Because the turtle is limited to positive movement, the length of each path is 80 units. He must go down 40 and right 40 units to reach the end point. Because his total path is 80 units and the grid is 40 units square, this is a simple matter of permutations. 80 choose 40 is around 1.07507209 x— 10^23. The problem becomes vastly more difficult when the turtle is allowed to travel in other directions.