Problem Solving!

One of the few requirements for this SLOG is that we document a problem solving episode of a "algorithmic, mathematical nature". I wanted to work on the paper folding problem since I had already given it some thought after the class discussion, but I was still having a hard time with it once I tried it at home. Instead, I went on the problem solving wiki that was provided to us, and chose a new problem from there. I settled on the Paper Slicing problem.

Understanding the problem

Given: number of straight lines
Required: number of regions

In this problem you are given a number that represents a number of straight lines. These lines theoretically divide up a piece of paper into several regions. The number of regions for 1, 2, and 3 lines were provided, and are listed below:

1 line = 2 regions
2 lines= 4 regions
3 lines = 7 regions

To better understand the problem, I created diagrams:

Creating the diagram with 3 lines helped me to realize that in order to create the most regions, the most existing regions must be intersected with the new line. 

Devising a Plan

Seeing the data in the form of a chart made it seem as though addition could be used to solve the problem. Previous problems like this that I had encountered suggested that a certain value added to the number of previous regions in order to find out the number of regions in the current state.

Carrying out the Plan

Using my addition plan I figured out what needed to be added to 2 (the number or regions created by 1 line) in order to arrive at the value 4 (the number of regions created by 2 lines). Clearly 2+2 = 4. I then saw if 4+2 = the number of regions created by 3 lines, but it was one value short. Clearly, adding 2 to the previous number of regions was not the solution to this problem. 

Looking Back

Clearly my first solution was not correct, but that didn't mean that addition wasn't the solution.

Devising a Second Plan

While executing my first plan, I realized that I was only off by 1 when using the +2 technique for 3 lines. This made me wonder if the value that I needed to add should increase each time a new line is added. This meant that I would +2, followed by +3 (2+1) for 3 lines.

Carrying our the Second Plan

1 line = 2 regions

2 lines = 2+2 = 4 regions

3 lines = 4+3 =7 regions

By increasing the number of regions added by 1 for every line I was able to solve the problem for a maximum of three lines.

Looking back

I wondered if my solution would apply if I had more data to work with. Having only 3 lines only gave me 2 opportunities to try my solution with. I decided that I needed one more opportunity before I would feel confident that I has solved the problem.

Devising a Third Plan

I was able to create a diagram with the maximum number of regions created by 4 lines:

I was able to create 11 regions with 4 lines. 

Carrying out the Third Plan

Now that I had a third piece of data, I tried out my formula again to see if my previous solution still worked.

1 line = 2 regions

2 lines = 2+2= 4 regions

3 lines = 4+3= 7 regions

4 lines = 7+4= 11 regions

Looking Back

My solution still worked with the new piece of data, and I was able to conclude that my solution was accurate.