It is always possible to ``fairly'' divide a cake among people using only vertical cuts. Furthermore, it is possible to
cut and divide a cake such that each person believes that *everyone* has received 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 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 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.

**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.

© 1996-9

1999-05-26