Here are some points you should keep in mind. Here rook can move in the right, left, vertical and horizontal direction. So it moves either 3 or 4 moves in the direction of the final position (i.e. upper right corner) or in opposite direction of the final position am assuming 'a'=number of 3 moves towards the final position either horizontally or vertically (i.e. towards the east or north side) 'b'=number of 3 moves away from the final position either horizontally or vertically (i.e. towards the west or south side) 'c'=number of 4 moves towards the final position either horizontally or vertically(i.e. towards the north or east side) 'd'=number of 4 moves away from the final position either horizontally or vertically(i.e. towards the south or west side) As per the given fact total number of moves =9 so we can say that a+b+c+d=9 where a>=0 and b>=0 and c>=0 and d>=0 (because the number of moves can not be negative) and no. of moves required from the initial position to the final position is 14 (7 to the east side and 7 to the north side) so 3*a+3*b+4*a+4*b=14 where a>=b and c>=d and (a-b)<3 and (c-d)<2. these constraints are because of the limit of the chess board. the number of solutions to the given equation can give you a final answer. comment please if you solve the given equation by satisfying all constraints. and if there are some other points you can comment