Circle Division by Chords


A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if n points on its circumference are joined by chords with no three internally concurrent. The answer is

g(n)=(n; 4)+(n; 2)+1

(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where (n; m) is a binomial coefficient. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (OEIS A000127). This sequence demonstrates the danger in making assumptions based on limited trials. While the series starts off like 2^(n-1), it begins differing from this geometric series at n=6.

See also

Cake Cutting, Circle Division by Lines, Cylinder Cutting, Ham Sandwich Theorem, Pancake Theorem, Pizza Theorem, Plane Division by Circles, Plane Division by Ellipses, Plane Division by Lines, Square Division by Lines, Torus Cutting

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Conway, J. H. and Guy, R. K. "How Many Regions." In The Book of Numbers. New York: Springer-Verlag, pp. 76-79, 1996.Guy, R. K. "The Strong Law of Small Numbers." Amer. Math. Monthly 95, 697-712, 1988.Noy, M. "A Short Solution of a Problem in Combinatorial Geometry." Math. Mag. 69, 52-53, 1996.Sloane, N. J. A. Sequence A000127/M1119 in "The On-Line Encyclopedia of Integer Sequences."Yaglom, A. M. and Yaglom, I. M. Problem 47 in Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, 1987.

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Circle Division by Chords

Cite this as:

Weisstein, Eric W. "Circle Division by Chords." From MathWorld--A Wolfram Web Resource.

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