Cake-cutting

Exact division

Exact division, also called consensus division, is a partition of a continuous resource ("cake") into some k pieces, such that each of n people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with k=3 and n=2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms: * Consensus halving – the cake should be partitioned into two pieces (k=2), and all agents agree that the pieces have equal values. * Consensus 1/k-division, for any constant k>1 - the cake should be partitioned into k pieces, and all agents agree that the pieces have equal values. Another term is consensus splitting. * 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. * -near-exact division, for any constant - the agents may disagree on the pieces values, but the difference between the values should be at most . Similarly, the approximate variants of the above-mentioned problems are called -consensus-halving, -consensus 1/k-division or -consensus-splitting, and -perfect-division. * Problem of the Nile - there are infinitely many agents. * Necklace splitting - the resource to divide is made of a finite number of indivisible objects ("beads"). When both n and k are finite, Consensus divisions always exist. However, they cannot be found by discrete protocols (with a finite number of queries). In some cases, exact divisions can be found by moving-knife protocols. Near-exact divisions can be found by discrete protocols. (Wikipedia).

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From playlist Polynomial Functions

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From playlist Prealgebra Chapter 1 (Complete chapter)

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From playlist Number Sense - Decimals, Percents, and Ratios

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From playlist EngageNY Grade 6 Module 2

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Demonstrating why synthetic division gives you the same result as traditional algebraic long division Watch the next lesson: https://www.khanacademy.org/math/algebra2/polynomial_and_rational/polynomial-remainder-theorem-tutorial/v/polynomial-remainder-theorem?utm_source=YT&utm_medium=Desc

From playlist Algebra II | High School Math | Khan Academy

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From playlist Prealgebra Chapter 1 (Complete chapter)

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From playlist Challenge Problems

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Tucker's lemma | Countable set | Almost surely | PPA (complexity) | Intermediate value theorem | Piecewise-constant valuation | Dimension | Hyperplane | Truthful cake-cutting | Robertson–Webb query model | Proportional division | Online algorithm | Jerzy Neyman | Problem of the Nile | Hobby–Rice theorem | Fair division among groups | Dice | Necklace splitting problem | Stromquist–Woodall theorem | Fair cake-cutting | Measure (mathematics) | Dubins–Spanier theorems | Envy-free cake-cutting