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© Copyright 1997, Jim Loy
A line
divides the plane into two pieces (regions). Draw another line. The plane is
now divided into three or four regions. It is three regions if the lines are
parallel, four if they intersect. For the purposes of this article, we want the
greatest number of regions. So, two lines, four regions. A third line divides
the plane into seven regions (See the diagram). The results, so far:
| lines | regions |
| 1 | 2 |
| 2 | 4 |
| 3 | 7 |
Using a little algebra, we can find an equation for the first three lines:
R(n) = an^2 + bn + c 2 = a + b + c 4 = 4a + 2b + c 7 = 9a + 3b + c ... R(n) = (n^2+n+2)/2
We see, from the diagram, that four lines produces eleven regions. So we see that our formula works for four lines. And we suspect that it is valid in general. How do we prove that?
Let's say that we've got n lines (for some arbitrary n). And we add an n+1th line. That line goes through region-line-region-line-...-line-region. It went through n lines and n+1 regions (assuming that all of the lines intersect). For each region that it went through, it added a region (split that region into two regions). So it added n+1 regions. So, we've just proved the rule: R(n+1)=R(n) + n + 1.
Our earlier formula, which works for some n, is:
R(n) = (n^2+n+2)/2
What is R(n+1), by that formula?
R(n+1) = [(n+1)^2 + (n+1) + 2]/2
= (n^2+3n+4)/2
= (n^2+n+2)/2 + n + 1
In other words:
R(n+1) = R(n) + n + 1
So, our formula follows the rule R(n+1)=R(n) + n + 1. This means that we have actually proved our formula, by Mathematical Induction. Our formula works for the first case (n=1). And we showed that if the formula works for some n, then it works for n+1.
By using Mathematical Induction, we have shown that if our formula works for n=1 (which it does), then it works for n=2. And if it works for n=2, then it works for n=3. And if it works for n=3, then it works for n=4, etc. In other words, it works for all, infinitely many cases, from 1 on up.
Our formula, R(n)=(n^2+n+2)/2 is an integer divided by 2. Does it always give us an integer for R(n)? It should, because we never end up with a fraction of a region. Well, if n is even, then n^2 is even and n^2+n+2 is even, we get an integer when we divide by 2. If n is odd, then n^2 is odd and n^2+n+2 is even, we get an integer when we divide by 2.
Incidentally, our formula works for n=0. With no lines, we find that the plane is divided into one region. In our proof, we could have saved a little effort, by using n=0 as the first case.
Notice: 100 lines divide the plane into 5051 regions. See my article, How To Be A Little Gauss, for an amazing "coincidence." Also notice that 1000 lines divide the plane into 500501 regions. This article was actually the inspiration for that article.