fair division

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. There are also similar methods of dividing collections of individually indivisible objects among two or more people when cash payments are used to even up the final division.

cake number | circle division by chords | circle division by lines | cube division by planes | cylinder cutting | envyfree | Frobenius-König theorem | ham sandwich theorem | pancake theorem | pizza theorem | space division by planes | square division by lines | torus cutting | voting paradoxes

cake | Alan D. Taylor | Bronisław Knaster | Hugo Dyonizy Steinhaus | Stefan Banach | Steven Brams