Circle Cutting
Determining the maximum number of pieces in which it is possible to divide a Circle for a given number of cuts iscalled the circle cutting, or sometimes Pancake Cutting, problem. The minimum number is always 
, where 
is thenumber of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cutcreates 2 regions, and the th cut creates new regions, so
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
Therefore,
| |  | |
|  | ||
|  | ||
|  | (4) |
Evaluating for
, 2, ... gives 2, 4, 7, 11, 16, 22, ... (Sloane's A000124).
A related problem, sometimes called Moser's Circle Problem, is to find the number of pieces into which a Circleis divided if points on its Circumference are joined by Chords with no three Concurrent. The answer is
 | |  | (5) |
|  | (6) |
(Yaglom and Yaglom 1987, Guy 1988, Conway and Guy 1996, Noy 1996), where
is a BinomialCoefficient. The first few values are 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (Sloane's A000127). This sequence andproblem are an example of the danger in making assumptions based on limited trials. While the series starts offlike 
, it begins differing from this Geometric Series at 
.See also Cake Cutting, Cylinder Cutting, Ham Sandwich Theorem, Pancake Theorem,Pizza Theorem, Square Cutting, Torus Cutting
References
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. SequencesA000124/M1041and A000127/M1119in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995. Yaglom, A. M. and Yaglom, I. M. Problem 47. Challenging Mathematical Problems with Elementary Solutions, Vol. 1. New York: Dover, 1987.
- Highest Weight Theorem
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