number of regions

This problem is not easy. This problem requires keen observational skills in order to solve. My approach was to solve a simpler problem. Of course, we could try to draw 7 arbitrary lines and stop when we can no longer improve on the current solution. I try to avoid the brute-force style of problem-solving because, as a problem-solver, I feel as if I have gained no more understanding.

Let's start with a simpler problem. Instead, let's ask the following question: What is the maximum regions in which 0 lines divide a plane? The answer may seem obvious, but solving the easier versions of a difficult problem may reveal some patterns that are generalizable to the more difficult one. By default, a plane is a region. Without any lines to divide that region, there is only one distinct region. 

Let's introduce 1 line that lies on the plane and see what happens now. Again, this is an easy problem. Regardless of how a line is oriented within the plane, a line will divide a plane into two distinct regions.It's unavoidable. Here is a diagram to represent this phenomenon.

How about 2 lines? This problem is slightly harder, but this problem is still manageable because there are few options. Wrestling with this problem for a short while reveals that we can maximize the number of regions by ensuring that the individual lines intersect at some point on the plane. Below reveals my solution for 2 lines. For 2 lines, the best we can do is divide the plane into 4 distinct sections or regions.

With 3 lines, there are even more options to explore, but this problem is still duable. By some trial and error, again we learn to avoid drawing parallel lines. While it is possible to construct a line that passes through the intersection point of the previous two lines, there is still a better solution. Here is my solution I discovered. It splits the plane into 7 regions. 

I decided to overachieve and constructed the 4th line to see if a pattern emerges. Careful counting reveals that there are 11 distinct regions. Don't miss any!

Let \(n\) represent the number of lines that lie on an arbitrary plane, and let \(L_n\) represent the maximum number of regions in which one can divide a plane with \(n\) lines. Below are the results condensed into a table.

I stared blankly at this table for some time and searched for any noticeable patterns, if any exist at all. I then realized that I could use a recursive definition to generate the nth term. Namely, \(L_n=L_{n-1}+n,n\geq1,L_0=1\). This is an awesome discovery! Of course, we should never take a pattern for granted, even if it seems to hold true for every n, so let's think about why this pattern will hold for larger values of n.

Observe that inserting a straight line into a region, whether bounded or unbounded, will divide an arbitrary region into two distinct regions. The total number regions after inserting an arbitrary line will be the number of regions of the region plus the number of times the line enters a region, which will happen n times.

Once and for all, let's finally get the answer to the question with the power of recursion!

\(L_n=L_{n-1}+n\\ L_7=L_6+7\\ L_6=L_5+6\\ L_5=L_4+5\\ L_4=11\\ L_5=16\\ L_6=22\\ L_7=29\)

Therefore, the maximum number of regions for 7 straight lines is 29.