Pizza theorem
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 multiple of 4 and greater than or equal to 8. Form "n" sectors of the disk with equal angles by choosing an arbitrary line through "p", rotating the line times by an angle of radians, and slicing the disk on each of the resulting lines. Number the sectors consecutively in a clockwise or anti-clockwise 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. History. The pizza theorem was originally proposed as a challenge problem by . The published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors. Generalizations. 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. answered a problem of 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 straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors. As 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. Related results. 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. See also. 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 three-dimensional objects whose two-dimensional version implies that any pizza, no matter how misshapen, can have its area and its crust length simultaneously bisected by a single carefully chosen straight-line cut, and whose three-dimensional version implies that a plane cut exists that equally shares base, tomato and cheese.