8 queens problem algorithm pdf

8 queens problem using backtracking algorithm pdf. 8 queens problem solution in daa. 8 queens problem solution. 8 queen problem in algorithm.

The 8-queens problem can be defined as follows: Place 8 queens on an (8 by 8) chess board such that none of the queens attacks any of the others. A configuration of 8 queens on the board is shown in figure 1, but this does not represent a solution as the queen in the first column is on the same diagonal as the queen in the last column. Figure 1:
Almost a solution of the 8-queens problem Searching for a Solution This problem can be solved by searching for a solution. The initial state is given by the empty chess board. Placing a queen on the board represents an action in the search problem. A goal state is a configuration where none of the queens attacks any of the others. Note that every goal
state is reached after exactly 8 actions. This formulation as a search problem can be improved when we realize that, in any solution, there must be exactly one queen in each of the columns. Thus, the possible actions can be restricted to placing a queen in the next column that does not yet contain a queen. This reduces the branching factor from
(initially) 64 to 8. Furthermore, we need only consider those rows in the next column that are not already attacked by a queen that was previously on the board. This is because the placing of further queens on the board can never remove the mutual attack and turn the configuration into a solution. The EightQueensApp Java Application
EightQueensApp.jar is a Java application that explores the above search space using a number of (uninformed) search strategies. Download the application and double-click it. Alternatively, run the command "java -jar EightQueensApp.jar" from the command line. Either should bring up a window that looks essentially like the one shown in figure 2.
Figure 2: The window of the application To search for a solution, first select a search strategy. Next, there are some configuration options for the search process. If the search space is to be searched as a graph, multiple paths leading to the same node will usually only be explored once. In a tree, search states that can be reached by multiple paths will
also be explored multiple times. However, given our formulation of the search problem, there can only be one path to every state.
The number of states to be generated can be limited to the given value, resulting in the search being abandoned at that point. For a depth-first search it is also possible to set a depth limit, meaning no states at a greater depth will be explored. Finally, a trace of the search can be written to the window and/or a text file. The operation performed by a
search engine consists of selecting a current search node and generating its successors, and the trace reflects this process. For example, the line: current state: 11: > indicates that the selected node contains the given state. In this state 2 queens are on the board. The following numbers then indicate the positions of all queens on the board, starting
with the first column. In the example, the first queen is in row 7 (numbering of rows starts from 0 here), and the second queen is in row 3. The remaining zeros can be ignored as there are only two queens on the board. 56974212589.pdf The lines following this one in the trace describe the successor states that have been generated, for example:
successor state: 59: successor state: 60: successor state: 61: So, expanding the given search node (11) resulted in three new search nodes (59, 60 and 61). Even with tracing off the window will display some information about the search (which is not part of the trace). It will show how many states have been explored (the goal test has been performed
and successors have been generated) and how many states have been generated (explored states plus fringe nodes). If a solution is found this will also be printed. Finally, the elapsed time taken to perform the search is printed. Note that writing the trace usually takes more time than searching itself. References S. Russell and P. Norvig. Artificial
Intelligence: A Modern Approach, chapter 3. Prentice Hall, 2nd edition, 2003. Wikipedia.org: Eight queens puzzle Reading time: 30 minutes | Coding time: 10 minutes You are given an 8x8 chessboard, find a way to place 8 queens such that no queen can attack any other queen on the chessboard. A queen can only be attacked if it lies on the same row,
or same column, or the same diagonal of any other queen. Print all the possible configurations. 9438435961.pdf To solve this problem, we will make use of the Backtracking algorithm. The backtracking algorithm, in general checks all possible configurations and test whether the required result is obtained or not. For thr given problem, we will explore
all possible positions the queens can be relatively placed at. The solution will be correct when the number of placed queens = 8. The time complexity of this approach is O(N!). Input Format - the number 8, which does not need to be read, but we will take an input number for the sake of generalization of the algorithm to an NxN chessboard.
foximinite.pdf Output Format - all matrices that constitute the possible solutions will contain the numbers 0(for empty cell) and 1(for a cell where queen is placed). Hence, the output is a set of binary matrices. Visualisation from a 4x4 chessboard solution : In this configuration, we place 2 queens in the first iteration and see that checking by placing
further queens is not required as we will not get a solution in this path. Note that in this configuration, all places in the third rows can be attacked.
As the above combination was not possible, we will go back and go for the next iteration. 28315887205.pdf This means we will change the position of the second queen. In this, we found a solution. Now let's take a look at the backtracking algorithm and see how it works: The idea is to place the queens one after the other in columns, and check if
previously placed queens cannot attack the current queen we're about to place. eldorado laurent gaudé mouvemen

If we find such a row, we return true and put the row and column as part of the solution matrix. If such a column does not exist, we return false and backtrack* Pseudocode START 1. język angielski podstawowe zwroty pdf begin from the leftmost column 2. if all the queens are placed, return true/ print configuration 3. check for all rows in the current
column a) if queen placed safely, mark row and column; and recursively check if we approach in the current configuration, do we obtain a solution or not b) if placing yields a solution, return true c) if placing does not yield a solution, unmark and try other rows 4.

if all rows tried and solution not obtained, return false and backtrack END Implementation Implementaion of the above backtracking algorithm : #include using namespace std; int board[8][8]; // you can pick any matrix size you want bool isPossible(int n,int row,int col){ // check whether // placing queen possible or not // Same Column for(int i=row-
1;i>=0;i--){ if(board[i][col] == 1){ return false; } } //Upper Left Diagonal for(int i=row-1,j=col-1;i>=0 && j>=0 ; i--,j--){ if(board[i][j] ==1){ return false; } } // Upper Right Diagonal for(int i=row-1,j=col+1;i>=0 && j>n; // could use a default 8 as well placeNQueens(n); return 0; } Output ( for n = 4): 1 indicates placement of queens 0 0 1 0 1 0 0 0 0
0 0 1 0 1 0 0 Explanation of the above code solution: These are two possible solutions from the entire solution set for the 8 queen problem. main() { call placeNQueens(8), placeNQueens(){ call nQueenHelper(8,0){ row = 0 if(row==n) // won't execute as 0 != 8 for(int j=0; j<8; j++){ { if(isPossible==true) { board[0][0] = 1 // board[row][0] = 1 call
nQueenHelper(8,row+1) // recur for all rows further print matrix when row = 8 if solution obtained and (row==n) condition is met } board[0][0] = 0 // backtrack and try for // different configurations } } } } for example, the following configuration won't be displayed Time Complexity Analysis the isPossible method takes O(n) time for each invocation
of loop in nQueenHelper, it runs for O(n) time the isPossible condition is present in the loop and also calls nQueenHelper which is recursive adding this up, the recurrence relation is: T(n) = O(n^2) + n * T(n-1) solving the above recurrence by iteration or recursion tree, the time complexity of the nQueen problem is = O(N!) Question :- You are given
an NxN maze with a rat placed at (0,0). Find and print all the paths that the rat can follow to reach its destination i.e (N-1,N-1). factional warfare eve guide The rat can move in all four directions (left,right,up,down). 20931509386.pdf
Value of every cell will be either 0 or 1. 0 represents a blocked cell, the rat cannot move through it, though 1 is an unblocked cell. Solve the problem using backtracking algorithm, Mathematical problem set on a chessboard abcdefgh8877665544332211abcdefghThe only symmetrical solution to the eight queens puzzle (up to rotation and reflection)
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. The problem was first posed in the mid-19th century. In the modern era, it is often used as an example
problem for various computer programming techniques. The eight queens puzzle is a special case of the more general n queens problem of placing n non-attacking queens on an n×n chessboard. Solutions exist for all natural numbers n with the exception of n = 2 and n = 3. Although the exact number of solutions is only known for n ≤ 27, the
asymptotic growth rate of the number of solutions is approximately (0.143 n)n. History Chess composer Max Bezzel published the eight queens puzzle in 1848.

Franz Nauck published the first solutions in 1850.[1] Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n×n squares. Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n-queens version. In 1874, S. Gunther proposed a method
using determinants to find solutions.[1] J.W.L. Glaisher refined Gunther's approach. In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm.[2] Constructing and counting solutions when n = 8 The problem of finding
all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 possible arrangements of eight queens on an 8×8 board,[a] but only 92 solutions. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute-force computational techniques. For example, by applying
a simple rule that chooses one queen from each column, it is possible to reduce the number of possibilities to 16,777,216 (that is, 88) possible combinations. Generating permutations further reduces the possibilities to just 40,320 (that is, 8!), which can then be checked for diagonal attacks. The eight queens puzzle has 92 distinct solutions. If solutions
that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below. A fundamental solution usually has eight variants (including its original form) obtained by rotating 90, 180, or 270° and then reflecting
each of the four rotational variants in a mirror in a fixed position. However, one of the 12 fundamental solutions (solution 12 below) is identical to its own 180° rotation, so has only four variants (itself and its reflection, its 90° rotation and the reflection of that).[b] Such solutions have only two variants (itself and its reflection). netunalunibaketupu.pdf
Thus, the total number of distinct solutions is 11×8 + 1×4 = 92. All fundamental solutions are presented below: abcdefgh8877665544332211abcdefghSolution 1 abcdefgh8877665544332211abcdefghSolution 2 abcdefgh8877665544332211abcdefghSolution 3 abcdefgh8877665544332211abcdefghSolution 4
abcdefgh8877665544332211abcdefghSolution 5 abcdefgh8877665544332211abcdefghSolution 6 abcdefgh8877665544332211abcdefghSolution 7 abcdefgh8877665544332211abcdefghSolution 8 abcdefgh8877665544332211abcdefghSolution 9 abcdefgh8877665544332211abcdefghSolution 10 abcdefgh8877665544332211abcdefghSolution 11
abcdefgh8877665544332211abcdefghSolution 12 Solution 10 has the additional property that no three queens are in a straight line.

Existence of solutions Brute-force algorithms to count the number of solutions are computationally manageable for n = 8, but would be intractable for problems of n ≥ 20, as 20! = 2.433 × 1018. If the goal is to find a single solution, one can show solutions exist for all n ≥ 4 with no search whatsoever.[3][4] These solutions exhibit stair-stepped
patterns, as in the following examples for n = 8, 9 and 10: abcdefgh8877665544332211abcdefgh Staircase solution for 8 queens abcdefghi 998877665544332211abcdefghi Staircase solution for 9 queens abcdefghij 1010998877665544332211abcdefghij Staircase solution for 10 queens The examples above can be obtained with the following formulas.
[3] Let (i, j) be the square in column i and row j on the n × n chessboard, k an integer. One approach[3] is If the remainder from dividing n by 6 is not 2 or 3 then the list is simply all even numbers followed by all odd numbers not greater than n.

Otherwise, write separate lists of even and odd numbers (2, 4, 6, 8 – 1, 3, 5, 7). If the remainder is 2, swap 1 and 3 in odd list and move 5 to the end (3, 1, 7, 5).
If the remainder is 3, move 2 to the end of even list and 1,3 to the end of odd list (4, 6, 8, 2 – 5, 7, 9, 1, 3). Append odd list to the even list and place queens in the rows given by these numbers, from left to right (a2, b4, c6, d8, e3, f1, g7, h5). For n = 8 this results in fundamental solution 1 above. A few more examples follow.
14 queens (remainder 2): 2, 4, 6, 8, 10, 12, 14, 3, 1, 7, 9, 11, 13, 5. 15 queens (remainder 3): 4, 6, 8, 10, 12, 14, 2, 5, 7, 9, 11, 13, 15, 1, 3. s10 manual transmission removal 20 queens (remainder 2): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 1, 7, 9, 11, 13, 15, 17, 19, 5. Counting solutions for other sizes n Exact enumeration There is no known formula for the
exact number of solutions for placing n queens on an n × n board i.e. the number of independent sets of size n in an n × n queen's graph. The 27×27 board is the highest-order board that has been completely enumerated.[5] The following tables give the number of solutions to the n queens problem, both fundamental (sequence A002562 in the OEIS)
and all (sequence A000170 in the OEIS), for all known cases. n fundamental all 1 1 1 2 0 0 3 0 0 4 1 2 5 2 10 6 1 4 7 6 40 8 12 92 9 46 352 10 92 724 11 341 2,680 12 1,787 14,200 13 9,233 73,712 14 45,752 365,596 15 285,053 2,279,184 16 1,846,955 14,772,512 17 11,977,939 95,815,104 18 83,263,591 666,090,624 19 621,012,754 4,968,057,848
20 4,878,666,808 39,029,188,884 21 39,333,324,973 314,666,222,712 22 336,376,244,042 2,691,008,701,644 23 3,029,242,658,210 24,233,937,684,440 24 28,439,272,956,934 227,514,171,973,736 25 275,986,683,743,434 2,207,893,435,808,352 26 2,789,712,466,510,289 22,317,699,616,364,044 27 29,363,495,934,315,694
234,907,967,154,122,528 Asymptotic enumeration In 2021, Michael Simkin proved that for large numbers n, the number of solutions of the n queens problem is approximately ( 0.143 n ) n {\displaystyle (0.143n)^{n}} .[6] More precisely, the number Q ( n ) {\displaystyle {\mathcal {Q}}(n)} of solutions has asymptotic growth Q ( n ) = ( ( 1 ± o ( 1 ) )
n e − α ) n {\displaystyle {\mathcal {Q}}(n)=((1\pm o(1))ne^{-\alpha })^{n}} where α {\displaystyle \alpha } is a constant that lies between 1.939 and 1.945.[7] (Here o(1) represents little o notation.) If one instead considers a toroidal chessboard (where diagonals "wrap around" from the top edge to the bottom and from the left edge to the right), it
is only possible to place n queens on an n × n {\displaystyle n\times n} board if n ≡ 1 , 5 mod 6. {\displaystyle n\equiv 1,5\mod 6.} In this case, the asymptotic number of solutions is[8][9] T ( n ) = ( ( 1 + o ( 1 ) ) n e − 3 ) n . {\displaystyle T(n)=((1+o(1))ne^{-3})^{n}.} Related problems Higher dimensions Find the number of non-attacking queens
that can be placed in a d-dimensional chess space of size n. More than n queens can be placed in some higher dimensions (the smallest example is four non-attacking queens in a 3×3×3 chess space), and it is in fact known that for any k, there are higher dimensions where nk queens do not suffice to attack all spaces.[10][11] Using pieces other than
queens On an 8×8 board one can place 32 knights, or 14 bishops, 16 kings or eight rooks, so that no two pieces attack each other. 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. The solution is also easy for rooks and kings. Sixteen kings can be placed on the board by
dividing it into 2-by-2 squares and placing the kings at equivalent points on each square. Placements of n rooks on an n×n board are in direct correspondence with order-n permutation matrices. representantes del renacimiento literario español Chess variations Related problems can be asked for chess variations such as shogi. For instance, the n+k
dragon kings problem asks to place k shogi pawns and n+k mutually nonattacking dragon kings on an n×n shogi board.[12] Nonstandard boards Pólya studied the n queens problem on a toroidal ("donut-shaped") board and showed that there is a solution on an n×n board if and only if n is not divisible by 2 or 3.[13] In 2009 Pearson and Pearson
algorithmically populated three-dimensional boards (n×n×n) with n2 queens, and proposed that multiples of these can yield solutions for a four-dimensional version of the puzzle.[14][better source needed] Domination Given an n×n board, the domination number is the minimum number of queens (or other pieces) needed to attack or occupy every
square. For n = 8 the queen's domination number is 5.[15][16] Queens and other pieces Variants include mixing queens with other pieces; for example, placing m queens and m knights on an n×n board so that no piece attacks another[17] or placing queens and pawns so that no two queens attack each other.[18] Magic squares In 1992, Demirörs,
Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa.[19] Latin squares In an n×n matrix, place each digit 1 through n in n locations in the matrix so that no two instances of the same digit are in the same row or column. Exact cover Consider a matrix with one primary column for each of the
n ranks of the board, one primary column for each of the n files, and one secondary column for each of the 4n − 6 nontrivial diagonals of the board. The matrix has n2 rows: one for each possible queen placement, and each row has a 1 in the columns corresponding to that square's rank, file, and diagonals and a 0 in all the other columns. Then the n
queens problem is equivalent to choosing a subset of the rows of this matrix such that every primary column has a 1 in precisely one of the chosen rows and every secondary column has a 1 in at most one of the chosen rows; this is an example of a generalized exact cover problem, of which sudoku is another example. n-queens completion The
completion problem asks whether, given an n×n chessboard on which some queens are already placed, it is possible to place a queen in every remaining row so that no two queens attack each other. This and related questions are NP-complete and #P-complete.[20] Any placement of at most n/60 queens can be completed, while there are partial
configurations of roughly n/4 queens that cannot be completed.[21] Exercise in algorithm design Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used as an example problem for various programming techniques, including nontraditional approaches such as constraint
programming, logic programming or genetic algorithms. Most often, it is used as an example of a problem that 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×n chessboard. The induction bottoms out with the
solution to the 'problem' of placing 0 queens on the chessboard, which is the empty chessboard. This technique can be used in a way that 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 solves the eight rooks puzzle by generating the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), and uses the elements of each permutation as
indices to place a queen on each row. Then it rejects those boards with diagonal attacking positions. This animation illustrates backtracking to solve the problem. A queen is placed in a column that is known not to cause conflict. If a column is not found the program returns to the last good state and then tries a different column. The backtracking
depth-first search program, 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 rook and diagonal attacks even on incomplete boards, it examines only 15,720 possible
queen placements. A further improvement, which examines only 5,508 possible queen placements, is to combine the permutation based method with the early pruning method: the permutations are generated depth-first, and the search space is pruned if the partial permutation produces a diagonal attack. Constraint programming can also be very
effective on this problem. min-conflicts solution to 8 queens An alternative to exhaustive search is an 'iterative repair' algorithm, which typically starts with all queens on the board, for example with one queen per column.[22] 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 easily finds a solution to even the 1,000,000 queens problem.[23][24] Unlike the backtracking search outlined above, iterative repair does not
guarantee a solution: like all greedy procedures, it may get stuck on a local optimum. (In such a 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.
As an alternative to backtracking, solutions can be counted by recursively enumerating valid partial solutions, one row at a time. Rather than constructing entire board positions, blocked diagonals and columns are tracked with bitwise operations. This does not allow the recovery of individual solutions.[25][26] Sample program The following program
is a translation of Niklaus Wirth's solution into the Python programming language, but does without the index arithmetic found in the original and instead uses lists to keep the program code as simple as possible. geodon medication guide By using a coroutine in the form of a generator function, both versions of the original can be unified to compute
either one or all of the solutions. Only 15,720 possible queen placements are examined.[27][28] def queens(n, i, a, b, c): if i < n: for j in range(n): if j not in a and i+j not in b and i-j not in c: yield from queens(n, i+1, a+[j], b+[i+j], c+[i-j]) else: yield a for solution in queens(8, 0, [], [], []): print(solution) In popular culture In the game The 7th Guest, the
8th Puzzle: "The Queen's Dilemma" in the game room of the Stauf mansion is the de facto eight queens puzzle.[29]: 48–49, 289–290 In the game Professor Layton and the Curious Village, the 130th puzzle: "Too Many Queens 5" (クイーンの問題5) is an eight queens puzzle.[30] See also Mathematical game Mathematical puzzle No-three-in-line problem
Rook polynomial Costas array Notes ^ The number of combinations of 8 squares from 64 is the binomial coefficient 64C8. ^ Other symmetries are possible for other values of n. For example, there is a placement of five nonattacking queens on a 5×5 board that is identical to its own 90° rotation. If n > 1, it is not possible for a solution to be equal to its
own reflection because that would require two queens to be facing each other. References ^ a b W. W.
Rouse Ball (1960) "The Eight Queens Problem", in Mathematical Recreations and Essays, Macmillan, New York, pp. 165–171. ^ O.-J. Dahl, E. W. Dijkstra, C. A. R. Hoare Structured Programming, Academic Press, London, 1972 ISBN 0-12-200550-3, pp. 72–82. ^ a b c Bo Bernhardsson (1991). "Explicit Solutions to the N-Queens Problem for All N".
SIGART Bull. 2 (2): 7.
doi:10.1145/122319.122322. S2CID 10644706. ^ E. J. Hoffman et al., "Construction for the Solutions of the m Queens Problem". Mathematics Magazine, Vol. 90931200417.pdf XX (1969), pp. 66–72. [1] ^ The Q27 Project ^ Sloman, Leila (21 September 2021). "Mathematician Answers Chess Problem About Attacking Queens". Quanta Magazine.
Retrieved 22 September 2021. ^ Simkin, Michael (28 July 2021). "The number of $n$-queens configurations". arXiv:2107.13460v2 [math.CO]. ^ Luria, Zur (15 May 2017). "New bounds on the number of n-queens configurations". arXiv:1705.05225v2 [math.CO]. ^ Bowtell, Candida; Keevash, Peter (16 September 2021). "The $n$-queens problem".
arXiv:2109.08083v1 [math.CO]. ^ J. Barr and S.
Rao (2006), The n-Queens Problem in Higher Dimensions, Elemente der Mathematik, vol 61 (4), pp. 133–137. how to hack bullet force crazy games ^ Martin S. Pearson. "Queens On A Chessboard – Beyond The 2nd Dimension" (php). Retrieved 27 January 2020.
^ Chatham, Doug (1 December 2018). "Reflections on the n +k dragon kings problem". Recreational Mathematics Magazine. 5 (10): 39–55. doi:10.2478/rmm-2018-0007.
^ G. Pólya, Uber die "doppelt-periodischen" Losungen des n-Damen-Problems, George Pólya: Collected papers Vol. IV, G-C. Rota, ed., MIT Press, Cambridge, London, 1984, pp. 237–247 ^ "Queens on a Chessboard - Beyond the 2nd Dimension". ^ Burger, A. P.; Cockayne, E.
J.; Mynhardt, C. M. (1997), "Domination and irredundance in the queens' graph", Discrete Mathematics, 163 (1–3): 47–66, doi:10.1016/0012-365X(95)00327-S, hdl:1828/2670, MR 1428557 ^ Weakley, William D. (2018), "Queens around the world in twenty-five years", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. descargar el ser uno
libro 7 pdf (eds.), Graph Theory: Favorite Conjectures and Open Problems – 2, Problem Books in Mathematics, Cham: Springer, pp. 43–54, doi:10.1007/978-3-319-97686-0_5, MR 3889146 ^ "Queens and knights problem". Archived from the original on 16 October 2005. Retrieved 20 September 2005. ^ Bell, Jordan; Stevens, Brett (2009). "A survey of
known results and research areas for n-queens".
Discrete Mathematics.
309 (1): 1–31.
doi:10.1016/j.disc.2007.12.043. ^ O. Demirörs, N. Rafraf, and M.M. Tanik. Obtaining n-queens solutions from magic squares and constructing magic squares from n-queens solutions. Journal of Recreational Mathematics, 24:272–280, 1992 ^ Gent, Ian P.; Jefferson, Christopher; Nightingale, Peter (August 2017). "Complexity of n-Queens Completion".
Journal of Artificial Intelligence Research. 59: 815–848. doi:10.1613/jair.5512. ISSN 1076-9757. ufw firewall guide
Retrieved 7 September 2017. ^ Glock, Stefan; Correia, David Munhá; Sudakov, Benny (6 July 2022). "The n-queens completion problem". Research in the Mathematical Sciences. 9 (41): 41. doi:10.1007/s40687-022-00335-1. PMC 9259550. PMID 35815227. S2CID 244478527. ^ A Polynomial Time Algorithm for the N-Queen Problem by Rok Sosic and
Jun Gu, 1990. Describes run time for up to 500,000 Queens which was the max they could run due to memory constraints. ^ Minton, Steven; Johnston, Mark D.; Philips, Andrew B.; Laird, Philip (1 December 1992).
"Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems". Artificial Intelligence. 58 (1): 161–205. doi:10.1016/0004-3702(92)90007-K. ISSN 0004-3702. S2CID 14830518. ^ Sosic, R.; Gu, Jun (October 1994). "Efficient local search with conflict minimization: a case study of the n-queens problem".
IEEE Transactions on Knowledge and Data Engineering.
6 (5): 661–668. doi:10.1109/69.317698. ISSN 1558-2191. ^ Qiu, Zongyan (February 2002). "Bit-vector encoding of n-queen problem". ACM SIGPLAN Notices.
37 (2): 68–70.
doi:10.1145/568600.568613. ^ Richards, Martin (1997). Backtracking Algorithms in MCPL using Bit Patterns and Recursion (PDF) (Technical report). University of Cambridge Computer Laboratory. UCAM-CL-TR-433. ^ Wirth, Niklaus (1976). Algorithms + Data Structures = Programs. Prentice-Hall Series in Automatic Computation. Prentice-Hall.
Bibcode:1976adsp.book.....W.
ISBN 978-0-13-022418-7. p. 145 ^ Wirth, Niklaus (2012) [orig. 2004]. "The Eight Queens Problem". Algorithms and Data Structures (PDF). Oberon version with corrections and authorized modifications. pp. 114–118. ^ DeMaria, Rusel (15 November 1993). The 7th Guest: The Official Strategy Guide (PDF). Prima Games.
ISBN 978-1-5595-8468-5. Retrieved 22 April 2021.
^ "ナゾ130 クイーンの問題5". ゲームの匠 (in Japanese). Retrieved 17 September 2021. Further reading Bell, Jordan; Stevens, Brett (2009). "A survey of known results and research areas for n-queens".
Discrete Mathematics. 309 (1): 1–31. doi:10.1016/j.disc.2007.12.043. Watkins, John J. (2004). Across the Board: The Mathematics of Chess Problems. Princeton: Princeton University Press. ISBN 978-0-691-11503-0.
Allison, L.; Yee, C.N.; McGaughey, M. (1988). "Three Dimensional NxN-Queens Problems". Department of Computer Science, Monash University, Australia. Nudelman, S. (1995). "The Modular N-Queens Problem in Higher Dimensions". Discrete Mathematics. 146 (1–3): 159–167. doi:10.1016/0012-365X(94)00161-5. Engelhardt, M. (August 2010). "Der
Stammbaum der Lösungen des Damenproblems (in German, means The pedigree chart of solutions to the 8-queens problem". Spektrum der Wissenschaft: 68–71. On The Modular N-Queen Problem in Higher Dimensions, Ricardo Gomez, Juan Jose Montellano and Ricardo Strausz (2004), Instituto de Matematicas, Area de la Investigacion Cientifica,
Circuito Exterior, Ciudad Universitaria, Mexico. Budd, Timothy (2002).
"A Case Study: The Eight Queens Puzzle" (PDF). An Introduction to Object-Oriented Programming (3rd ed.). Addison Wesley Longman. pp. 125–145.
ISBN 0-201-76031-2.
Wirth, Niklaus (2004) [updated 2012]. "The Eight Queens Problem". Algorithms and Data Structures (PDF). Oberon version with corrections and authorized modifications. pp. 114–118. External links The Wikibook Algorithm Implementation has a page on the topic of: N-queens problem Weisstein, Eric W. "Queens Problem". MathWorld. queens-cpm on
GitHub Eight Queens Puzzle in Turbo Pascal for CP/M eight-queens.py on GitHub Eight Queens Puzzle one line solution in Python Solutions in more than 100 different programming languages (on Rosetta Code) Retrieved from "