The ancient Greeks raised the famous problem of trisecting an arbitrary angle with a compass and an unmarked ruler, which was proved impossible. Such a construction is possible if a marked ruler is used instead. In this article, the possible geometric constructions by a compass and a marked ruler are studied.
People sometimes need to distribute or select things randomly. Some do this by drawing lots, some by throwing dice, and others by drawing “ghost leg”. Our group finds that the input and output (elements to be arranged) can be considered as a sequence while the “ghost leg” itself as a permutation. So each horizontal line leads to a transformation of neighboring elements. Also these lines can be transformed and deleted under some conditions without affecting the input and output sequences. And thus a simplified one is obtained.
Our objective in this project is to systemize and solve “ghost leg” mathematically so that result can be obtained quickly and systematically.
A shopping mall can be divided into different regions with different numbers of people. After a person has finished shopping in a shop the person may want to go to other levels but there are only limited numbers of lifts in certain positions. In our project, we attempted to find the best position for installing lifts in a shopping mall such that the total walking distances for people to reach the lifts can be minimized.
The aim of our project was to prove our conjecture that the product of consecutive positive integers is never a square. In our investigation, we had developed three approaches to prove it.
In the first approach, we used the fact that a number lying between two consecutive squares is never a square to prove that the product of eight consecutive integers is never a square. Then we made use of relatively prime-ness of consecutive integers to prove the rest.
In the second approach, we had used Bertrand’s Postulate Theorem to obtain a beautiful theorem that the product of consecutive positive integers is never a square if there is a prime number among them. Besides we had found some interesting results from this theorem.
When we started our project, we thought that our conjecture had not been proved. However, we found later in a website that our conjecture has already been proved by two famous mathematicians P. Erdos and J.L. Selfridge in 1939. Although our conjecture was proved, we didn’t give up but tried our best to develop our third approach.
In the third approach, we had referred to an academic journal written by P. Erdos and J.L. Selfridge and knew that the square-free parts of consecutive integers are distinct. By counting, we arrived at a necessary condition for the product not to be a square. Unluckily, we then discovered the limitations of the third approach when the number of consecutive integers is very large. It may be due to the roughness of our estimation. Although we couldn’t complete the proof of our conjecture, we all enjoyed the process of formulating conjectures and thinking new ideas of solving problems through the cooperation among our team members in the past few months.