This program uses backtracking to count the number of paths from [1][1] to [N][N] in an NxN array.

robot_movements.cpp

#include <iostream>

using namespace std;
#define N 4
#define LEGAL 1
#define ILLEGAL 0
int counter = 0;

void count_paths(int x, int y, int n, int m, int array[N+2][N+2]){
  if(x == n && y == m){
    counter++;
  }
  else {
    int dx = 1, dy = 0;
    for(int i = 0; i < 4; i++){
      if(array[x+dx][y+dy] == LEGAL){
        array[x+dx][y+dy] = ILLEGAL;
        count_paths(x + dx, y + dy, n, m, array);
        array[x+dx][y+dy] = LEGAL;
      }
      swap(dx,dy);
      if(i == 1)
        dx = -dx;
    }
  }
}

int main(){
  int array[N+2][N+2];
  
  for(int i = 1; i < N+1; i++)
    for(int j = 1; j < N+1; j++)
      array[i][j] = LEGAL;

  for(int i = 0; i < N+2; i++)
    array[0][i] = array[i][0] = array[N+1][i] = array[i][N+1] = ILLEGAL;

  array[1][1] = ILLEGAL; //Set start cell to be seen.
  count_paths(1,1,N,N,array);
  cout << counter << endl;
}