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Topic: N Queens Problem

	Eight queens puzzle - Wikipedia, the free encyclopedia
		The eight queens puzzle is the problem of putting eight chess queens on an 8×8 chessboard such that none of them is able to capture any other using the standard chess queen's moves.
		20 queens (remainder 8): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 1, 7, 5, 11, 9, 15, 13, 19, 17.
		Most often, it is 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 n−1 queens problem.
	en.wikipedia.org /wiki/Eight_queens_puzzle (1715 words)

	Problems I Played With
		The goal is to place n queens on the board so that no two queens are on the same row, column, or diagonal.
		When n is eight, i.e., a normal chess board is used, the problem is known as the eight queens problem.
		The n queens problem is one of NP-hard problems, and it has been using as an example in AI research or programming.
	www.cs.mcgill.ca /~emal-a/queen.htm (273 words)

	Data Set from the n-queens Problem
		Every queen on a checker square can reach the other squares that are located on the same horizontal, vertical, and diagonal line.
		You also see that repositioned queens are mostly represented by small and clear spots, indicating that the algorithms try to keep queens placed early in the process and reposition those that have been placed late, rather than choosing at random a queen to reposition when a conflict occurs (depth-first strategy).
		We can notice that at depth 27, numerous trials have been made to place the queen: steps 27, 28, 46, 47, 77, 78, 88 and 89 are trials to place a queen on the same row at a different column.
	www2.ilog.com /preview/Discovery/samples/nqueens (1966 words)

	The N Queens Problem
		It then places a queen in the second column and moves it until it finds a place where it cannot be hit by the queen in the first column.
		It then places a queen in the third column and moves it until it cannot be hit by either of the first two queens.
		If the queen there is at the end of the column it removes that queen as well and goes to the preceding column.
	www.math.utah.edu /~alfeld/queens/queens.html (1607 words)

	Table of Content
		The N Queens problem is an old one that has been around for more than a century.
		The queens' problem is really a puzzle but, surprisingly, there are some practical applications such as parallel memory storage schemes, VLSI testing, traffic control, and deadlock prevention.
		"N queens are to be placed on a n by n chess board such that no queen threatens any other queen." As we all know, in Chess, a queen is the most powerful piece.
	www.cs.mcgill.ca /~ramadan/projects/605/project-summery.html (877 words)

	n-queens problem (Site not responding. Last check: 2007-11-07)
		Problem: Arrange n queens on an n × n chessboard so that no two queens attack each other.
		; further analysis is complicated by the fact that placing a queen in a particular rank reduces the number of unattacked squares in other (empty) ranks by either one, two, or three, depending on the arrangement of the other queens on the board.
		The basic idea behind the algorithm is to put queens in the unattacked squares, one rank at a time, using lists of lists of board configurations (a data structure computer geeks like to call a "tree") to keep track of what we've tried and what doesn't work.
	www.jayharris.info /website/programs/n-queens/nqueens.htm (1011 words)

	Encyclopedia: Eight queens puzzle (Site not responding. Last check: 2007-11-07)
		In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation.
		In mathematics, a magic square (å¹»æ–¹) of order n is an arrangement of nÂ² numbers in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.
		A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
	www.nationmaster.com /encyclopedia/Eight-queens-puzzle (3847 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Since each queen has its own array stacked on the previous one, it is assumed that no two queens will be in the same row (which is one less thing to check).
		The problem with this in terms of the modified queen portion of the program is that there is only ONE queen per array per board.
		I came up with this way of doing the 8 queens problem realizing that it was simply one less direction of attack that I had to check for and would save a good amount of time.
	members.visi.net /~spyvsspy/files/cplus/cmsc246/readme_8q2.txt (441 words)

	LETAVEC and RUGGIERO--The n-Queens Problem
		The n-queens problem, originally introduced in 1850 by Carl Gauss, may be stated as follows: find a placement of n queens on an n×n chessboard, such that no one queen can be taken by any other.
		While it has been well known that the solution to the n-queens problem is n, numerous solutions have been published since the original problem was proposed.
		(1969), we define the chessboard as an n×n matrix of square elements for n 4; each square is identified as an ordered pair (i,j), where i and j are the row and column numbers of the square, respectively.
	ite.pubs.informs.org /Vol2No3/LetavecRuggiero/index.php (865 words)

	Info on Queens problem (Site not responding. Last check: 2007-11-07)
		Therefore, the objective is to place n queens on an n by n board in such a way that no two queens are on the same row, column or diagonal.
		The problem was generalized to n by n boards in 1850 by Franz Nauck.
		A useful manipulative for the queens problem is a chessboard.
	projects.cs.dal.ca /agatemath/amof/e_queeI.htm (665 words)

	The N Queens Problem (Site not responding. Last check: 2007-11-07)
		N queens on a chessboard in such a way that no two queens can take each other.
		To solve the N Queens Problem with a Visual Prolog program, you must record which rows, columns, and diagonals are unoccupied, and also make a note of where the queens are placed.
		Queens are placed one at a time until every row and column is occupied.
	www.aistudy.com /program/prolog/visual_prolog/The%20N%20Queens%20Problem.htm (475 words)

	n-queens problem: hill-climbing algorithms (Site not responding. Last check: 2007-11-07)
		Complexity is a measure of the computational resources — primarily time and memory (space) — an algorithm requires to run to completion, and because the search tree grows exponentially with the size of the chessboard, tree searches exponentially complex in both time and space.
		Each queen must be moved to a specific column (relative to the others), and this is unlikely to happen when we are just moving the queens around at random.
		The global minimum is the solution to the problem; a local minimum is an area where the algorithms can't make any progress (because all the successor nodes have a higher value than the current node) but that also isn't a solution.
	www.jayharris.info /website/programs/n-queens/hill-climbing.htm (2114 words)

	Knight's Tour Notes, Part Cc: Chronology 1900 to Present
		Problem 736 (WJ), 985-986 (TRD), 987-991 (DHH), 1061 (TRD), 1062-63 (WJ), 1064 (HP).
		Problems 1132-35 (FD), 1303-06, 1449-52, 1525-28, 1593-96 (all TRD).
		Problems 1674 -77 (TRD), 1704 (SHH, squares in a circle), 1705 (TRD, squares in a circle), 1813-1816, 1834-37, 1917-20 (all TRD).
	www.ktn.freeuk.com /cc.htm (7794 words)

	Math Games: Chessboard Tasks
		Problem 5 is the non-dominating queens problem, and a favorite puzzle of both Martin Gardner and me. (Solve it!) It's a favorite problem of Mario Velucchi.
		For placing n queens on an n×n board, the maximal number of non-dominated squares is still unsolved for n>17.
		Erich Friedman recently explored the non-dominating problem for bishops and knights.
	www.maa.org /editorial/mathgames/mathgames_04_11_05.html (1996 words)

	The N-Queens Problem
		The original eight queens problem consisted of trying to find a way to place eight queens on a chessboard so that no queen would attack any other queen.
		An alternate way of expressing the problem is to place eight "anythings" on an eight by eight grid such that none of them share a common row, column, or diagonal.
		For higher orders of "N", the problem has to be broken into parts with each part delegated to a separate computer.
	www.durangobill.com /N_Queens.html (791 words)

	A Method of Solving N Queens Problem or Sorting Using CCM
		Queens to apply rules are randomly selected when executing the prigram.
		The sum of the LEFs of all the combinations of the queens is calculated.
		So the third queen, which has nothing to do with the column swap operation itself, is added to the rule as shown in the right figure.
	www.kanadas.com /ccm/queens-sort/method.html (695 words)

	The N Queens Problem and Sorting Using CCM
		The N queens problem is a constraint satisfaction problem to place N queens on N by N ``chess board'' and to satisfy a condition that no queen can take any other queens.
		If you are interested in the methods of N queens or sorting, see the method page.
		There are applets that solve the N queens problem using conventional method (based on backtracking).
	www.kanadas.com /ccm/queens-sort/index.html (831 words)

	ICS 461 Homework #4, Due Friday, 11 April 2003
		Assume each of the N queens is constrained to one row, and the queens are numbered from 0 to N-1, corresponding to rows 0 to N-1.
		That is place Queen 0, then Queen 1, etc. Of course, when a placement violates a constraint, the program should backtrack (move back up the search tree to make the next legal placement).
		This would describe placing the queen in row 0 onto column 1, the queen in row 1 onto column 3, the queen in row 2 onto column 0, and the queen in row 3 onto column 2.
	www2.hawaii.edu /~cpowley/ICS461/nqueens.html (623 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The problem is to place n queens on ; an nxn chessboard so that no two queens conflict by being in the same row, ; column, or diagonal.
		Each queen is placed in a separate column.
		A solution is ; represented by giving the row numbers of each of the queens in order by column.
	home.earthlink.net /~mathewp/experience/code/scheme/nqueens.txt (177 words)

	Backtracking (Site not responding. Last check: 2007-11-07)
		In the case of the N-queens, the constraint is that all N queens be placed in non-attacking position.
		For the N-Queens problem, a placement is "promising" if the placed queen is not in the same column or same diagonal as one of the already positioned queens.
		When none of the children nodes of a placed queen leads to a promising solution, the algorithm backtracks and removes the last queen placed.
	www.cs.vassar.edu /~cs241/teneyck/Backtracking.htm (1450 words)

	n-Queens
		Kraitchik, M. "The Problem of the Queens" and "Domination of the Chessboard." ?10.3 and 10.4 in Mathematical Recreations.
		Gauss's Arithmetization of the Problem of 8 Queens.
		An Analytical Evidence for Kalé's Heuristic for the N Queens Problem.
	www.liacs.nl /home/kosters/nqueens.html (1062 words)

	Info on Queens problem
		The problem is to find all ways of placing n non-taking queens on a n by n board.
		A queen attacks all cells in its same row, column, and either diagonal.
		The number of solutions for n = 1,2,...,15, is 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184.
	www.theory.cs.uvic.ca /~cos/inf/misc/Queen.html (138 words)

	Chess Guide > Eight Queens
		The eight queens puzzle is the problem of putting eight chess queens on an 8x8 chessboard such that none of them is able to capture any other using the standard chess queen's moves.
		The eight queens problem has 92 distinct solutions, or 12 distinct solutions if symmetry operations such as rotations and reflections of the board are taken into consideration.
		The eight queens puzzle is a good example of a simple but non-trivial problem that can be solved by a recursive algorithm, by phrasing the n-queen problem inductively in terms of adding a single queen to any solution to the (n-1)-queen problem.
	chess-guide.fateback.com /problems-puzzles/puzzles/eight_queens.html (932 words)

	Recursion: N Queens Lab
		In this lab you will implement an algorithm that places N queens on an N x N board (like a chess board) in such a way that no queen can attack another queen.
		method to add a queen to any arbitrary column (your choice) of the correct row for that queen, display the environment, and then remove the queen and redisplay the environment.
		if any queens have already been placed on the board in locations that are on the diagonal from the location parameter.
	max.cs.kzoo.edu /AP/MBS/RelatedAssignments/NQueens/NQueensLab.shtml (388 words)

	CS 1622 (Winter 1998-1999) Lab 4
		The N Queens Problem is one in which you try to find a way to place N queens on a NxN chess board in such a way that no queen attacks any other.
		One strategy to solve this problem is to start with the first column, finding a position to place a queen in that column.
		You should then print "Placing Queen K at R,K" followed by a newline where K is the column number and R is the row you are trying.
	www.duluth.umn.edu /~rmaclin/cs1622/lab4.html (688 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		# ############################################################################ # # This program displays the solutions to the non-attacking n- # queens problem: the ways in which n queens can be placed on an # n-by-n chessboard so that no queen can attack another.
		A positive # integer can be given as a command line argument to specify the # number of queens.
		For example, # # iconx queens -n8 # # displays the solutions for 8 queens on an 8-by-8 chessboard.
	www.cs.arizona.edu /icon/library/src/progs/queens.icn (350 words)

	NQueensStackIteration (This is the program which gives solutions to N Queens Problem. N is given by the user as input.) (Site not responding. Last check: 2007-11-07)
		NQueensStackIteration (This is the program which gives solutions to N Queens Problem.
		N is given by the user as input.)
		Java application prints solution for N Queens positions in ChessBoard so that no 2 queens attack each other.
	www.cs.sunyit.edu /~roys2/NQueensStackIteration.html (324 words)

	dBforums - solving the n queens problem with a n digit number
		n queens on a n*n chess board on each column so that no one may kill
		queens are not on the same rows and not on the same diagonals,
		> queens are not on the same rows and not on the same diagonals,
	www.dbforums.com /t668095.html (543 words)

	Java Programming: N-Queens Problem: My Code Doesn't Work With n = 5, Sometimes Other n Too...
		I have an n sized int array containing a permutation of 0...n.
		I use this to conceptualize an nXn chessboard, using the array index i for the rows of the board, and array[i] as the columns.
		This total includes attack for queens a and b, for a to b and b to a, so a pair of attacking queens adds 2 to the count.
	www.experts-exchange.com /Programming/Programming_Languages/Java/Q_21325259.html (840 words)

	Clustor Case Studies: N Queens Problem: Conclusions (Site not responding. Last check: 2007-11-07)
		To facilitate this, the dataprep program would also have to be altered so that it could pipe into the program queen.
		Instead of using the ranum files to supply random numbers, Clustor could be used generate a random number seed for use with Nqasyned.c, which could then produce it's own random numbers.
		Another useful refinement would have been to let the queen program accept a string that tells it the location of the ranum files.
	www.csse.monash.edu /cluster/RJK/Glenn/conclusion.html (196 words)

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