I seem to remember the "heap paradox" being a very old one (given a heap and repeatedly removing a single grain of sand, when does it stop being a heap?). Yet I don't ever recall hearing a solution to it. No doubt there are different views of things, but is there at least a generally accepted solution to this paradox?

You asked, "Is there at least a generally accepted solution to this paradox?" Not by a long shot! The paradox of the heap (and its cousins that use other vague concepts) is in my opinion one of the greatest unsolved intellectual problems. It has generated a huge philosophical literature, and it's very much a topic of current philosophical debate, but I have yet to see a proposed solution that even comes close to being satisfactory. For starters, you might take a look at these entries from the Stanford Encyclopedia of Philosophy: SEP, " Sorites Paradox " SEP, " Vagueness " Best wishes as you work your way through this daunting -- but inescapable -- problem! I think you'll find it repays your careful thought even if you don't end up much closer to a satisfying solution.