8 Queens Puzzle — Base Camp Math (2024)

8 Queens Puzzle

To win the 8 queens puzzle, you need to find a way to place eight queens on an 8x8 chessboard so that no queen can capture any other queen.

The 8 queens problem is a classic puzzle in chessboard mathematics, where the goal is to place eight queens on a standard 8x8 chessboard in such a way that no queen can capture any other queen. This means that no two queens should be placed on the same row, column, or diagonal. The problem was first posed in the mid-1800s and has since been studied extensively in the fields of mathematics and computer science.

The problem is significant because it is an example of a combinatorial optimization problem, which is a type of problem that requires finding the best solution from a large set of possibilities. It is also a classic problem in the study of algorithms and has been used as a benchmark for testing the efficiency of various algorithms.

Solving the 8 queens problem requires a combination of mathematical reasoning and algorithmic thinking. There are several methods for solving the problem, including backtracking, genetic algorithms, and simulated annealing.

To win the 8 queens puzzle, you need to find a way to place eight queens on an 8x8 chessboard so that no queen can capture any other queen. This means that no two queens can be placed on the same row, column, or diagonal.

One way to solve the problem is to use a backtracking algorithm. This involves placing queens on the board one at a time and checking if the placement is valid. If a queen is placed in a position where it can capture another queen, the algorithm backtracks and tries a different position for the previous queen.

Here are the steps to solve the 8 queens problem using a backtracking algorithm:

  1. Start by placing a queen in the first row of the first column.

  2. Move to the second column and place a queen in the first row of that column.

  3. Continue placing queens in the next columns, starting in the first row and moving downwards.

  4. If you reach a point where you cannot place a queen in any row of a particular column without violating the constraints of the puzzle, backtrack to the previous column and try a different row for the queen in that column.

  5. Repeat steps 3-4 until all eight queens have been placed on the board.

  6. Once you have placed all eight queens on the board, you have solved the puzzle.

Note that there are many different ways to solve the 8 queens problem, and the specific algorithm you use may vary depending on your preferences and experience level.

The objective of the eight queens puzzle is to place eight chess queens on an 8x8 chessboard in a way that no two queens threaten each other. This means that there should not be two queens on the same row, column, or diagonal. This problem is a subset of the more general n queens problem, which involves placing n non-attacking queens on an n×n chessboard. Except for n=2 and n=3, solutions are available for all natural numbers n. Although the exact number of solutions is known only for n ≤ 27, the growth rate of solutions is approximately (0.143 n)n.

The eight queens puzzle was first introduced by chess composer Max Bezzel in 1848. Franz Nauck presented the first solutions to the problem in 1850 and also extended it to the n queens problem. Since then, many mathematicians, including Carl Friedrich Gauss, have contributed to the problem. In 1972, Edsger Dijkstra used this problem as an example of structured programming and presented a detailed description of a depth-first backtracking algorithm.

Finding all the solutions to the 8-queens problem can be computationally intensive, as there are over 4 billion possible arrangements of eight queens on an 8x8 board, but only 92 solutions. There are various ways to reduce computational requirements, such as applying a rule that chooses one queen from each column, which reduces the number of possibilities to 16.8 million. By generating permutations and checking for diagonal attacks, the possibilities can be further reduced to just 40,320.

There are 92 distinct solutions to the eight queens puzzle. However, if solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, then there are only 12 fundamental solutions. Each fundamental solution has eight variants obtained by rotating and reflecting it. One of the fundamental solutions has only four variants, and such solutions have only two variants. Hence, the total number of distinct solutions is 92.

8 Queens Puzzle — Base Camp Math (2024)

FAQs

Is there a solution to the 8 Queens problem? ›

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.

What is the solution to the n-queens problem? ›

If we place a queen in the center of the board, then it can always move to any other spot on the board in one move. So placing a queen at the center of the board does not let us place a second queen. So we can try placing a queen in the corner of the board. Now we have to cells that the queen cannot reach in one move.

How many possible solutions exist for an 8 problem? ›

The 8 Queen Problem is a puzzle in which 8 queens must be placed on an 8x8 chessboard so that no two queens threaten each other. It is a classic problem in computer science and mathematics. There are 92 solutions to this problem.

Which algorithm is used to solve the 8 queens problem? ›

Explanation: This pseudocode uses a backtracking algorithm to find a solution to the 8 Queen problem, which consists of placing 8 queens on a chessboard in such a way that no two queens threaten each other.

What is the prize for the 8 queens problem? ›

The underlying problem is one of the most major unsolved problems in computer science and mathematics. Known as P versus NP, it is one of the seven Millennium Prize Problems which carry the million dollar reward for their first correct solution.

How many combinations for n queens? ›

But anybody in their right mind trying to place 8 queens in a way that they don't attack each other wouldn't try to place two queens into the same row, but would put one queen each into row 1, row 2, row 3, ..., row 8. So the number of reasonable positions is only 88 or 16,777,216, about 264 times smaller.

How many solutions are there for the 8 queen problem? ›

Explanation: For 8*8 chess board with 8 queens there are total of 92 solutions for the puzzle.

Is N queens a hard problem? ›

Following our paper, we now understand that the reason why the n-queens completion problem is so much harder than the version with an empty board is that it is an example of an NP-complete problem.

Which technique is used to solve N queen problem? ›

The backtracking algorithm is used to solve the N queen problem, in which the board is recursively filled with queens, one at a time, and backtracked if no valid solution is found. At each step, the algorithm checks to see if the newly placed queen conflicts with any previously placed queens, and if so, it backtracks.

What is the 8 queens chess puzzle? ›

"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.

How to put 8 queens on a chessboard without threatening each other? ›

Placing queens on a chessboard using the knight's move to separate them can be quite a good strategy for playing eight queens. If you remove the black knights from Figure 1a and replace the four white knights with four queens, then no two queens are threatening each other (Figure 1b).

Is N queens np complete? ›

We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem.

What is the complexity of the 8 queens problem? ›

A simple brute-force solution would be to generate all possible chess boards with 8 queens. Accordingly, there would be N^2 positions to place the first queen, N^2 – 1 position to place the second queen and so on. The total time complexity, in this case, would be O(N^(2N)), which is too high.

What is the initial state of the 8 queens problem? ›

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.

What is the objective function in 8 queens problem? ›

The objective function will count the number of queens that are positioned in a place where they cannot be attacked. Given that queens move vertically, it's reasonable to say that no queen should be placed in the same vertical and thus we can represent the position of each queen in a simple array of 8 positions.

How many solutions are possible for the 8 queen problem? ›

Explanation: For 8*8 chess board with 8 queens there are total of 92 solutions for the puzzle.

What is the brute-force solution to the 8 queens problem? ›

The simplest solution to the 8-Queens puzzle is to use a brute-force search algorithm, which considers all 648 = 281,474,976,710,656 possible blind placements of eight queens, and then removing all placements that place two queens, either on the same space, or in mutually attacking positions.

Which of the following methods can be used to solve the N-queens problem? ›

The backtracking algorithm is used to solve the N queen problem, in which the board is recursively filled with queens, one at a time, and backtracked if no valid solution is found.

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