How to Solve It by George Polya

• The first step to solving a problem is: understand the problem. After the problem-solver has completed this phase of the problem, he goes on to the next step, which is: devise a plan. After these two steps are accomplished, the next step becomes relatively easy, which is: carry out the plan.
• One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles.
• The student should acquire as much experience of independent work as possible. BUt if he is left alone with his problem without any help or with insufficient help, he may make no progress at all.
• Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. [...] Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.
• Understand the problem. It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire.
• If you cannot solve the proposed problem try to solve first some related problem.
• Consider your problem from various sides and seek contacts with your formerly acquired knowledge.
• Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts.
• In solving a proposed problem, we can often use the solution of a simpler analogous problem; we may be able to use its method, or its result, or both.
• An auxiliary theorem is a theorem whose proof we undertake in the hope of promoting the solution of our original problem.
• An auxiliary problem is a problem which we consider, not for its own sake, but because we hope that it's condensation may help us to solve another problem, our original problem. The original problem is the end we wish to attain, the auxiliary problem a means by which we try to attain our end.
• Human superiority consists in going around an obstacle that cannot be overcome directly, in devising a suitable auxiliary problem when the original problem appears insoluble. To devise an auxiliary problem is an important operation of the mind.
• The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself.
• In scientific cowork, it is necessary to apportion wisely determination to outlook. You do not take up a problem, unless it has some interest; you settle down to work seriously if the problem seems instructive; you throw in your whole personality if there is a great promise. If your purpose is set, you stick to it, but you do not make it unnecessarily difficult for yourself. You do not despise little successes, on the contrary, you seek them: If you cannot solve the proposed problem try to solve first some related problem.
• Incomplete understanding of the problem, owing to lack of concentration, is perhaps the most widespread deficiency in solving problems. With respect to devising a plan and maintaining a general idea of the solution two opposite faults are frequent. Some students rush into calculations and constructions without any plan or general idea; others wait clumsily for some idea to come and cannot do anything that would accelerate its coming.
• Assume what is required to be done as already done.
• Generalization may be useful in the solution of problems.
• If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again.
• Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.
• Induction is the process of discovering general laws by the observation and combination of particular instances.
• Inventor’s paradox: The more ambitious plan may have more chances of success. This sounds paradoxical. Yet, when passing from one problem to another, we may often observe that the new more ambitious problem is easier to handle than the original problem.
• The more comprehensive theorem may be easier to prove, the more general problem may be easier to solve.
• Speaking and thinking are closely connected, the use of words assists the mind. Certain philosophers and philologists went a little further and asserted that the use of words is indispensable to the use of reason.
• Reductio ad absurdum shows the falsity of an assumption by deriving from it a manifest absurdity.
• Indirect proof establishes the truth of an assertion by showing the falsity of the opposite assumption.
• Both “reductio ad absurdum” and indirect proof are effective tools of discovery which present themselves naturally to an intent mind.
• The intelligent problem-solver tries first of all to understand the problem as fully and as clearly as he can. Yet understanding alone is not enough; he must concentrate upon the problem, he must desire earnestly to obtain its solution. If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem.
• How to solve it:
• First: You have to understand the problem.
• Second: Find the connection between the data and the known. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
• Third: Carry out the plan.
• Fourth: Examine the solution obtained.