Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

What is the greatest number

What is the greatest number of parts the plane can be divided by:
n straight lines ?
n circles ?

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is the greatest number

For the lines:

There are two different "types" of divisions.  What are they?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: What is the greatest number

Straight Lines in a plane at funny angles:

I drew it in MSPaint and got this:

2n + BowlingPinNumber(n-2)

2n + (n-2)(n-1)/2

For the circles I am getting:

2n + twiceBowlingPinNumbers(n-2)

2n + (n-2)(n-1)

but I can't prove these, sorry.

Last edited by John E. Franklin (2008-07-18 07:11:09)

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igloo myrtilles fourmis

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: What is the greatest number

With straight lines the answer fits a quadratic function. Find the first few examples and solve for the polynomial coefficients.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: What is the greatest number

You guys aren't saying anything about how it is you're drawing these lines.  That's the hard part.  Figuring out the function once you know how to optimally draw them is easy.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Identity

Member

Registered: 2007-04-18

Posts: 934

Re: What is the greatest number

The nth optimal line can intersect (n-1) other lines. The nth, line, by simply existing, creates an extra region by dividing the current one into two, and for every line it intersects, it creates another region.
So the nth optimal line creates 1+(n-1) = n regions.

The empty plane already has 1 region, so for n lines, you will get 1+(1+2+3+...+n) regions.

This is equal to

regions.

For the circles, the nth optimal circle will intersect each of the other (n-1) circles twice. If you draw several examples you will see that each time the new circle intersects the others, a new region is created (this also accounts for the region created by the new circle itself). So the nth circle will create 2(n-1) = 2n-2 new regions.

The first circle is a bit of a tricky case, so we'll just start the sum with the 2nd circle and add 2 to the end. For n circles you will get (2(2)-2+...+2(n)-2)+2 regions.

This is equal to

!!!!!!

Last edited by Identity (2008-07-18 17:13:21)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: What is the greatest number

Sorry I'm not good at explaining it like Identity, but our
answers are the same if you expand mine.  Mine was
mostly guesswork, but Identity seems to have a good
description.  All I can say is there's stuff in the
middle of the plane, and stuff around the outside like
petals on a flower, for the straight lines, anyway.

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igloo myrtilles fourmis

cushydom

Member

Registered: 2008-04-23

Posts: 10

Re: What is the greatest number

for lines I get:
1+((n-2)(n-1)/2)