Cake Cutting

It is always possible to ``fairly'' divide a cake among $n$ people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes that everyone has received $1/n$ of the cake according to his own measure. Finally, if there is some piece on which two people disagree, then there is a way of partitioning and dividing a cake such that each participant believes that he has obtained more than $1/n$ of the cake according to his own measure.

Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a Circle into $n$ equal Area pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the Frobenius-König Theorem.

See also Circle Cutting, Cylinder Cutting, Envyfree, Frobenius-König Theorem, Ham Sandwich Theorem, Pancake Theorem, Pizza Theorem, Square Cutting, Torus Cutting

References

Brams, S. J. and Taylor, A. D. ``An Envy-Free Cake Division Protocol.'' Amer. Math. Monthly 102, 9-19, 1995.

Brams, S. J. and Taylor, A. D. Fair Division: From Cake-Cutting to Dispute Resolution. New York: Cambridge University Press, 1996.

Dubbins, L. and Spanier, E. ``How to Cut a Cake Fairly.'' Amer. Math. Monthly 68, 1-17, 1961.

Gale, D. ``Dividing a Cake.'' Math. Intel. 15, 50, 1993.

Jones, M. L. ``A Note on a Cake Cutting Algorithm of Banach and Knaster.'' Amer. Math. Monthly 104, 353-355, 1997.

Rebman, K. ``How to Get (At Least) a Fair Share of the Cake.'' In Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 22-37, 1979.

Steinhaus, H. ``Sur la division pragmatique.'' Ekonometrika (Supp.) 17, 315-319, 1949.

Stromquist, W. ``How to Cut a Cake Fairly.'' Amer. Math. Monthly 87, 640-644, 1980.