![]() However, when the pizza is divided unevenly, the diner who gets the most pizza area actually gets the least crust.Īs Hirschhorn et al. 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 cut-point lying in the latter's interior), and since the disks bounded by both circles are partitioned evenly so is their difference. 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. 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. 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. 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. Generalizations 12 sectors: green area = orange area ![]() Frederickson (2012) gave a family of dissection proofs for all cases (in which the number of sectors is 8, 12, 16. ![]() They show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent piece in an even-numbered sector, and vice versa. The published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors.Ĭarter & Wagon (1994a) provide an alternative proof by dissection. The pizza theorem was originally proposed as a challenge problem by Upton (1967). The sum of the areas of the odd-numbered sectors equals the sum of the areas of the even-numbered sectors ( Upton 1968). Number the sectors consecutively in a clockwise or anti-clockwise fashion. 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 2 π / n radians, and slicing the disk on each of the resulting n / 2 lines. Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. It shows that if two people share a pizza sliced into 8 pieces (or any multiple of 4 greater than 8), and take alternating slices, then they will each get an equal amount of pizza, irrespective of the central cutting point. The theorem is so called because it mimics a traditional pizza slicing technique. In elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way. ![]() Proof without words for 8 sectors by Carter & Wagon (1994a). Equality of areas of alternating sectors of a disk with equal angles through any interior point Example of application of the theorem with eight sectors: by cutting the pizza along the blue lines and, alternately taking one slice each, proceeding clockwise or counterclockwise, two diners eat the same amount (measured in area) of pizza.
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