Normal arguments regarding maximising expected utility, presuppose that the distribution of conditions is independent of your strategy.
The counterintuitive (and many would say: counterfactual) element of Newcomb's paradox is that the distribution is said to depend on your strategy. You can understand the game easily enough for two strategies:
Suppose the predictor actually has the described power, and you both believe it does and trust that it will provide the $1M if you intend to take the contents of only the one box. Then, if only for selfish reasons, you'd be a fool to plan to take the second box. And after the reveal, it doesn't matter that you could then grab the second box as well: by hypothesis, you are in that circumstance because of a prediction, assumed accurate, that you won't choose to.
If the predictor actually has the described power, but you either don't believe it does or don't trust it to provide the $1M, you will be receptive to taking both boxes. The predictor, knowing this (by hypothesis) and also knowing whether you would choose to take the second box given the opportunity, will leave the opaque box empty if so. It would in effect be producing a self-fulfilling prophecy, but only because it knows enough about your intentions to do so, and because it has for some reason resolved only to give you $1M under specific circumstances.
Your strategy, in the game, would obviously depend on whether you think such a predictor is possible, and honest. If you don't, you should take both boxes; if you do believe in the predictor, you shouldn't.
If you believe in the predictor but get an empty box, that's tough: you were wrong either about it's ability to predict (if you go away empty-handed) or its intent to act in good faith (if you intended to take only one box but take both after re-evaluating your relationship with the predictor). However, the premise of the puzzle disallows this contingency. You may not believe that the game can be realised in fact, but the premise of the question is that the predictor is both perfect at prediction and honest. And in this case, any doubt of the player is a matter only of not knowing, or accepting, that premise.
If you accept the premise of the question, it's obvious how you ought to act. The only question, then, is whether you accept the premise of the question! If you don't, then it's not that Newcomb's paradox is unsolvable: it's that you refuse to regard it as worthy of serious consideration. And that's bound up with issues of philosophical naturalism (computational complexity in particular), and psychology.