The Feynman Method Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!” ...
Finding counter examples to conjectures can be notoriously hard (pun? I think not). This is an area of creativity that mostly goes unappreciated. Here’s a personal anecdote: my father once came up with an algorithm to solve a hugely constrained version of the traveling salesman problem. The greedy proof was slightly hand-wavy, and I felt it would be an easier thing to find a counter example where his algorithm wouldn’t find the optimal tour. Of course, I was just trying to tell him that he couldn’t have solved TSP (or even approximated it). I learnt two lessons that day. ...