A frog is sitting k meters North of the water's edge of an infinite straight river that runs West to East. A flat area for frog jumping extends North from the water's edge for m meters, and just North of that is a vertical wall that prevents frogs from jumping any farther North. Our frogs jump once per second, but can only jump exactly 1 meter in any of the 4 cardinal directions (N, S, E, or W) which is done randomly with equal probability. At the end of any hop, frogs can only be an integer number of meters from the water from 0 to m, inclusive.
Consider 2 cases:
(case 1) Magoo Frog, who is very nearsighted, cannot see the wall even when he is m meters from the river. He will still try to jump in any of the 4 directions with equal probablity. If he happens to be at m meters and then tries to jump North, he will hit the wall and slide harmlessly down to where he started that jump, still m meters from the river. This "wasted" jump will take 1 second just like all the other jumps.
(case 2) Michigan Frog, who has good vision, will see the wall when he is m meters from the river, and he will jump with 1/3 probability either South, East, or West. If his distance from the water is < m, he jumps in any of the 4 directions.
Determine the expected values E_Magoo(k,m) and E_Michigan(k,m) for the time needed to reach the river.
Analytic and computer simulations welcome.