Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and call $p$ the probability that each second the flea moves clockwise.
What is the probability $P_{m,t}$ of the flea passing through $1\le m\le n$ vertices after $t$ seconds?
I got interested in this problem after proving that when $p=1/2$ and $n=5$ we have $P_{5,28}>1/2.$