circle chord division

[jasen]

Place 6 points along a unit circle, in such a way that when you draw all lines connecting every pair of points, no more than two lines pass through any interior point. How many regions does this divide the unit disk into?

[CaptainEd]

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Now I get 31. picture shows additional subdivision of affected regions created by lines connecting to 6th spot

 

[BobbyGo]

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32

# of Segments -> X

# of Points on Circle -> Z

X = 2^(Z-1)

Unless you are only counting the segments that are made up of straight sides and not the segments that utilize part of the circle.

In that case: 26

X = 2^(Z-1) - Z

 

Edited October 21, 2015 by BobbyGo

[jasen]

  On 10/21/2015 at 1:05 PM, BobbyGo said:

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wrong! You are trapped. Check again.

[CaptainEd]

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I can see 30, but not more

[jasen]

  On 10/22/2015 at 9:19 PM, CaptainEd said:

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Right !

This case is one of the famous misleading sequence

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For n=1, you get 1 region.
For n=2, you get 2 regions.
For n=3, you get 4 regions.
For n=4, you get 8 regions.
For n=5, you get 16 regions.

See a pattern?

Yes, but if you conjecture that n points produces 2^n-1 regions, you would be wrong!

The correct answer is a little more subtle: 
it is the sum of the first 5 binomial coefficients for power (n-1):

              1                        = 1
            1   1                      = 2
          1   2    1                   = 4
        1   3   3     1                = 8
      1   4   6    4     1             = 16 
    1   5   10  10    5     1          = 31
  1   6  15   20   15    6     1       = 57
1   7  21   35   35   21    7     1    = 99