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Counting solutionsThe following table gives the number of solutions for n queens, both unique and distinct .
Related problems Using pieces other than queens » For example, on an 8×8 board one can place 32 knights, or 14 bishops, or 16 kings, so that no two pieces attack each other. Fairy chess pieces have also been substituted for queens. In the case of knights, an easy solution is to place one on each square of a given color, since they move only to the opposite color. Domination » Given an n×n board, find the domination number, which is the minimum number of queens (or other pieces) needed to attack or occupy every square. For the 8×8 board, the queen's domination number is 5. Queens and knights problem Latin squares » In an n×n matrix, place each digit 1 through n in n locations in the matrix such that no two instances of the same digit are in the same row or column. The eight queens puzzle as an exercise in algorithm designconstraint programming, logic programming or genetic algorithms. Most often, it's used as an example of a problem which can be solved with a recursive algorithm, by phrasing the n queens problem inductively in terms of adding a single queen to any solution to the problem of placing n−1 queens on an n-by-n chessboard. The induction bottoms out with the solution to the 'problem' of placing 0 queens on an n-by-n chessboard, which is the empty chessboard.This technique is much more efficient than the naïve brute-force search algorithm, which considers all 648 = 248 = 281,474,976,710,656 possible blind placements of eight queens, and then filters these to remove all placements that place two queens either on the same square (leaving only 64!/56! = 178,462,987,637,760 possible placements) or in mutually attacking positions. This very poor algorithm will, among other things, produce the same results over and over again in all the different permutations of the assignments of the eight queens, as well as repeating the same computations over and over again for the different sub-sets of each solution. A better brute-force algorithm places a single queen on each row, leading to only 88 = 224 = 16,777,216 blind placements. It is possible to do much better than this. One algorithm generates the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), uses the elements of each permutation as indices to place a queen on each row, guaranteeing no rook attacks. Then it rejects those boards with diagonal attacking positions. The backtracking depth-first search program below, a slight improvement on the permutation method, constructs the search tree by considering one row of the board at a time, eliminating most nonsolution board positions at a very early stage in their construction. Because it rejects diagonal attacks even on incomplete boards, it examines only 15,720 possible queen placements. Constraint programming is even more effective on this problem. An 'iterative repair' algorithm typically starts with all queens on the board, for example with one queen per column. It then counts the number of conflicts (attacks), and uses a heuristic to determine how to improve the placement of the queens. The 'minimum-conflicts' heuristic — moving the piece with the largest number of conflicts to the square in the same column where the number of conflicts is smallest — is particularly effective: it solves the 1,000,000 queen problem in less than 50 steps on average. This assumes that the initial configuration is 'reasonably good' — if a million queens all start in the same row, it'll obviously take at least 999,999 steps to fix it. A 'reasonably good' starting point can for instance be found by putting each queen in its own row and column such that it conflicts with the smallest number of queens already on the board. Note that 'iterative repair', unlike the 'backtracking' search outlined above, doesn't guarantee a solution: like all hillclimbing 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 breadth-first search. An animated version of the recursive solutionThis animation uses backtracking to solve the problem. A queen is placed in a column that's known not to cause conflict. If a column isn't found the program returns to the last good state and then tries a different column.Algorithms that solve the eight queens puzzle implemented in different programming languages are found in the eight queens puzzle solutions article. Further Information Get more info on 'Eight Queens Puzzle'.
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