Circle division by chords, sometimes known as Moser's circle problem, asks for the number of regions into which a closed disk is divided
by joining every pair of distinct points on its circumference
by a chord, with no three chords
internally concurrent. For a nonnegative
integer,
the answer is
(1)
(2)
(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where is a binomial coefficient.
For ,
2, ..., 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 sequence starts off like
,
it begins differing from this geometric sequence
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. https://doi.org/10.1080/0025570X.1996.11996383.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.
distinct points on its circumference
by a chord, with no three chords
internally concurrent. For a nonnegative
integer 
,
the answer is
is a binomial coefficient.
For 
,
2, ..., 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 sequence starts off like
,
it begins differing from this geometric sequence
at 
.
