Good Ways to Solve Difficult Math Problems, What are They ?

 I'm going to copy in an answer from another related question question here, because I think it can help.

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I have two pieces of advice.

First, math is not a spectator sport. You can watch it for entertainment, but the only way to learn the game is to get out and play. It's not always about formulas, and sometimes it's hard. Sometimes you fail. Sometimes you spend hours crafting a beautiful and elegant solution to the wrong problem. When that happens: Get up, dust yourself off, turn to a new page in the notebook, and try again. When you beat a problem that has punished you, celebrate! Fist pumps, Woot, Hoo-rah, the whole deal. It is awesome to win.

Second, I'd like to offer a snippet from the book "How To Solve It: A New Aspect of Mathematical Method" by George Pulyas. Don't be put off by the title, this book is very readable. The examples, except for one, all use high-school level geometry or less. Even if you don't understand the math of an example or two it is still very useful.

Here are the most valuable two pages of that book:

Understanding the Problem
First. You have to understand the problem.

What is the unknown? What are the data? What is the condition?
Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?

Draw a figure. Introduce suitable notation.
Seperate the various parts of the condition. Can you write them down?

Devising a plan
Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

Have you seen it before? Or have you seen the same problem in a slightly different form?
Do you know a related problem? Do you know a theorem that could be useful?
Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Could you restate the problem? Could you restate it still differently? Go back to definitions.

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve another part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the data appropriate to determine the unknown? Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?

Did you use all the data? Did you use the whole condition? Have you taken into account all the essential notions involved in the problem?

Carrying out the plan
Third. Carry out your plan.

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

Looking back
Fourth. Can you check the result? Can you check the argument?
Can you derive the result differently? Can you see it at a glance?
Can you use the result, or the method, for some other problem?

Writer : Elizabeth Greene