Clash Of Queens Hack (44)

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We first place queen q1 in the first acceptable position (1, 1). Next, we place 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).

We first place queen q1 in the first acceptable position (1, 1). Next, we place 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.

Here, we have to first write the program to find out the positions of the queens. Then, we can write the main program in which the positions of the queens are stored in memory (RAM). Then, the program will read the positions from the memory and try to place the queens so that the minimum number of enemy queens can be captured. For the solution, we need to place the queens in all possible orders.

The above argument is feasible for a smaller number of queens, but as the number of queens increases, the problem becomes difficult. The number of solutions for the 16-queens problem is 78. This figure is obtained by dividing the total number of possible solutions by the total number of partial solutions. That is, for a partial solution, we have to backtrack from the position of the queen placed at (i, j) to the position of the queen placed at (i-1, j). That is, we have to choose the positions for the queens in a particular order. For a solution, we have to choose the positions for the queens in all possible orders. Thus for the number of solutions, we need to multiply the 78 partial solutions by 78! i.e. 78!.

Now, you see that a queen placed at the square (i, j) would capture the jth row enemy and could not be captured by a queen placed at square (i-1, j) and the same goes for the column. So, we take out the row and column from the board and solve the same problem on a 8 by 8 board. You can find this solution at the page, https://q1.com 827ec27edc