This is what I was going to say; it's consistent with the apparent time symmetry, and is the only solution that makes sense if we accept the problem as stated. But it seems like the wrong answer intuitively, because it means that every strategy is equal, as long as the probability of opening the box on a given day is in the half-open interval (0,0.5]. I'd certainly be happier with, say, p=0.01 than p=0.5, (and so would everyone else, apparently) which suggests that I don't actually have a real-valued utility function. This might be a good argument against real-valued utility functions in general (bounded or not). Especially since a lot of the proposed solutions here "fight the hypothetical" by pointing out that real agents can only choose from a finite set of strategies.