First off, assuming this problem - or one similar enough to directly apply - hasn't already got an official theorem or conjecture proving it one way or the other, I'd be content with simply being pointed in the direction of whatever maths would help me understand and learn about the problem myself. In light of this, I should say that the highest level of maths I studied have been a high school statistics class and intro to trig.
Anyway, so the question proper. What first got me thinking about this was playing old D&D-based CRPGs like Baldur's Gate and KotOR where things like weapon damage are listed as "2-8" or "3-10". But these number ranges actually correspond to different combinations of dice (two 4-sided dice, and one 8-sided dice adding 2 to the result, respectively; often referred to as 2d4 and 1d8+2 in RPG circles, for those who don't know), and I'm wondering if it's possible to derive a single combination of dice from any given (valid) pair of numbers.
Further, I'd be interested to know what sorts of constraints there might be on being able to derive things this way. Like is it only possible with specific "sizes" of dice, or so long as all the dice in a given pool are the same size, etc. Which is kind of why, if someone hasn't already written a paper on the subject, I'd be content with simply being directed to the sorts of math that would help me solve these problems myself, rather than simply being given an answer (though I certainly wouldn't decline one if it is offered).