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  [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right):

  1. Isolate a toy model case x of major problem X.
  2. Solve model case x using method A.
  3. Try using method A to solve the full problem X.
  4. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”.
  5. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far.
  6. Conjecture that P is true for all instances of problem X.
  7. Discover a family of counterexamples y, y’, y”, … to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed.
  8. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see whether method A can work in the absence of P.
  
  (... 15 more steps)