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 2^n-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 binomial coefficients for power (n-1):

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

References: John H. Conway and Richard K. Guy, The Book of Numbers, Springer-Verlag, 1996, pp.76-79. Subjects:    geometry, combinatorics Level:    Medium Fun Fact suggested by:   Francis Su Suggestions? Use this form. 4.25 current rating Click to rate this Fun Fact...     *   Awesome! I totally dig it!     *   Fun enough to tell a friend!     *   Mildly interesting     *   Not really noteworthy and see the most popular Facts! Get the Math Fun Facts iPhone App!

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