$\begingroup$What is the greatest number of regions a plane can be divided into using $n$ straight lines? What about $n$ circles?
Can you generalize this into 3-dimensional space, planes and spheres?
For lines, I got $U_{n+1}=U_n+n,$ with $U_0=1.$ And for circles, I got $U_{n+1}=U_n+2n,$ with $U_0=1$ and $U_1=2.$
Am I right so far?
$\endgroup$04 Answers
$\begingroup$Lines
For a plane with $n$ lines, consider what happens when you add a line. It divides all the regions through which it passes into two, thus adding one region for each region it passes through. The number of regions it passes through is the number of lines it crosses plus one, that is the number of points it creates plus one for itself. Thus, the number of regions added to the original one, $\binom{n}{0}$, is the number of points, $\binom{n}{2}$, plus the number of lines, $\binom{n}{1}$. Thus, the maximum number of regions is $$ \binom{n}{0}+\binom{n}{1}+\binom{n}{2}=\frac{n^2+n+2}{2} $$
Circles
For a plane with $n$ circles, consider what happens when you add a circle. It divides all the regions through which it passes into two, thus adding one region for each region it passes through. The first circle divides the plane into two. The number of regions it passes through is at least one, if it doesn't intersect any other circles, and up to the number of crossings with other circles. For $n$ circles there can be up to $2\binom{n}{2}$ crossings. Thus, the first circle divides the plane into two, $2\binom{n}{0}$, then the rest of the circles add the number of crossings. Thus, the maximum number of regions is $$ 2\binom{n}{0}+2\binom{n}{2}=n^2-n+2 $$
$\endgroup$6$\begingroup$if plane divided by lines, recursion function is $R(n+1)=R(n)+n+1$, where $R(0)=1$, $n\geq0$ Solving recurrence $R(n)=\frac{n^2+n+2}{2}$
if plane divided by circles, recursion function is $R(n+1)=R(n)+2n$, where $R(1)=2$, $n>0$ Solving recurrence $R(n)=n^2-n+2$
For a plane divided by lines: $$l_{n+1}=l_{n}+n+1$$
For space divided by planes: $$s_{n+1}=s_{n}+l_{n}$$
$\endgroup$1$\begingroup$Start with an empty plane. That’s one region. Then add $n$ parallel lines. You add $n$ more regions. Then tilt one line. Every time it hits another line a new region is created. Tilt all lines. You get $n$ choose 2 new regions for each pair of lines hitting each other. The sum is the formula above.
Likewise, in 3-D space start with an empty space which is one region. Then add $n$ parallel vertical walls, which add $n$ new regions. Then rotate the walls while keeping them vertical. They will meet in $n$ choose 2 vertical lines creating as many new regions. Then tilt the walls and they will meet in $n$ choose 3 points creating as many new regions (look up in the room you’re in right now!). The sum is the formula above.
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