Linked Questions

2
votes
2answers
14k views

How many regions do $n$ lines divide the plane into? [duplicate]

Suppose you draw $n \ge 0$ distinct lines in the plane, one after another, none of the lines parallel to any other and no three lines intersecting at a common point. The plane will, as a result, be ...
3
votes
1answer
1k views

The number of regions into which a plane is divided by n lines in generic position [duplicate]

Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r(n)$ be the number of regions the plane is divided into after ...
0
votes
1answer
101 views

Into how many regions do they divide the plane? [duplicate]

Suppose that n lines in general position are given in a plane. (General position means that no two lines are parallel, and no three lines have a common point.) Into how many regions do they divide the ...
0
votes
0answers
38 views

How can you show that each new line drawn on a plane creates another $k+1$ regions? [duplicate]

The questions is If $n$ straight lines are drawn on a plane such that each line intersects every other line and no three lines have a common point of intersection, then the plane is divided into $$...
0
votes
0answers
26 views

How many closed polygons $n$ lines make on a plane?(not duplicate) [duplicate]

How many closed polygons will $n$ coplanar lines, all intersecting each other (no three concurrent), form on the plane? Can we find a recursion formula for this?? Is there a way out?
0
votes
0answers
25 views

Prove simple induction [duplicate]

Suppose we we have n straight lines on the plane such that no two of them are parallel and no three of them go through the same point. Prove that the number of different regions that are created by ...
79
votes
6answers
6k views

20 circles in the plane, all passing through the origin

Suppose I draw $20$ circles in the plane, all passing through the origin, but no two tangent at the origin. Also, except for the origin, no three circles pass through a common point. How many regions ...
7
votes
3answers
5k views

Greatest number of planes we can get when dividing with lines and circles

What is the greatest number of parts a plane can be divided into using $n$ infinite straight lines? What about $n$ circles? Can you generalise this into 3-dimensional space, planes and spheres? For ...
1
vote
1answer
735 views

Maximum number of pieces of pizza when making 7 cuts

If we have a circular pizza then the maximum number of pieces we can get by making $7$ cuts in it? The fact that I know the solution only got me the way to find it but it was like a kid trying to ...
0
votes
1answer
535 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
2
votes
1answer
64 views

Into how many finite regions does $n$ lines in general position divide the plane?

I have $n$ lines in the plane such that no two are parallel and no three lines intersect in a common point. Into how many finite regions do the lines divide the plane? I came up with this ...