Examples of when a "new way of thinking" led to a solution I was reading William Thurston's On Proof and Progress in Mathematics in which he discusses the value of the different ways people think about the same mathematical structure. He claims that many mathematical solutions are the result of different ways of thinking about the same underlying mathematics.
What are some examples of problems that seem very difficult, but yield to a new way of thinking about the relevant structures? 
Answers should include a description of the problem, the “way of thinking”, and the rigorous solution.

The probabilistic method has been successfully used for proving the existence of mathematical objects non constructively, by proving that the probability of choosing an object in that class is not zero.
