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.

9.Discover several counterexamples in which method A fails, in which the cause of failure can be definitively traced back to P. Abandon efforts to modify method A.

10.Realise that special case y is related to (or at least analogous to) a problem z in another field of mathematics. Look up the literature on z, and ask experts in that field for the latest perspectives on that problem.

11.Learn that z has been successfully attacked in that field by use of method B. Attempt to adapt method B to solve y.

12.After much effort, an adapted method B’ is developed to solve y.

13.Repeat the above steps 1-12 with A replaced by B’ (the outcome will of course probably be a little different from the sample storyline presented above). Continue doing this for a few years, until all model special cases can be solved by one method or another.

14.Eventually, one possesses an array of methods that can give partial results on X, each of having their strengths and weaknesses. Considerable intuition is gained as to the circumstances in which a given method is likely to yield something non-trivial or not.

15.Begin combining the methods together, simplifying the execution of these methods, locating new model problems, and/or finding a unified and clarifying framework in which many previous methods, insights, results, etc. become special cases.

16.Eventually, one realises that there is a family of methods A^* (of which A was the first to be discovered) which, roughly speaking, can handle all cases in which property P^* (a modern generalisation of property P) occurs. There is also a rather different family of methods B^* which can handle all cases in which Q^* occurs.

17.From all the prior work on this problem, all known model examples are known to obey either P^* or Q^*. Formulate Conjecture C: all cases of problem X obey either P^* or Q^*.
18.Verify that Conjecture C in fact implies the problem. This is a major reduction!

19.Repeat steps 1-18, but with problem X replaced by Conjecture C. (Again, the storyline may be different from that presented above.) This procedure itself may iterate a few times.
20.Finally, the problem has been boiled down to its most purified essence: a key conjecture K which (morally, at least) provides the decisive input into the known methods A^*, B^*, etc. which will settle conjecture C and hence problem X.

21.A breakthrough: a new method Z is introduced to solve an important special case of K.

22.The endgame: method Z is rapidly developed and extended, using the full power of all the intuition, experience, and past results, to fully settle K, then C, and then at last X.

23.The technology developed to solve major problem X is adapted to solve other related problems in the field. But now a natural successor question X’ to X arises, which lies just outside of the reach of the newly developed tools… and we go back to Step 1.?