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 will produce beautiful 3D graphics (indeed, you will see the Sierpinski gasket when you look from a certain view point). We will also present some new results for the chocolate problem, a problem which was studied in a previous paper and related to Nim. The authors make substantial use of Mathematica in their research of combinatorial games.

Author Bio

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.