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Ижжж║4ю·/*Yb┘Dear editor.
We are going to submit our article "Variants of the Game of Nim that have Inequalities as Conditions".
This article has been written by a freshman at university and two high school students. Our sponsor is Dr. Ryohei Miyadera who has Ph.D.in mathematics and mathematics education.
The research has been completed by September 13 of 2009 at Kwansei Gakuin high school in Japan.
A biographical sketch.
Toshiyuki Yamauchi is now a student at Science & Technology Department of Kwansei Gakuin University . He loves mathematics and physics. He represented Japan at ISEF 2009, but Japanese team cancelled participation, because of swine flu.
Yamauchi was very disappointed with the decision of Japanese team.
Yamauchi has already published two articles at Undergraduate Math Journal. He has also published several articles at other math journals.
His dream is to be a good scientist.
Taishi Inoue is a student at Kwansei Gakuin. He loves mathematics very much and his dream is to be a good mathematician.
Yuuki Tomari is a student at Kwansei Gakuin. He loves art and films. His dream is to be a producer at broadcasting company.
Abstract
In this article the authors are going to present several combinatorial games that are variants of the game of Nim. They are very different from the traditional game of Nim, since the coordinates of positions of the game satisfy inequalities.
These games have very interesting mathematical structures. For example, the lists of P-positions of some of these variants are subsets of the list of P-positions of the traditional game of Nim. The authors are sure that they were the first people who treated variants of the game of Nim conditioned by inequalities.
Some of these games produce beautiful 3D graphs, and you can discover the Sierpinski gasket when you look at them from certain view point. The graphs are in the last part of this article, and they are full color graphics, but you can see them clearly in gray color, too.
We have studied this chocolate problem, and have presented our result in еcite{suurikaiseki2009}.
In this article the authors are going to present new results of the research of the chocolate problems.
The authors have used Mathematica a lot in their research of combinatorial games.
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