Chess Board Problems

Question

There is a chess board of dimension n * n. You are given with 2 squares on that board S(x1,y1) ;M(x2,y2). S is a fixed point. M can move diagonally. It can move any number of steps or jumps in 1 move . Find the minimum number of moves M needs to reach S

My Approach: We can just count diagonal blocks, But I'm confused with jumps. Can anybody explain what they meant by jumps?

Answer 1

I think jump here refers to the case when the chess the piece can move diagonally more than 1 step. Example if at (1,1) then it can go to (3,3) in a single step.

I have coded a bactracking algorithm assuming the above case. The basic idea here to get all possible moves to reach the destination (x,y) co-ordinate. It checks all valid moves at a given position and prints the path followed to reach here. construct_candidates() will give you all valid candidate co-ordinates for the current position. It checks the bounds and also validates we have not visited the chess block earlier, if these conditions are satisfied, then it is a valid candidate for the move.

You can easily modify it to keep track of the shortest possible path followed.

#define N 4 /* Chess Board Dimension */
#define TRUE     1   
#define FALSE    0

#define START_X  0
#define START_Y  0
#define TARGET_X 1
#define TARGET_Y 3

typedef short int bool;

typedef struct point_ {
    int x;
    int y;
} point_t;


bool is_candidate_valid (point_t *a, int k, int new_x, int new_y)
{
    int i;
    /* Check bounds */
    if ((new_x < 0) || (new_x > (N-1)) ||
        (new_y < 0) || (new_y > (N-1))) {
        return FALSE;
    }

    /* Check if this new position is already in the path followed */

    for (i = 0; i < k; i++) {
        if (a[i].x == new_x && a[i].y == new_y) {
            return FALSE;
        }
    }
    return TRUE;
}

void construct_candidates (point_t *a, int k, point_t *candidates, int *n_candidates)
{
    int delta;
    *n_candidates = 0;
    int new_x, new_y;

    for (delta = 1; delta <= (N-1); delta++)  {

        new_x = a[k-1].x + delta;
        new_y = a[k-1].y + delta;

        if (is_candidate_valid (a, k, new_x, new_y) == TRUE) {
             candidates[*n_candidates].x = new_x;
             candidates[*n_candidates].y = new_y;
             (*n_candidates)++;
        }

        new_x = a[k-1].x + delta;
        new_y = a[k-1].y - delta;

        if (is_candidate_valid (a, k, new_x, new_y) == TRUE) {
             candidates[*n_candidates].x = new_x;
             candidates[*n_candidates].y = new_y;
             (*n_candidates)++;
        }

        new_x = a[k-1].x - delta;
        new_y = a[k-1].y + delta;

        if (is_candidate_valid (a, k, new_x, new_y) == TRUE) {
             candidates[*n_candidates].x = new_x;
             candidates[*n_candidates].y = new_y;
             (*n_candidates)++;
        }

        new_x = a[k-1].x - delta;
        new_y = a[k-1].y - delta;

        if (is_candidate_valid (a, k, new_x, new_y) == TRUE) {
             candidates[*n_candidates].x = new_x;
             candidates[*n_candidates].y = new_y;
             (*n_candidates)++;
        }
    }
}

bool is_a_solution (point_t *a, int k)
{
    if (a[k-1].x == TARGET_X && a[k-1].y == TARGET_Y) {
        return TRUE; /* Actual Solution found */
    }
    if (k == (N*N)) {
        return TRUE; /* No Solution found */
    }
    return FALSE;
}

void process_solution (point_t *a, int k)
{
    int i;

    if (k == (N*N)) {
        return; /* No solution Possible */
    }

    for (i = 0; i < k; i++) {
        printf ("(%d, %d) ", a[i].x, a[i].y);
    }
    printf ("\n");
}


void backtrack (point_t *a, int k)
{
    int i, n_candidates;
    point_t candidates[4*(N-1)];

    if (is_a_solution (a, k) == TRUE) {
        process_solution (a, k);
        return;
    }

    construct_candidates (a, k, candidates, &n_candidates);
    for (i = 0; i < n_candidates; i++) {
        a[k].x = candidates[i].x;
        a[k].y = candidates[i].y;

        backtrack (a, k + 1);
    }
}

int main()
{
    point_t a[N*N];
    /* Fill up the initial position */
    a[0].x = START_X;
    a[0].y = START_Y;

    backtrack (a, 1);
}

Output:   

(0, 0) (1, 1) (2, 2) (3, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (1, 1) (2, 2) (3, 1) (1, 3) 
(0, 0) (1, 1) (2, 2) (1, 3) 
(0, 0) (1, 1) (2, 0) (3, 1) (2, 2) (1, 3) 
(0, 0) (1, 1) (2, 0) (3, 1) (1, 3) 
(0, 0) (1, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (1, 1) (0, 2) (1, 3) 
(0, 0) (1, 1) (0, 2) (2, 0) (3, 1) (2, 2) (1, 3) 
(0, 0) (1, 1) (0, 2) (2, 0) (3, 1) (1, 3) 
(0, 0) (1, 1) (3, 3) (2, 2) (3, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (1, 1) (3, 3) (2, 2) (3, 1) (1, 3) 
(0, 0) (1, 1) (3, 3) (2, 2) (1, 3) 
(0, 0) (2, 2) (3, 3) (1, 1) (2, 0) (3, 1) (1, 3) 
(0, 0) (2, 2) (3, 3) (1, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (2, 2) (3, 3) (1, 1) (0, 2) (1, 3) 
(0, 0) (2, 2) (3, 3) (1, 1) (0, 2) (2, 0) (3, 1) (1, 3) 
(0, 0) (2, 2) (3, 1) (2, 0) (1, 1) (0, 2) (1, 3) 
(0, 0) (2, 2) (3, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (2, 2) (3, 1) (1, 3) 
(0, 0) (2, 2) (1, 3) 
(0, 0) (2, 2) (1, 1) (2, 0) (3, 1) (1, 3) 
(0, 0) (2, 2) (1, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (2, 2) (1, 1) (0, 2) (1, 3) 
(0, 0) (2, 2) (1, 1) (0, 2) (2, 0) (3, 1) (1, 3) 
(0, 0) (3, 3) (2, 2) (3, 1) (2, 0) (1, 1) (0, 2) (1, 3) 
(0, 0) (3, 3) (2, 2) (3, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (3, 3) (2, 2) (3, 1) (1, 3) 
(0, 0) (3, 3) (2, 2) (1, 3) 
(0, 0) (3, 3) (2, 2) (1, 1) (2, 0) (3, 1) (1, 3) 
(0, 0) (3, 3) (2, 2) (1, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (3, 3) (2, 2) (1, 1) (0, 2) (1, 3) 
(0, 0) (3, 3) (2, 2) (1, 1) (0, 2) (2, 0) (3, 1) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 2) (3, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 2) (3, 1) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 0) (3, 1) (2, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 0) (3, 1) (1, 3) 
(0, 0) (3, 3) (1, 1) (2, 0) (0, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (0, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (0, 2) (2, 0) (3, 1) (2, 2) (1, 3) 
(0, 0) (3, 3) (1, 1) (0, 2) (2, 0) (3, 1) (1, 3)