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       #Post#: 120--------------------------------------------------
       MISLEADING SEQUENCE
       By: eba95 Date: July 30, 2010, 6:55 am
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       Misleading Sequence
       Place n 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?
       Let's see:
       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 2n-1 regions, you
       would be wrong!
       Presentation Suggestions:
       Draw some pictures and have
       students form conjectures. This is a
       good example for stressing why
       proofs are so important for
       mathematicians.
       The Math Behind the Fact:
       The correct answer is a little more
       subtle: it is the sum of the first 5
       binomial coefficients for power (n-1):
       ((n-1) CHOOSE 0) + ((n-1) CHOOSE 1) +
       ((n-1) CHOOSE 2) + ((n-1) CHOOSE 3) +
       ((n-1) CHOOSE 4).
       See the reference for an explanation
       of where this formula comes from.
       This yields 31 regions for n=6, 57
       regions for n=7, 99 regions for n=8.
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