I'm not going to argue for what it takes to win the prize. Obviously, chances of me doing it are slim to none, so I never bothered to look into the actual rules. My point is, if I somehow arrived at an algorithm that somehow finds Hamiltonian paths in all inputs, I will make a lot of money in various ways. The millenium prize is besides the point.

You say "No, it is not obviously possible. It is possible if and only if P = NP." So you are trying to tell me that it is not POSSIBLE for me to guess all answers right in a row for infinity of tries? In what sense do you mean that? Because surely probability theory will disagree with you.

Regarding the description of the non-deterministic TM, again, I get all that. I'm not saying the same TM WILL work for us in the future past the current try. I'm saying, assume we are at time t_now. It is theoretically POSSIBLE to have such a configuration of the random number generator that it will answer correctly every time. and if it is POSSIBLE, than we can just stumble upon it, if nothing else. All I'm saying is, let's take this POSSIBILITY seriously and picture it in real world. Now this magic guess of ours may not work tomorrow for new input (to your point about convincing you about all input). But that's not important! what is important is, there will be a POSSIBILITY tomorrow of finding another configuration that will not only correctly solve all previous inputs, but also the current ones. We may never be able to find a reliable algorithm of this sort, but we are guaranteed to be sure that there is one. How it works and what the mathematics are behind it and why this particular set of random generator settings produce this mysterious and baffling property of always finding correct hamiltonian paths is an entirely different question. It's not the same question -- its an entirely different question. What I'm arguing towards is that there IS such an algorithm right now, not how or why it would in the future. But in the future we are GUARANTEED to find another such algorithm, even if we have to abandon our current one, because it will be probable. (sorry for all the caps)