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Proof: [3] [4]: Cor.2 A finite algorithm has value-data only about a finite number of pieces. I.e. there is only a finite number of subsets of the cake, for which the algorithm knows the valuations of the partners. Suppose the algorithm has stopped after having value-data about subsets.
A full-size commercial sheet cake pan is 18 by 24 inches (46 cm × 61 cm) or 18 by 26 inches (46 cm × 66 cm) in size. [5] A half-sheet is half that size, and a quarter-sheet or 9-by-13-inch (23 cm × 33 cm) pan, which usually results in 16 to 24 servings of cake, is one-quarter the size.
A. Assume that the cake is the 1-dimensional interval [0,1], and that the value of the entire cake for each of the partners is normalized 1. In each step, the algorithm asks a certain partner either to evaluate a certain interval contained in [0,1], or to mark an interval with a specified value.
Efficient cake-cutting is a problem in economics and computer science. It involves a heterogeneous resource, such as a cake with different toppings or a land with different coverings, that is assumed to be divisible - it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among ...
1 / 4 tsp freshly grated nutmeg; 4 large eggs; 3 / 4 cup light brown sugar; 1 1 / 2 unsalted butter, softened; pinch of salt; 1 / 8 tsp ground cloves; 12 oz pitted plump Medjool dates; 1 / 2 tsp cinnamon; 1 1 / 2 tsp baking soda; 1 cup cake flour; 1 cup all-purpose flour; 3 / 4 cup water; 2 tbsp brewed espresso; 2 tbsp dark rum
In a proportional cake-cutting, each person receives a piece that he values as at least 1/n of the value of the entire cake. In the example cake, a proportional division can be achieved by giving all the vanilla and 4/9 of the chocolate to George (for a value of 6.66), and the other 5/9 of the chocolate to Alice (for a value of 5). In symbols:
Last diminisher is the earliest proportional division procedure developed for n people: . One of the partners is asked to draw a piece which he values as at least 1/n. The other partners in turn have the option to claim that the current piece is actually worth more than 1/n; in that case, they are asked to diminish it such that the remaining value is 1/n according to their own valuation.
Ask partner #1 cut the cake to k pieces that he values as 1/k. Ask partner #2 to trim pieces as needed (using at most k-1 cuts) such that each piece has a value of at most 1/k. These new pieces of course still have a value of at most 1/k for partner #1. Continue with partners #3, #4, …, #n.