Circle Regions
There are distinct points chosen at random on the circumference of a circle, and each point is joined to every other point by a chord, dividing the circle into different regions.
True or False: Assuming that three or more chords do not intersect at the same point inside the circle, the equation holds true for every positive integer value of .
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False. Counter-example: When n = 6, R = 31.
This is a classic problem demonstrating the dangers of inductive reasoning, because although R = 2 n − 1 holds true for positive integers n from 1 ≤ n ≤ 5 , it is not true for n ≥ 6 . The number of points n and the number of regions R are actually related by the equation R = 1 + ( 2 n ) + ( 4 n ) , 1 for the starting region of the circle, ( 2 n ) for the number of possible chords that can be drawn (each new chord adds a new region), and ( 4 n ) for the number of possible quadrilaterals that can be drawn (each new quadrilateral has two intersecting diagonals that adds a new region).