Here, I solve the problem of induction for you.* [ *Based on: “Explanationist Aid for the Theory of Inductive Logic,” British Journal for the Philosophy of Science 60 (2009): 1-31. ] This was a long and somewhat technical paper, so I’m going to give a baby version.
Do you have an easier version to understand than the baby version?
I take issue with your cube example. I don't think the number of molecules in the cube is causally basic. More causally basic is the manufacturer's intent to produce cubes in which certain measures are within specified bounds.
But then your (a) and (b) cases still should yield different answers if all you know is that in one case they're measuring side lengths and in the other they're measuring volume.
Actually, you could take this a little further (I understand this drifts off topic) and say that 1/2 is the better answer for both cases because in order make the assumption we're asked to make (in either case), the manufacturer would *need* to measure the lengths of edges (and indeed angles) to ensure that it is actually a cube rather than some other parallelepiped. And then the case (b) degenerates into case (a), because if the manufacturer is sure it's a cube, there's no point in measuring the volume because in verifying it's a cube, the side length constraint (between 0 and 2 inches) is sufficient to meet the volume condition.