8 Puzzle with Backtracking

Question

I was reading this book from Skiena, Programming Challenges and after the backtracking chapter there was a question about solving the 15-puzzle with backtracking, which I reduce it to 8-puzzle just experimenting. I have this recursive code and I am wondering whether it have a chance to find the solution ever. The code is kind of ugly (be warned):

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int arr[20][20]={
    {3,1,2},
    {4,0,5},
    {6,8,7}
};

int moveX[20]={1,1,1,0,0,-1,-1,-1};
int moveY[20]={0,1,-1,1,-1,0,1,-1};
int depth=0;

int findRow(){
    int i,j;
    for(i=0;i<4;i++){
        for(j=0;j<4;j++){
            if(arr[i][j]==0){
                return i;
            }
        }
    }
}

int findCol(){
    int i,j;
    for(i=0;i<3;i++){
        for(j=0;j<3;j++){
            if(arr[i][j]==0){
                return j;
            }
        }
    }
}

void print(){
    int i,j;
    for(i=0;i<3;i++){
        for(j=0;j<3;j++){
            printf("%i ",arr[i][j]);
        }
        printf("\n");
    }
    printf("\n");
}

int isReady(){
    if(arr[0]==1 && arr[1]==2 && arr[2]==3 && arr[3]==4 && arr[4]==5 && arr[5]==6 && arr[6]==7 && arr[7]==8){
        return 1;
    }
    else return 0;
}

void perm(int row,int col,int n){
    if(n>=9){
        print();
        if(isReady())
            printf("Finished");
        depth++;

        return;
    }

    int i=0;int diffX,diffY,temp;
    int r=findRow();
    int c=findCol();
    temp=arr[r][c];
    arr[r][c]=arr[row][col];
    arr[row][col]=temp;
    for(i=0;i<8;i++){
        diffX=row+moveX[i];
        diffY=col+moveY[i];
        if(diffX>=0 && diffX<4 && diffY>=0 && diffY<4){
            perm(diffX,diffY,n+1);
        }
    }
    temp=arr[r][c];
    arr[r][c]=arr[row][col];
    arr[row][col]=temp;
}

int main()
{
    perm(0,0,0);
    return 0;
}

My question is, is there a chance with this code to find the solution and second, can anybody how the puzzle can be solved in reasonable time?

Answer 1

You have five problems. First, the isReady function is incorrect. It should look like this:

int isReady(){
    if(arr[0][0]==1 && arr[0][1]==2 && arr[0][2]==3 &&
            arr[1][0]==4 && arr[1][1]==5 && arr[1][2]==6 &&
            arr[2][0]==7 && arr[2][1]==8){
        return 1;
    }
    else return 0;
}

Second, you are exceeding your puzzle bounds with diffX and diffY. You need to change this:

    if(diffX>=0 && diffX<4 && diffY>=0 && diffY<4){

to this:

    if(diffX>=0 && diffX<3 && diffY>=0 && diffY<3){

Third, your findRow function also exceeds the puzzle bounds. Change all of the 4 to 3.

Fourth, you should check your victory condition only after you have made your move. So move your victory check below the swap:

temp=arr[r][c];
arr[r][c]=arr[row][col];
arr[row][col]=temp;
// This victory check is now below the move.
if(n>=9){
    print();
    if(isReady())
        printf("Finished");
    depth++;

    return;
}

Fifth, you should change your initial call from this:

perm(0,0,0);

to this:

perm(1,1,0);

The way you have it, you are always forcing a move to the upper left as your first move. The modified way keeps the 0 in the center so it doesn't force your first move. When I ran this code with all of the modifications I made, it found 3 solutions. When I further modified the code to check for solutions at any depth, it found 2 solutions at depth 8 and 3 solutions at depth 9.