Question
a. Which is the last solution to the five-queens problem found by the backtracking algorithm?
b. Use the board’s symmetry to find at least four other solutions to the problem.
Answer
a.
The five-queens problem asks for a placement of five queens on a chessboard such that no two queens threaten each other.
- The last solution found by the backtracking algorithm may vary depending on the specific implementation and the order in which the algorithm explores the search space.
- Without further information, it is not possible to determine the exact last solution found by the backtracking algorithm for the five-queens problem.
b.
Utilizing the board's symmetry, we can find at least four other solutions to the five-queens problem by mirroring or rotating the board. Here are four additional solutions:
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Solution 1:
- Queen positions: (1, 1), (2, 3), (3, 5), (4, 2), (5, 4)
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Solution 2 (Mirror of Solution 1):
- Queen positions: (1, 5), (2, 3), (3, 1), (4, 4), (5, 2)
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Solution 3 (Rotate Solution 1 by 180 degrees):
- Queen positions: (5, 5), (4, 3), (3, 1), (2, 4), (1, 2)
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Solution 4 (Rotate Solution 2 by 180 degrees):
- Queen positions: (5, 1), (4, 3), (3, 5), (2, 2), (1, 4)
These solutions are obtained by reflecting and rotating the board, maintaining the constraint of no two queens threatening each other. By utilizing the board's symmetry, we can derive multiple valid solutions from a single solution, expanding the total number of solutions to the five-queens problem.