We generally think that the past is settled and nothing we do in the present can change it (barring time travel or backwards causation). In other words, it seems irrational to direct our efforts toward trying to affect the past. For example, if an organism evolved a mutation that caused it to spend time and energy trying to influence the past instead of the future, it's reasonable to assume that the organism wouldn't survive very long. So far, so good.
However, a wrinkle in the assumption that it's irrational to direct our action toward the past is provided by Newcomb's Paradox. In this paradox, we are asked to stipulate an infallible (or near infallible) Predictor that can predict your actions. You're then presented with two boxes, Box A and Box B. Box A is transparent and you can see that it contains $1000. Box B is opaque, but you're told that it either contains $1,000,000 or nothing. You're then told that you can take both boxes or take Box B only. If the Predictor predicted that you would take both boxes, Box B contains nothing, but if the Predictor predicted that you would take Box B only, it contains $1,000,000.
Newcomb's Paradox is a common problem in decision theory, but it also pushes our intuitions about actions affecting events in the past. From Wikipedia:
The crux of the paradox is in the existence of two contradictory arguments, both being seemingly correct.
A powerful intuitive belief, that past events cannot be affected. My future action cannot determine the fate of an event that happened before the action.
Newcomb proposes a way of doing precisely this — affecting a past event. The prediction of the Predictor establishes equivalence between my choice (of renouncing the open box) and the content of the closed box, which was determined in the past. Since I can affect the future event, I can also affect the past event, which is equivalent to it.
Your 'solution' to the problem will likely depend on whether or not you think it's rational to try to affect something that's past and settled. After all, either the money is already there or it's not. Newcomb's Paradox is fun partly because it causes us to question this intuition, but in another sense, it can sharpen it (at least it does for me).
So let's say that our intuition that our actions can't affect the past survives Newcomb's Paradox. This intuition, if sound, has implications for determinism. If determinism is true, then all of our actions are determined by the prior causal history of the universe. A Predictor, in possession of all the facts about the universe, could in principle predict our future actions, much like in the case of Newcomb's boxes. The causal history of the universe, then, would be different depending on whether I chose both boxes or Box B only. But this also seems like a case of affecting the past, for the following reason.
Let's say I want the causal history of the universe to lead to my getting $1,000,000. Then I would choose Box B only. This seems to entail that my choice caused the past to be a certain way rather than another. But that doesn't seem possible. In fact, regardless of what choice I make, it doesn't seem like the past would change at all. Again, the matter has already been settled ex hypothesi. Thus, based on our intuition about the immutability of the past, it seems that we cannot coherently regard our actions as being determined by the prior causal history of the universe. This suggests that determinism is false.