An Interesting Elementary Mathematics Problem

  On Thursday, Mrs. Choo always appoints three oral reporters to give their reports simultaneously at three small groups. After all the persons have done once, they all should shift their places and give another report to a different group. Suppose there are three reporters named A, B and C and three groups--1, 2 and 3. Now, suppose who are giving their oral reports at group 1, 2 and 3 are respectively A, B and C. After that, everyone should shift their places. And there are just two possibilities—BCA or CAB (Note: everyone should shift his place). Then suppose there are N people who are giving their oral reports respectively at N places. And after they all have done once, everyone should shift their place. Here comes the question: How many possibilities are there in this situation? It seems like a simple math problem and we have done lots of exercises similar to it when we are in senior high school. But I can promise you that it is an elementary but a little difficult problem which is also interesting. I realized this problem after three people finishing their oral reports on Thursday last week. Having thought it for a long time, I solved this problem with Wu Yu’s help. The answer is “ i=2∑N N!/i!* (-1)^i ” in term of N. I am very happy to find this problem and solve it eventually. From this, you can also see the strong relationship between life and math. The appearance of math was based on life and was to make people's lives more convenient. Although sometimes the development of math seemed detached from all the materials we discover in reality, all the important progresses in math originated from the progresses in human's perspectives towards the world. For example, the development from calculus to analysis was greatly influenced by the development in industries. Mathematics is powerful and mysterious. I love it.