I came across the problem stated below a while ago, and because I have an affinity for dodecahedrons, I spent some time with it. It confused me, because I was not sure what it was asking for. The key phrase is "expected value", which is a technical term, and I was not sure what it meant in this case. In gambling, "expected value" is a statement of the return you will get if you bet the entire board, the complement of the house edge. In most casinos, this is a value of about 5/6; in state run lotteries, it is about 1/2. But I couldn't find an intuitive meaning of the phrase for this kind of problem. I worked through the posted solutions, did a number of relevant probability calculations, and still didn't understand what the question wanted. Eventually I fell back to brute force, and set up a spreadsheet that ran 1000 ants through 200 moves, and found the answer. For this type of problem, "expected value" is equal to the number on which the average number of turns to complete converges over infinite iterations. I am not at all sure that this is a useful value, but I learned some new tricks along the way.
By the way, the answer is 35...
An ant starts on one vertex of a dodecahedron. Every second he randomly walks along one edge to another vertex. What is the expected value of the number of seconds it will take for him to reach the vertex opposite to the original vertex he was on?
Clarification: Every second he chooses randomly between the three edges available to him, including the one he might have just walked along.