Geometry Question?

Question

One line can divide a plane into two regions. Two lines can divide a plane into four regions. Three lines intersecting at a point can divide a plane into six regions, BUT you can get more regions than that if the lines don't intersect at a single point.

The question: If you use SIX LINES, what is the MAXIMUM number of regions into which you can divide the plane?

10 points will go to the person with the correct answer and I will not choose a best answer until next Monday at the earliest. If possible, I would really appreciate a sketch of the number of regions you came up with the answer. Thank you for any help I can get.

Answer 1

1 line makes a max of 2 regions
2 lines make 4 regions (previous+2)
3 lines make 7 regions (previous +3)
4 lines make 11 regions (previous +4)
note the pattern to go from 1 to 2 add 2
So 5 lines make 16 regions ( previous + 5)
6 lines 22 regions (previous +6)
I drew it out on scrap paper. To maximize the nunmber of regions you want to cross as many lines as possible so don't intersect at a common point.

Answer 2

The answer is 22.

One line divides the plane into 2.

Adding a 2nd line produces 2 more regions, bringing the total to 4.

Adding a 3rd line produces a maximum of 3 more regions, if the new line cuts the 2 existing lines. This brings the total to 7.

Adding a 4th line produces a maximum of 4 more regions, if the new line cuts the 3 existing lines. This brings the total to 11.

Adding a 5th line produces a maximum of 5 more regions, if the new line cuts the 4 existing lines. This brings the total to 16.

Adding a 6th line produces a maximum of 6 more regions, if the new line cuts the 5 existing lines. This brings the total to 22.

Answer 3

I'm not computer literate enough to post a diagram here, but if you email [e-mail] I'll email one to you.

The number is 57, I think. It seems to be the same as dividing a circle into regions using non-concurrent lines, and many years ago I worked out the formula for this.

The interesting thing is that if you start listing the number of regions for various numbers of points, starting with 1 point, the numbers of regions are
1, 2, 4, 8, 16, so just when you think aha! they're all powers of 2, the next number is 31, and you think "have I lost one? Surely it's 32." But it isn't. And the next one is 57.

The formula is actually
f(n) = 1 + (n/24)*(n-1)*(n^2 - 5n + 18).

I think for your problem, for n to be the number of lines we might have to replace n by n+1 in my formula, giving

f(n) = 1 + (n/24)*(n+1)*(n^2 - 3n + 14)

and f(6) = 57.


Yeah, OK, I just drew the diagram and sahsjing is right (though I hadn't seen his answer before drawing it). My formula relates to taking a circular region, marking a few points on it, and joining all those points. So if you have 3 points there are 3 lines (4 regions), but put in a fourth point there are 6 lines (8 regions), and a fifth point gives 10 lines (16 regions) etc -- as long as no three lines are concurrent, of course.

Answer 4

If you draw n lines which divide the plane in q_n regions. When you draw a new line it will meet at most the previous lines in n distinct points. These points divide the new line in at most (n+1) segments, 2 of them infinite. Each of these segments correspond to a region visited by the new line which divides it in 2, thus adding at most (n+1) regions.
Hence q_1=2, q_2 less than 4, q_3 less than 7, q_4 less than 11, q_5 less than 16, and q_6, less than 22. If no more than 2 lines meet at any given point and if they all cut each other, then the upper bound is attained. That means you can just take anything, except for the obvious counterexamples.

Answer 5

7.

Draw 2 lines in an X. Make the 3rd line go through both the other 2 beyond the point that they intersect. You have something like an anarchy symbol you are looking at. This makes 6 outside sections and a triangle in the middle = 7.

Answer 6

Max. no of regions formed by “n” number of lines has the general formula:
“2” raised to the power of “n”.
perhaps you have to use 6-D geometry for that as like 2-D & 3-D geometry

Answer 7

# of lines # of maximum regions
1 line: 2 regions
2 lines: 4 regions
3 lines: 7 regions
4 lines: 11 regions
5 lines: 16 regions
6 lines: 22 regions

Can you see the pattern?

If you add the nth line, you add n more regions.

Answer 8

Roflwtime!