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Chapter 5: Problem 8

Find at least two instances of the \(n\) -Queens problem that have no solutions.

Short Answer

Expert verified
In the case of the 2-Queens and 3-Queens problem, no placements will satisfy the problem's conditions due to tightly packed rows, columns, and diagonals. Thus, there exist no solutions.

Step by step solution

01

2-Queens Problem

Attempt to place two queens on a 2x2 chess board, while ensuring they do not threaten each other. A queen threatens another if it shares the same row, column, or diagonal. But, in a 2x2 chess board, no matter how the queens are placed, they always share at least one common row, column, or diagonal.
02

3-Queens Problem

Attempt to place three queens on a 3x3 chess board. Given the same conditions, no matter how the queens are placed, they will always share at least one common row, column, or diagonal. As such, there is no possible solution for this problem.

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Most popular questions from this chapter

Suppose we have a solution to the \(n\) -Queens problem instance in which \(n=4 .\) Can we extend this solution to find a solution to the problem instance in which \(n=5 ?\) Can we then use the solutions for \(n=4\) and \(n=5\) to construct a solution to the instance in which \(n=6\) and continue this dynamic programming approach to find a solution to any instance in which \(n>4 ?\) Justify your answer.

Use the Backtracking algorithm for the Sum-ofSubsets problem (Algorithm 5.4) to find all combinations of the following numbers that sum to \(W=52:\) \\[ w_{1}=2 \quad w_{2}=10 \quad w_{3}=13 \quad w_{4}=17 \quad w_{5}=22 \quad w_{6}=42 \\] Show the actions step by step.

Given an \(n \times n \times n\) cube containing \(n^{3}\) cells, we are to place \(n\) queens in the cube so that no two queens challenge each other (so that no two queens are in the same row, column, or diagonal). Can the \(n\) -Queens algorithm (Algorithm 5.1) be extended to solve this problem? If so, write the algorithm and implement it on your system to solve problem instances in which \(n=4\) and \(n=8\).

Suppose that to color a graph properly we choose a starting vertex and a color to color as many vertices as possible. Then we select a new color and a new uncolored vertex to color as many more vertices as possible. We repeat this process until all the vertices of the graph are colored or all the colors are exhausted. Write an algorithm for this greedy approach to color a graph of \(n\) vertices. Analyze this algorithm and show the results using order notation.

Implement the Backtracking algorithm for the \(n\) Queens problem (Algorithm 5.1) on your system, and run it on problem instances in which \(n=4\) \(8,10,\) and 12.
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