Find three other winning sequences of moves for the

Problem 11E Chapter 10.1

Discrete Mathematics with Applications | 4th Edition

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Problem 11E

Find three other winning sequences of moves for the vegetarians and the cannibals in Example.ExampleUsing a Graph to Solve a Problem: Vegetarians and Cannibals The following is a variation of a famous puzzle often used as an example in the study of artificial intelligence. It concerns an island on which all the people are of one of two types, either vegetarians or cannibals. Initially, two vegetarians and two cannibals are on the left bank of a river. With them is a boat that can hold a maximum of two people. The aim of the puzzle is to find a way to transport all the vegetarians and cannibals to the right bank of the river. What makes this difficult is that at no time can the number of cannibals on either bank outnumber the number of vegetarians. Otherwise, disaster befalls the vegetarians!Solution A systematic way to approach this problem is to introduce a notation that can indicate all possible arrangements of vegetarians, cannibals, and the boat on the banks of the river. For example, you could write (vvc/Bc) to indicate that there are two vegetarians and one cannibal on the left bank and one cannibal and the boat on the right bank. Then (vvccB/) would indicate the initial position in which both vegetarians, both cannibals, and the boat are on the left bank of the river. The aim of the puzzle is to figure out a sequence of moves to reach the position (/Bvvcc) in which both vegetarians, both cannibals, and the boat are on the right bank of the river.Construct a graph whose vertices are the various arrangements that can be reached in a sequence of legal moves starting from the initial position. Connect vertex x to vertex y if it is possible to reach vertex y in one legal move from vertex x. For instance, from the initial position there are four legal moves: one vegetarian and one cannibal can take the boat to the right bank; two cannibals can take the boat to the right bank; one cannibal can take the boat to the right bank; or two vegetarians can take the boat to the right bank. You can show these by drawing edges connecting vertex (vvccB/) to vertices (vc/Bvc), (vv/Bcc), (vvcBc), and (cc/Bvv). (It might seem natural to draw directed edges rather than undirected edges from one vertex to another. The rationale for drawing undirected edges is that each legal move is reversible.) From the position (vc/Bvc), the only legal moves are to go back to (vvccB/) or to go to (vvcB/c). You can also show these by drawing in edges. Continue this process until finally you reach (/Bvvcc). From Figure 10.1.3 it is apparent that one successful sequence of moves is (vvccB/) (vc/Bvc) (vvcB/c) (c/Bvvc) (ccB/vv) (/Bvvcc).Figure

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ISBN: 9780495391326

The full step-by-step solution to problem: 11E from chapter: 10.1 was answered by , our top Math solution expert on 07/19/17, 06:34AM. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. Since the solution to 11E from 10.1 chapter was answered, more than 228 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This full solution covers the following key subjects: vegetarians, bank, cannibals, boat, Right. This expansive textbook survival guide covers 131 chapters, and 5076 solutions. The answer to “Find three other winning sequences of moves for the vegetarians and the cannibals in Example.ExampleUsing a Graph to Solve a Problem: Vegetarians and Cannibals The following is a variation of a famous puzzle often used as an example in the study of artificial intelligence. It concerns an island on which all the people are of one of two types, either vegetarians or cannibals. Initially, two vegetarians and two cannibals are on the left bank of a river. With them is a boat that can hold a maximum of two people. The aim of the puzzle is to find a way to transport all the vegetarians and cannibals to the right bank of the river. What makes this difficult is that at no time can the number of cannibals on either bank outnumber the number of vegetarians. Otherwise, disaster befalls the vegetarians!Solution A systematic way to approach this problem is to introduce a notation that can indicate all possible arrangements of vegetarians, cannibals, and the boat on the banks of the river. For example, you could write (vvc/Bc) to indicate that there are two vegetarians and one cannibal on the left bank and one cannibal and the boat on the right bank. Then (vvccB/) would indicate the initial position in which both vegetarians, both cannibals, and the boat are on the left bank of the river. The aim of the puzzle is to figure out a sequence of moves to reach the position (/Bvvcc) in which both vegetarians, both cannibals, and the boat are on the right bank of the river.Construct a graph whose vertices are the various arrangements that can be reached in a sequence of legal moves starting from the initial position. Connect vertex x to vertex y if it is possible to reach vertex y in one legal move from vertex x. For instance, from the initial position there are four legal moves: one vegetarian and one cannibal can take the boat to the right bank; two cannibals can take the boat to the right bank; one cannibal can take the boat to the right bank; or two vegetarians can take the boat to the right bank. You can show these by drawing edges connecting vertex (vvccB/) to vertices (vc/Bvc), (vv/Bcc), (vvcBc), and (cc/Bvv). (It might seem natural to draw directed edges rather than undirected edges from one vertex to another. The rationale for drawing undirected edges is that each legal move is reversible.) From the position (vc/Bvc), the only legal moves are to go back to (vvccB/) or to go to (vvcB/c). You can also show these by drawing in edges. Continue this process until finally you reach (/Bvvcc). From Figure 10.1.3 it is apparent that one successful sequence of moves is (vvccB/) (vc/Bvc) (vvcB/c) (c/Bvvc) (ccB/vv) (/Bvvcc).Figure” is broken down into a number of easy to follow steps, and 461 words.

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