Come up with a list of potential approaches, order them by roughly how promising you think they are and then try them all in that order. Obviously limit this by

Come up with a list of potential approaches, order them by roughly how promising you think they are and then try them all in that order. Obviously limit this by how much time/energy you can offer it somehow. When you're done, see if you can think of more approaches. If you can't, you probably can't solve this problem and it's time to call it a day. But that's ok, because there's almost certainly some higher level goal you're aiming to achieve. So think about other ways that you could achieve that without solving your original problem. Maybe there's a way to solve a similar problem that has looser constraints which is easier.