**Ryohei Miyadera
(Kobe University)
Daisuke Minematsu
Satoshi Hashiba
(Kwansei Gakuin)**

**0. Abstract**

Please see the beautiful figure bellow. We made this figure by calculating , and substituting the number with colours. The theme of this article is to explain the reason why we can make this beautiful figure.The following function is the mathematical structure behind the figure.

Perhaps this article can give one of the most beautiful examples of the binomial Theorem. We did most of the research with high school level mathematics, but I am sure that the result is quite new to mathematicians and teachers, and the result can teach students how beautiful a simple theorem can be.

Here we are going to introduce the result of a joint research of high school students and a mathematician. Our research was very similar to a detective work, and at last we found an answer to our own questions after a lot of trials, errors and reasoning. Many of the ideas were proposed by high school students, and backed up by the knowledge of the mathematician.

We are going to write this article as a story of finding a theorem. By finding new facts we gradually go nearer and nearer to the conclusion.

Our result suggests that a good use of computer algebra system can open up a completely new aspect in mathematics education, because this kind of activity can give students true joy of mathematics. The joy of finding new things is surprisingly strong, and we were thrilled to find attractive pictures.

We used a computer algebra system Mathematica. Even if you do not know anything about computer algebra systems, you can understand our article. Although our farvorite is Mathematica, other people can do the same kind of things using other computer algebras, and some of them are free now.

- 1. Introduction.
- 2. The observation of facts and questions.
- 3. The answer to the previous questions.
- 4. Related facts.
- 5. An open problem.