# Rat in a Maze with multiple steps or jump allowed?

The rat in maze problem is one of the well-known problem of the backtracking. Here we will see that problem with little variation. Suppose one NxN maze M is given. The starting point is top left corner M[0, 0], and the destination is right bottom corner M[N – 1, N - 1]. One rat is placed at the starting point. Our goal is to find a path from starting point to ending point that can be by the rat to reach the destination. Here the rat can jump (The variation). Now there are some constraints

• The rat can move either towards the right or towards the down.
• Maze with 0 in cell indicates that the cell is blocked.
• Non-zero cells are indicating valid path.
• The number inside the cell indicates the number of maximum jump the rat can make from that cell.

## Algorithm

#### ratInMaze

begin
if destination is reached, then
print the solution matrix
else
1. Place the current cell inside the solution matrix as 1
2. Move forward or jump (check max jump value) and recursively check if move leads to solution or not.
3. If the move taken from the step 2 is not correct, then move down, and check it leads to the solution or not
4. If none of the solutions in step 2 and 3 are correct, then make the current cell 0.
end if
end

## Example

#include <iostream>
#define N 4
using namespace std;
void dispSolution(int sol[N][N]) {
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++)
cout << sol[i][j] << " ";
cout << endl;
}
}
bool isSafe(int maze[N][N], int x, int y) { //check whether x,y is valid or not
// when (x, y) is outside of the maze, then return false
if (x >= 0 && x < N && y >= 0 && y < N && maze[x][y] != 0)
return true;
return false;
}
bool ratMazeSolve(int maze[N][N], int x, int y, int sol[N][N]) {
if (x == N - 1 && y == N - 1) { //if destination is found, return true
sol[x][y] = 1;
return true;
}
if (isSafe(maze, x, y)) {
sol[x][y] = 1; //mark 1 into solution matrix
for (int i = 1; i <= maze[x][y] && i < N; i++) {
if (ratMazeSolve(maze, x + i, y, sol)) //move right
return true;
if (ratMazeSolve(maze, x, y + i, sol)) //move down
return true;
}
sol[x][y] = 0; //if the solution is not valid, then make it 0
return false;
}
return false;
}
bool solveMaze(int maze[N][N]) {
int sol[N][N] = { { 0, 0, 0, 0 },
{ 0, 0, 0, 0 },
{ 0, 0, 0, 0 },
{ 0, 0, 0, 0 }
};
if (!ratMazeSolve(maze, 0, 0, sol)) {
cout << "Solution doesn't exist";
return false;
}
dispSolution(sol);
return true;
}
main() {
int maze[N][N] = { { 2, 1, 0, 0 },
{ 3, 0, 0, 1 },
{ 0, 1, 0, 1 },
{ 0, 0, 0, 1 }
};
solveMaze(maze);
}

## Output

1 0 0 0
1 0 0 1
0 0 0 1
0 0 0 1

Updated on: 31-Jul-2019

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