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The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of C n for n = 0, 1, 2, ... are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, ... (sequence A000125 in the OEIS).
An equitable cake allocation cannot be found using a finite protocol in the Robertson–Webb query model, even for 2 agents. [8] Moreover, for any ε > 0: A connected ε-equitable cake-cutting requires at least Ω(log ε −1) queries. [9] For 2 agents, an O(log ε −1) protocol exists. [5]
Symmetric fair cake-cutting is a variant of the fair cake-cutting problem, in which fairness is applied not only to the final outcome, but also to the assignment of roles in the division procedure. As an example, consider a birthday cake that has to be divided between two children with different tastes, such that each child feels that his/her ...
The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence. When a circle is cut n times to produce the maximum number of pieces, represented as p = f (n), the n th cut must be considered; the number of pieces before the last cut is f (n − 1), while the number of pieces added by the last cut is n.
Perfect division – the number of pieces equals the number of agents: the cake should be partitioned into n pieces, and all agents agrees that all pieces have equal values. ε {\displaystyle \varepsilon } -near-exact division , for any constant ε > 0 {\displaystyle \varepsilon >0} , the agents may disagree on the pieces values, but the ...
These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2 n − 1, where each digit position is an item from the set of n. Given 3 cards numbered 1 to 3, there are 8 distinct combinations , including the empty set:
20 Classic Cake Recipes Straight From Grandma's Kitchen. Corey Williams. August 27, 2024 at 12:50 PM ... Everyone in the family has been trying to recreate this delicious coffee cake for years."
Rather, as explained under combinations, the number of n-multicombinations from a set with x elements can be seen to be the same as the number of n-combinations from a set with x + n − 1 elements. This reduces the problem to another one in the twelvefold way, and gives as result