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. The eight queens puzzle is an example of the more general n queens problem of placing n non-attacking queens on an n×n chessboard, for which solutions exist for all natural numbers n with the exception of n = 2 and n = 3.
Chess composer Max Bezzel published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850.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.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
The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 (i.e., 64C8) possible arrangements of eight queens on an 8×8 board, 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 constrains each queen to a single column (or row), though still considered brute force, 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 are then 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, should a solution be equivalent to its own 90° rotation (as happens to one solution with five queens on a 5×5 board), that fundamental solution will have only two variants (itself and its reflection). Should a solution be equivalent to its own 180° rotation (but not to its 90° rotation), it will have four variants (itself and its reflection, its 90° rotation and the reflection of that). If n > 1, it is not possible for a solution to be equivalent to its own reflection because that would require two queens to be facing each other. Of the 12 fundamental solutions to the problem with eight queens on an 8×8 board, exactly one (solution 12 below) is equal to its own 180° rotation, and none is equal to its 90° rotation; thus, the number of distinct solutions is 11×8 + 1×4 = 92.
All fundamental solutions are presented below:
Solution 10 has the additional property that no three queens are in a straight line. Solutions 1 and 8 have a 4-queen line.
These 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. These solutions exhibit stair-stepped patterns, as in the following examples for n = 8, 9 and 10:
The examples above can be obtained with the following formulas.Let (i, j) be the square in column i and row j on the n × n chessboard, k an integer.
For n = 8 this results in fundamental solution 1 above. A few more examples follow.
The following tables give the number of solutions for placing n queens on an n × n board, both fundamental (sequence A002562 in the OEIS ) and all (sequence A000170 in the OEIS ).
The six queens puzzle has fewer solutions than the five queens puzzle.
There is no known formula for the exact number of solutions, or even for its asymptotic behaviour. The 27×27 board is the highest-order board that has been completely enumerated.
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. 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.
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.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 finds a solution to 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 will 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 so that it conflicts with the smallest number of queens already on the board.
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.
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.
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.
The following is a Pascal program by Niklaus Wirth in 1976.It finds one solution to the eight queens problem.
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. pp48-49)(
A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square, the tour is closed; otherwise, it is open.
Two mathematical objects a and b are called equal up to an equivalence relation R
A chessboard is the type of used for the game of chess, on which the chess pawns and are placed. A chessboard is usually square in shape, with an alternating pattern of squares in two colours. Though usually played on a surface, a tangible board is not a requirement to play the game. Traditionally wooden boards are made of unstained light and dark brown woods. To reduce cost, many boards are made with veneers of more expensive woods glued to an inner piece of plywood or chipboard. A variety of colours combinations are used for plastic, vinyl, and silicone boards. Common dark-light combinations are black and white, as well as brown, green or blue with buff or cream. Materials vary widely; while wooden boards are generally used in high-level games; vinyl, plastic, and cardboard are common for less important tournaments and matches and for home use. Decorative glass and marble boards are rarely permitted for games conducted by national or international chess federations. When they are permitted, they must meet various criteria
The queen is the most powerful piece in the game of chess, able to move any number of squares vertically, horizontally or diagonally, combining the power of the rook and bishop. Each player starts the game with one queen, placed in the middle of the first next to the king. Because the queen is the strongest piece, a pawn is promoted to a queen in the vast majority of cases.
A chess puzzle is a puzzle in which knowledge of the pieces and rules of chess is used to solve logically a chess-related problem. The history of chess puzzles reaches back to the Middle Ages and has evolved since then.
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement.
Backtracking is a general algorithm for finding all solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution.
The 15 puzzle is a sliding puzzle having 15 square tiles numbered 1–15 in a frame that is 4 tiles high and 4 tiles wide, leaving one unoccupied tile position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The goal of the puzzle is to place the tiles in numerical order.
In computer science, the min conflicts algorithm is a search algorithm or heuristic method to solve constraint satisfaction problems.
In computer science, dancing links is a technique for reverting the operation of deleting a node from a circular doubly linked list. It is particularly useful for efficiently implementing backtracking algorithms, such as Donald Knuth's Algorithm X for the exact cover problem. Algorithm X is a recursive, nondeterministic, depth-first, backtracking algorithm that finds all solutions to the exact cover problem. Some of the better-known exact cover problems include tiling, the n queens problem, and Sudoku.
In mathematics, given a collection of subsets of a set X, an exact cover is a subcollection of such that each element in is contained in exactly one subset in . One says that each element in is covered by exactly one subset in . An exact cover is a kind of cover.
Sudoku puzzles can be studied mathematically to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use of combinatorics and group theory.
This is a glossary of Sudoku terms and jargon. It is organized thematically, with links to references and example usage provided as (). Sudoku with a 9×9 grid is assumed, unless otherwise noted.
Board representation in computer chess is a data structure in a chess program representing the position on the chessboard and associated game state. Board representation is fundamental to all aspects of a chess program including move generation, the evaluation function, and making and unmaking moves as well as maintaining the state of the game during play. Several different board representations exist. Chess programs often utilize more than one board representation at different times, for efficiency. Execution efficiency and memory footprint are the primary factors in choosing a board representation; secondary considerations are effort required to code, test and debug the application.
In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge represents a legal move from one square to another. The same graphs can be defined mathematically as the Cartesian products of two complete graphs or as the line graphs of complete bipartite graphs.
The mutilated chessboard problem is a tiling puzzle proposed by philosopher Max Black in his book Critical Thinking (1946). It was later discussed by Solomon W. Golomb (1954), Gamow & Stern (1958) and by Martin Gardner in his Scientific American column "Mathematical Games". The problem is as follows:
Suppose a standard 8×8 chessboard has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares?
In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged.
A mathematical chess problem is a mathematical problem which is formulated using a chessboard and chess pieces. These problems belong to recreational mathematics. The most known problems of this kind are Eight queens puzzle or Knight's Tour problems, which have connection to graph theory and combinatorics. Many famous mathematicians studied mathematical chess problems; for example, Thabit, Euler, Legendre and Gauss. Besides finding a solution to a particular problem, mathematicians are usually interested in counting the total number of possible solutions, finding solutions with certain properties, as well as generalization of the problems to N×N or rectangular boards.
The following outline is provided as an overview of and topical guide to chess:
In mathematics, the telephone numbers or the involution numbers are a sequence of integers that count the ways n telephone lines can be connected to each other, where each line can be connected to at most one other line. These numbers also describe the number of matchings of a complete graph on n vertices, the number of permutations on n elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with n cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values
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