Solution
Cheese in the Center
Show the problem again
A mouse lives in a 3 × 3 grid of rooms; doors connect rooms that share a wall. Each minute the mouse leaves its room through a door chosen uniformly at random. It starts in a corner room, and there is cheese in the center room. What is the expected number of moves until the mouse first enters the center room? Give an exact number.
Worked solution
The answer is 6. By symmetry there are two kinds of non-center rooms: corners (2 doors, both to edge rooms) and edge rooms (3 doors: two corners and the center). Let C and E be the expected remaining moves. Then C = 1 + E and E = 1 + (2/3)C. Substituting, E = 1 + (2/3)(1 + E), so E/3 = 5/3, E = 5, and C = 6.
Source: Classic exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.