N-Queens Problem
This problem is to find an arrangement of N queens on a chess board, such that no queen can attack any other queens on the board. The chess queens can attack in any direction as horizontal, vertical, horizontal and diagonal way. A binary matrix is used to display the positions of N Queens, where no queens can attack other queens.
WORKING
It can be seen that for n =1, the problem has a trivial solution, and no solution exists for n =2 and n =3.
So first we will consider the 4 queens problem and then generate it to n - queens problem.
Given a 4 x 4 chessboard and number the rows and column of the chessboard 1 through 4.
Since, we have to place 4 queens such as q1 q2 q3 and q4 on the chessboard, such that no two queens attack each other.
In such a conditional each queen must be placed on a different row, i.e., we put queen "i" on row "i."
Now, we place queen q1 in the very first acceptable position (1, 1).
Next, we put queen q2 so that both these queens do not attack each other.
We find that if we place q2 in column 1 and 2, then the dead end is encountered.
Thus the first acceptable position for q2 in column 3, i.e. (2, 3) but then no position is left for placing queen 'q3' safely.
So we backtrack one step and place the queen 'q2' in (2, 4), the next best possible solution.
Then we obtain the position for placing 'q3' which is (3, 2).
But later this position also leads to a dead end, and no place is found where 'q4' can be placed safely.
Then we have to backtrack till 'q1' and place it to (1, 2) and then all other queens are placed safely by moving q2 to (2, 4), q3 to (3, 1) and q4 to (4, 3).
That is, we get the solution (2, 4, 1, 3). This is one possible solution for the 4-queens problem.
For another possible solution, the whole method is repeated for all partial solutions.
The other solutions for 4 - queens problems is (3, 1, 4, 2) i.e.
The implicit tree for 4 - queen problem for a solution (2, 4, 1, 3) is as follows:
It can be seen that all the solutions to the 4 queens problem can be represented as 4 - tuples (x1, x2, x3, x4) where xi represents the column on which queen "qi" is placed.
In this way it can be done to all the another numbers.
COMPLEXITY