 Pizza_theorem  Pizza  List_of_circle_topics  Proof_without_words  Fair_division  Dissection_puzzle  Fourth_normal_form  Calculus  Tommaso_Buscetta  The_Zero_Theorem  Remarkable_Theorem  The_Smelly_Car  Join_dependency  Slice  1986_in_organized_crime  Giovanni_Falcone  List_of_University_of_California,_Berkeley_alumni  Matt_Damon_filmography  Harold_J._Morowitz  Riki_Lindhome  Baseball_Prospectus  List_of_Johns_Hopkins_University_people  The_Age_of_Spiritual_Machines  Male_prostitution_in_the_arts  Zits  List_of_people_from_Michigan  List_of_common_misconceptions  List_of_Purdue_University_people  List_of_fictional_books  Matt_Damon  List_of_Google_hoaxes_and_easter_eggs 
In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way.
Let p be an interior point of the disk, and let n be a number that is divisible by four and greater than or equal to eight. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n/2 − 1 times by an angle of 2π/n radians, and slicing the disk on each of the resulting n/2 lines. Number the sectors consecutively in a clockwise or anticlockwise fashion. Then the pizza theorem states that:
The pizza theorem is so called because it mimics a traditional pizza slicing technique. It shows that, if two people share a pizza sliced in this way by taking alternating slices, then they each get an equal amount of pizza.
The pizza theorem was originally proposed as a challenge problem by Upton (1968); the published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors. Carter & Wagon (1994a) provide an alternative proof by dissection: they show how to partition the sectors into smaller pieces so that each piece in an oddnumbered sector has a congruent piece in an evennumbered sector, and vice versa. Frederickson (2012) has given a family of dissection proofs for all cases (in which the number of sectors is 8, 12, 16, ...).
The requirement that the number of sectors be a multiple of four is necessary: as Don Coppersmith showed, dividing a disk into four sectors, or a number of sectors that is not divisible by four, does not in general produce equal areas. Mabry & Deiermann (2009) answered a problem of Carter & Wagon (1994b) by providing a more precise version of the theorem that determines which of the two sets of sectors has greater area in the cases that the areas are unequal. Specifically, if the number of sectors is 2 (mod 8) and no slice passes through the center of the disk, then the subset of slices containing the center has smaller area than the other subset, while if the number of sectors is 6 (mod 8) and no slice passes through the center, then the subset of slices containing the center has larger area. An odd number of sectors is not possible with straightline cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.
Mabry & Deiermann (2009) also observe that, when the pizza is divided evenly, then so is its crust (the crust may be interpreted as either the perimeter of the disk or the area between the boundary of the disk and a smaller circle having the same center, with the cutpoint lying in the latter's interior), and since the disks bounded by both circles are partitioned evenly so is their difference. However, when the pizza is divided unevenly, the diner who gets the most pizza area actually gets the least crust.
As Hirschhorn et al. (1999) note, an equal division of the pizza also leads to an equal division of its toppings, as long as each topping is distributed in a disk (not necessarily concentric with the whole pizza) that contains the central point p of the division into sectors.
Hirschhorn et al. (1999) show that a pizza sliced in the same way as the pizza theorem, into a number n of sectors with equal angles where n is divisible by four, can also be shared equally among n/4 people. For instance, a pizza divided into 12 sectors can be shared equally by three people as well as by two; however, to accommodate all five of the Hirschhorns, a pizza would need to be divided into 20 sectors.
Cibulka et al. (2010) and Knauer, Micek & Ueckerdt (2011) study the game theory of choosing free slices of pizza in order to guarantee a large share, a problem posed by Dan Brown and Peter Winkler. In the version of the problem they study, a pizza is sliced radially (without the guarantee of equalangled sectors) and two diners alternately choose pieces of pizza that are adjacent to an alreadyeaten sector. If the two diners both try to maximize the amount of pizza they eat, the diner who takes the first slice can guarantee a 4/9 share of the total pizza, and there exists a slicing of the pizza such that he cannot take more. The fair division or cake cutting problem considers similar games in which different players have different criteria for how they measure the size of their share; for instance, one diner may prefer to get the most pepperoni while another diner may prefer to get the most cheese.
Part of a series on 
Pizza 

Main articles

Pizza tools

Related topics

Other mathematical results related to pizza slicing involve the lazy caterer's sequence, a sequence of integers that counts the maximum number of pieces of pizza that one can obtain by a given number of straight slices, and the ham sandwich theorem, a result about slicing threedimensional objects whose twodimensional version implies that any pizza, no matter how misshapen, can have its area and its crust length simultaneously bisected by a single carefully chosen straightline cut.
 deutsch  english  español  français  русский 