Question

I need an algorithm to find out all the possible positions of a group of pieces in a chessboard. Like finding all the possible combinations of the positions of a number N of pieces.

For example in a chessboard numbered like cartesian coordinate systems any piece would be in a position

(x,y) where 1 <= x <= 8 and 1 <= y <= 8

I’d like to get an algorithm which can calculate for example for 3 pieces all the possible positions of the pieces in the board. But I don’t know how can I get them in any order. I can get all the possible positions of a single piece but I don’t know how to mix them with more pieces.

for(int i = 0; i<= 8; i++){
    for(int j = 0; j<= 8; j++){
        System.out.println("Position: x:"+i+", y:"+j);
    }
}

How can I get a good algoritm to find all the posible positions of the pieces in a chessboard?

Thanks.

Answer 1

You got 8×8 board, so total of 64 squares.

Populate a list containing these 64 sqaures [let it be list], and find all of the possibilities recursively: Each step will “guess” one point, and invoke the recursve call to find the other points.

Pseudo code:

choose(list,numPieces,sol):
   if (sol.length == numPieces): //base clause: print the possible solution
       print sol
       return
   for each point in list:
       sol.append(point) //append the point to the end of sol
       list.remove(point)
       choose(list,numPieces,sol) //recursive call
       list.add(point)  //clean up environment before next recursive call
       sol.removeLast()

invoke with choose(list,numPieces,[]) where list is the pre-populated list with 64 elements, and numPieces is the pieces you are going to place.

Note: This solution assumes pieces are not identical, so [(1,2),(2,1)] and [(2,1),(1,2)] are both good different solutions.

EDIT:

Just a word about complexity, since there are (n^2)!/(n^2-k)! possible solutions for your problem – and you are looking for all of them, any algorithm will suffer from exponential run time, so trying to invoke it with just 10 pieces, will take ~400 years

[In the above notation, n is the width and length of the board, and k is the number of pieces]