Circles and Regions

Geometry Level 3

Let X be the maximum number of regions that a circle with 2014 chords can have. Find the sum of the digits in X.


The answer is 20.

Alex Segesta by Alex Segesta , 20, USA

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2 solutions

Erlo Villegas
Mar 23, 2014

The maximum regions that a circle could have by intersecting an "n" number of chords is given by

Max No. of Regions X=[(n/2) (n+1)] + 1

Substitute 2014 for n
X =2029106

Sum of digits is 20.

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where did you get the formula sir, please elaborate? @Alex Segesta and @ErLo Villegas

Mardokay Mosazghi - 6 years, 4 months ago

Ive discovered a formula for this using quadratic functions: f(n)=(1/2)(n^2)+(1/2)(n)+1 This can be derived by the fact that 1 chord can cut the circle at most 2 regions; 2 chords, 4 regions and 3 chords, 7 regions; Its equivalent on saying that f(1)=2 , f(2)=4 and f(3)=7 Then prove this to be a quadratic function then find the general formula

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