A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if points on its circumference
are joined by chords with no three internally concurrent.
The answer is
(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where 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 , it begins differing from this geometric
series at .
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. Monthly95,
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,