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 Does it continue? 1: Chord regions (Posted on 2017-09-13)
In his paper The Strong Law of Small Numbers Richard Guy states "There aren't enough small numbers to meet the demands made of them."

It's a great list of 35 examples where the pattern noted early on may or may not continue. Unfortunately, if you read it, you will give away a series of around 10 puzzles I plan to create from it.

Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Place n points around a circle so that no three of the C(n,2) chords joining them are concurrent. Count the number of regions into which the chords partition the circle.

n=0, 1 region
n=1, 2 regions (a single chord)
n=2, 4 regions (the chords form a triangle)
n=3, 8 regions
n=4, 16 regions

A pattern has emerged. Does it continue?

 No Solution Yet Submitted by Jer No Rating

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 n = 6 | Comment 5 of 9 |

You can only have at most 31 regions for 6 points on circle.

Let A, B, C, D, E are the first 5 points in that order, which form 16 regions as desired. Now let F be the 6-th point. Then by drawing FA and FE, you increase one region each, because they cross no other cords. By drawing FB and FD, you increase 4 regions each, because they both can cross 3 other cords. FC can cross 4 other cords, so 5 extra regions. Therefore, the number of extra regions due to 6-th point is 1+4+5+4+1 = 15

 Posted by chun on 2017-09-13 14:04:19

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