A non-intuitive(?) fact
This seems counter-intuitive, at least to me, but if you can find an intuitive explanation for it, please comment!
Fact: Let G be a cycle (connected graph where every vertex has degree 2: you know what I mean) of length n (i.e., n vertices and n edges). Start a random walk on G from any node u. Then the probability that v is the last node visited (i.e., the random walk hits all other nodes before hitting v) is the same for every v other than u!
[Such a property is obviously true for the complete graph, and apparently it's true only for cycles and complete graphs.]
I do not know a probabilist’s proof; what I could think of is a distinctly ComputerSciencey (Dynamic Programming / recurrence relation) proof.