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Six strategies for any problem
Simplify and borrow
Start by simplifying. Almost every problem comes wrapped in extraneous data, and bringing it down to the main issues shows what you are trying to do. The art is removing everything from a problem except what makes it interesting. Next, surround the problem with answers to similar questions already solved and look for what those answers have in common. It is much easier to make two small jumps than one big jump in any kind of mental thinking.
Restate and break it up
Restate the question. Change the words and the viewpoint to break loose from the mental blocks holding you in one way of looking at it, and do not get trapped by the sunk cost of work already done. Someone green to a problem sometimes solves it on the first attempt because the biases that build up over time do not constrain them. Then break an overwhelming problem into small pieces. Many mathematical proofs were found by extremely roundabout routes, where a mathematician proves many great results that seem to lead nowhere and ends up at the back door of the solution to the original problem.
Invert and stretch
Invert. If your premises will not prove your conclusion, imagine the conclusion is already true and see what happens, then try proving the premises instead. Finally, take an idea and see how far it will stretch. Someone always comes along and starts generalizing it, so you might as well do it yourself.
How does working backward solve problems?
The mathematician George Pólya's 1945 book How to Solve It lists working backward as a core heuristic. You start from the desired result and ask what would have to be true one step earlier, repeating until you reach something already known. The method goes back to the Greek mathematician Pappus of Alexandria, who called it analysis. Reversing the chain of steps then gives a direct solution.
Why do newcomers solve what experts miss?
In 1942 the psychologist Abraham Luchins ran water-jar experiments in which people learned a long procedure for measuring out amounts of water. Many kept using that procedure even when a much shorter one worked. This effect is now called Einstellung, German for a fixed mindset or set. People new to a problem have no practiced routine to fall back on, so they more often notice the simple route.
Why does inverting a problem work so well?
Carl Jacobi, the 19th-century German mathematician, was known for the advice 'man muss immer umkehren,' which means one must always invert. Asking what would make a result fail, or what a conclusion would imply, can expose structure that forward reasoning misses. Proof by contradiction, used since Greek mathematics, assumes the opposite of a claim and follows it to an absurdity.
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