Random Walk Simulation

1. Consider the scene of limited two-dimensional space: a) In the 55 grid map shown in FIG 1, a miner starts working from the central red dot at time t=0 and moves one space around (up/down/left/right) with equal probability (1/4) every unit of time for exploration. If the miner reaches the boundary, he will move one grid in the direction of possible movement with equal probability (1/3) at the next moment. The treasure is located at the position of the yellow five-pointed star. When the miner reaches any treasure position, it is considered a successful treasure hunt and stops working immediately. Find the average working time of miners for treasure hunting, and verify the results through simulation. b) Extend the two-dimensional map in a) to a 77 grid map as shown in FIG. 2. Treasure location, miner's starting position, working rules and stopping conditions of the miners remain unchanged. If the miner reaches the four corners of the map before the treasure hunt is successful, the miner will move one space in the direction of possible movement with equal probability (1/2) at the next moment. Find the average working time of miners for treasure hunting, and verify the results through simulation. 2. Consider the scenario of an infinite two-dimensional space. Extend the finite map in Problem 1 to an infinite map. The treasure location, miner's starting position, working rules and stopping conditions remain unchanged. Is it possible for miners to stop working within a limited time? If yes, find the average working time of miners for treasure hunting; if not, please give the reason