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Eight queens puzzle

Eight queens puzzle. Eight queens puzzle.

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Eight queens puzzle

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  1. Eight queens puzzle

  2. Eight queens puzzle • The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard such that none of them are able to capture any other using the standard chess queen's moves. The queens must be placed in such a way that no two queens would be able to attack each other. Thus, a solution requires that no two queens share the same row, column, or diagonal. 1 History

  3. Computationally expensive - brute force • ietry every possibility • The problem can be quite computationally expensive as there are 4,426,165,368 (or 64!/(56!8!)) possible arrangements. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute force computational techniques.

  4. Permutation and combinations. • For example, just by applying a simple rule that constrains each queen to a single column (or row), though still considered brute force, it is possible to reduce the number of possibilities to just 16,777,216 (8^8) possible combinations, which is computationally manageable for n=8, but would be intractable for problems of n=20. We can do better 8! = 40,320. Backtracking eliminates further with only 15,720.

  5. Heuristics (rules of thumb) • Extremely interesting advancements for this and other toy problems is the development and application of heuristics (rules of thumb) that yield solutions to the n queens puzzle at an astounding fraction of the computational requirements.

  6. Counting solutions • The following table gives the number of solutions for placing n queens on an n × n board, both unique and distinct • n: 1 2 3 4 5 6 7 8 • unique: 1 0 0 1 2 1 6 12 • distinct: 1 0 0 2 10 4 40 92 • Note that the six queens puzzle has fewer solutions than the five queens puzzle. • There is currently no known formula for the exact number of solutions.

  7. guarantee a solution • Note that 'iterative repair', unlike the 'backtracking' search outlined above, does not guarantee a solution: like all non-hillclimbing (i.e., greedy) procedures, it may get stuck on a local optimum (in which case the algorithm may be restarted with a different initial configuration). On the other hand, it can solve problem sizes that are several orders of magnitude beyond the scope of a depth-first search.

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