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: Introduction of the theory

Combinatorial Games and Beautiful Graphs Produced by them.

Masakazu Naito, Toshiyuki Yamauchi, Hiroshi Matsui, Taishi Inoue, Yuuki Tomari,
Koichiro Nishimura, Takuma Nakaoka, Daisuke Minamatsu and Ryohei Miyadera

Kwanseigakuin High School, Kobe City, Japan

In this article the authors are going to present several combinatorial games that are variants of the game of Nim. These games have very interesting mathematical structures. They are very different from the traditional game of Nim, since the coordinates of positions of the game satisfy inequalities. The authors are sure that this article is the first one to treat variants of the game of Nim that have inequalities as conditions.

Some of these games produce beautiful 3D graphs, and you can discover the Sierpinski gasket when you look at them from certain view point.

If you are interested in beautiful graphs, please read the section "Beautiful graphs produced by variants of Nim"in this article.

The authors have used computer algebra system Mathematica a lot in their research of combinatorial games.