Dissection of Plane with Lines

Problem 1. Let the maximum number of sections of the plane you can divide with $2n$ lines be the function $f(n)$. Let the maximum number of sections of the plane you can divide with $n$ pairs of parallel lines be $g(n)$. Prove that $f(n) > g(n)$ for $n > 0$.

Problem 2. Let the maximum number of sections of the plane you can divide with $2n$ lines be the function $f(n)$. Let the maximum number of sections of the plane you can divide with $n$ pairs of perpendicular lines be $g(n)$. Prove that $f(n)=g(n)$ for $n > 0$.

If any of these two problems are actually wrong, feel free to disprove it in the comments.

Note by Daniel Liu
5 years, 2 months ago

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$\textbf{Problem 1.}$

Let $f(n)$ denote the number of regions a plane is divided by n lines....

Clearly $f(1) = 2$ , $f(2) = 4$ and $f(3) = 7$.

Now let us think about the recursive definition of the function $f(n)$,when the nth line enters the [plane it intersect each of the previous $n-1$ lines at one point only and in process creates $n$ new regions.Hence $f(n) = f(n-1) + n$

Solving this and putting $f(1) = 2$, we get, $f(n)\ = 1+\frac{n(n+1)}{2}$

So clearly the number of regions a plane is divided by $2n$ lines is $f(2n) = 1+n(2n+1) = 1+ 2n^{2} + n$.

Now let us the consider n pairs of parallel lines...

By inspection we can say that $g(1) = 3$ and $g(2) = 9$,

Now let us consider the nth pair of parallel lines being drawn...each line intersects the previous $2n - 2$ lines to form $2n-1$ regions!!

So the recursive definition is $g(n) = g(n-1) + 2(2n-1)$

Solving this recurrence relation we get $g(n) = 1 + 2n^{2}$. This relation's validity can easily be proved by induction.

Hence $f(n) = 1+2n^{2} + n > 1+2n^{2} > g(n)$.

- 5 years, 2 months ago

I think perpendicular lines will also follow the same recursion as $f(n)$ and hence the 2nd identity also correct....

- 5 years, 2 months ago

Here you have no where mentioned whether the lines can coincide wholly with each other.......

- 5 years, 2 months ago

If I understand what you are saying, then yes, more than three lines can intersect at one place. However, that case will not give the maximum number of sections.

- 5 years, 2 months ago