Introduction(1).doc

Introduction
When we were young, we always chopped some food into pieces for fun, like biscuit sticks, cakes, cheese etc. When we were chopping them, we tried to divide them into as much as possible pieces so the food always looked chippy. As we grew up, we learned more knowledge and came to realize that the number of pieces when an object is cut can be a tricky and interesting problem. Thus, today in this task I will investigate the maximum number of pieces obtained when an n-dimensional object is cut.
To begin with, I decide to investigate the “biscuit stick-chopping” problem, which means to quest for the maximum number of a segment obtained from cuts.
To make it more clearly, I use diagrams to explain this.
This is a segment.
If we separate it with only one cut, we will get this:
1 cut made the segment had 2 patches. It is the unique way and with 1 cut, the maximum segments we can get is 2.
Then, we added one more cut, the segment e like this:
2 cuts broke the segment into 3 parts. It is also the only way and through this way we got the maximum pieces divided by 2 cuts, 3 pieces.
We used 3 cuts to keep on our investigation, the segment would be the pattern shown below:

It’s obvious that the segment was divided into 4 sections and this was we only can do. The segment can be cut into most 4 parts by 3 cuts.
Now, I wondered if we put 4 cuts, whether the segment is separated into 5 parts.
My diagram proved that 4 cuts can divide the segment into most 5 parts, and it was the one and only way.
From above, I can conclude that, for a finite one-dimensional object, the maximum number of segments obtained from n cuts followed a rule which is:

means the pieces we can get, means the times we cut.
Then, I wondered if this rule fitted the two-dimensional objects, so I extend the investigation to see the maximum number of regions () obtained when chords are drawn.
Our investigation began with the supposition that I have a circle in front of me and