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 a conjecture. 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 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.

How to Cite this Page: 
Su, Francis E., et al. “Misleading Sequence.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

John H. Conway and Richard K. Guy, The Book of Numbers, Springer-Verlag, 1996, pp.76-79.

Fun Fact suggested by:  
Francis Su

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