Algorithm for finding all paths in a NxN grid

Question

Imagine a robot sitting on the upper left hand corner of an NxN grid. The robot can only move in two directions: right and down. How many possible paths are there for the ro

Answer 1

Below is the code in Java to count all the possible paths from top left corner to bottom right corner of a NXN matrix.

public class paths_in_matrix {

    /**
     * @param args
     */
    static int n=5;
    private boolean[][] board=new boolean[n][n];
    int numPaths=0;
    paths_in_matrix(){
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                board[i][j]=false;
            }
        }
    }
    private void togglePiece(int i,int j){
        this.board[i][j]=!this.board[i][j];
    }
    private boolean hasBeenVisited(int i,int j){
        return this.board[i][j];
    }
    private boolean exists(int i,int j){
        return i < n && i > -1 && j < n && j > -1;
    }
    private boolean viablePosition(int i,int j){
        return exists(i, j) && !hasBeenVisited(i,j);
    }
    private void traversePaths(int i,int j){
        //BASE CASE: if reached (n - 1, n - 1), count as path and stop. 
        if (i == (n - 1) && j == (n - 1)) {
          this.numPaths++;
          return;
        }
        this.togglePiece(i, j);
        //RECURSIVE CASE: if next point is a viable position, go there and make the same decision

        //go right if possible
        if (this.viablePosition(i, j + 1)) {
          traversePaths(i, j + 1);
        }
      //go left if possible
        if (this.viablePosition(i, j - 1)) {
          traversePaths( i, j - 1);
        }

        //go down if possible
        if (this.viablePosition(i + 1, j)) {
          traversePaths( i + 1, j);
        }

        //go up if possible
        if (this.viablePosition(i - 1, j)) {
          traversePaths(i - 1, j);
        }

        //reset the board back to the way you found it after you've gone forward so that other paths can see it as a viable position for their routes
        this.togglePiece(i, j);

    }
    private int robotPaths(){

        traversePaths(0,0);
        return this.numPaths;
    }
    public static void main(String[] args) {
        paths_in_matrix mat=new paths_in_matrix();
        System.out.println(mat.robotPaths());
    }

}