1, 2, 4, 8, 16, ... and the pattern that breaks

Question

Place points around a circle and connect every pair with a straight chord (positioned so no three chords cross at the same interior point). Count the regions the chords cut the disk into:

1 point: 1 region
2 points: 2 regions
3 points: 4 regions
4 points: 8 regions
5 points: 16 regions

The pattern screams "doubling," so 6 points should give 32.

Actually count the regions for 6 points. Is it really 32?

Accepted Answer

No! Six points give 31 regions, not 32. The doubling pattern is a famous trap. Count with a reliable formula. The number of regions created by all chords of n points on a circle (in "general position," no three chords meeting at one interior point) is R(n) = 1 + \binom{n}{2} + \binom{n}{4}. Here \binom{n}{2} counts the chords and \binom{n}{4} counts the interior crossing points (each set of 4 points gives exactly one crossing of the two "diagonal" chords). Check it against the small cases: - n=5: 1 + \binom{5}{2} + \binom{5}{4} = 1 + 10 + 5 = 16. - n=6: 1 + \binom{6}{2} + \binom{6}{4} = 1 + 1…

1, 2, 4, 8, 16, ... and the pattern that breaks

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