One of the best problem-solving strategies I was ever taught was to respond to things like this with "well, what if it wasn't?" Don't try and fight their reasoning about why it's impossible, because you won't win - instead, step around it by getting them to ignore the part they're/you're getting stuck up on.
It takes a bit of persistence and convincing the other party to engage in what they consider a flight of mental fancy, but once you get over the "impossible" part and start figuring out how to solve the rest of the problem, I find that a solution that avoids the initially-impossible part tends to present itself. That, or you'll figure out why what you thought was impossible isn't. The beauty of this is that it works equally well when you're trying to convince someone else of something you know to be right and when you're trying to figure out a solution yourself.
The other easy strategy I use all the time (and this is more for concrete problems, as opposed to intellectual/theoretical ones) is start by deciding if the problem is one that can easily be solved by throwing money at it. If it can be, then it's a much easier question - you just have to determine if the problem is worth the money it would cost to fix it, which is simpler. If the problem can't be solved with money, then you know where to start to find a solution (enumerating all the non-monetary resources you have and seeing what might work).
It takes a bit of persistence and convincing the other party to engage in what they consider a flight of mental fancy, but once you get over the "impossible" part and start figuring out how to solve the rest of the problem, I find that a solution that avoids the initially-impossible part tends to present itself. That, or you'll figure out why what you thought was impossible isn't. The beauty of this is that it works equally well when you're trying to convince someone else of something you know to be right and when you're trying to figure out a solution yourself.
The other easy strategy I use all the time (and this is more for concrete problems, as opposed to intellectual/theoretical ones) is start by deciding if the problem is one that can easily be solved by throwing money at it. If it can be, then it's a much easier question - you just have to determine if the problem is worth the money it would cost to fix it, which is simpler. If the problem can't be solved with money, then you know where to start to find a solution (enumerating all the non-monetary resources you have and seeing what might work).