex and LateX2HTML
Retouch by Photoshop
Project:
Knowledge of Mathematica programming language
Knowledge of Java language
Knowledge of C++ language
Hardware Tools:
MacBook with Mac OSX 10.4.8
iMac with OSX 10.4
Sony Vaio Notebook computer with Window XP
ThinkPad with Windows XP
Panasonic notebook with Windows Vista
Source of Idea:
Dr. Miyadera and his students have been doing the research of mathematics, and they have published many articles around the world. We have learned many things from them, and we have studied many problems from our point of view.
The Wolfram Research that is famous for its product Mathematica has been helping us by providing softwares and inviting us to their headquarter to study.
We have discovered new theorems and formulas for variants of Josephus problem, combinatorial games.
We have made beautiful graphs made by variants of Josephus Problems.
This time we used Latex and LateX2HTML to make HTML with mathematical symbols and equations.
Awards Received:
The First Prize in Canada Wide Virtual Science Fair 2008.
The best prize in the Japan Wide Science Paper Contest for High School Students 2007. This contest was sponsored by Kogakuin University in Tokyo.
We got the best prize and 300000 yen (approximately 2900 dollars) .
We got the best prize in the Japan Science and Engineering Challenge 2008 sponsored by Asahi News Paper.
We have been selected by Wolfram Research as featured contributors in their demonstration project.
Our results have been introduced in the Mathworld.
See “Josephus problem”, “Russian roulette” and “Pascal's triangle ” in the web page of it.
Copyright:
All the computer graphics are made by the authors using Mathematica.
Thanks:
We would like to express our gratitude to professor R. Cowen of New York City University for his wonderful suggestions. He attended the Mathematica Symposium when we presented our research and gave us great encouragement. We learned many things from his book.
We would like to thank Wolfram Research Ltd. who encouraged us and provided us with software and technical advice.
We would also like to thank Mr. Daisuke Minematsu, Mr.Hiroshi Matsui, Mr. Toshiyuki Yamauchi and Dr. Ryohei Miyadera for helping us in our research and for making home pages.
Reviews of our research.
We think that they deserve the best prize in this contest.
Kogakuin University (Japan).
The well-directed use of Mathematica as a teaching and research tool for students, proves evey day to be very stimulating for mathematical concepts understanding, and hands-on problem-solving creativity.
Dr. Miyadera and his student's work tend to prove even further, that the younger the students, the more creative and insightful their achievements can be, relatively to their age, academic education level and experience.
As a company with educational vocation, we have a high respect and regard for Dr. Miyadera's initiative, and support and encourage as many initiatives of that kind Wordwide, Mathematica is involving as many different student profiles and motivations, in as many projects as imagination allows.
Wolfram Research Asia Ltd.,