A spider and a bug are located on the opposite corners of a regular cube. At each time instant they are allowed to move only along the edges of the cube. Each movement along an edge is equally likely. Calculate the mean time when they meet.
In this problem I tried to solve through markov process as there are 4 states. However, according to me they never meet if they are in opposite diagonal. So is the mean time will be infinite? Can some body comment in this regards?
It appears that the bug is too dull to realize that it is running into the spider, and that the spider's belly is full so it really doesn't care whether it can eat this bug. However, given their behavior, we can frame equations thus.
When they take the first step, although each has $3$ possibilities, so $9$ possibilities in all, a little thought would show that due to symmetry, it is enough to consider that the bug takes some particular step, and the spider has $3$ choices.
Denoting the initial position with a distance of $3$ between them by a, positions with a distance of $1$ between them as b, with one step taken by each, we get the equation
$a = 1 + \frac13\cdot a + \frac23\cdot b$
[With $1$ set of moves, either we return to initial state with probability $\frac13$, or move ahead $2$ steps with probability $\frac23$]
All we need to do now is to frame the equation after the second set of moves, and solve. [Remember that here you'll need to consider $9$ possibilities ]