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Mathematics consists of proving the most obvious thing in the least obvious way.
—  George Polya
Observe also (what modern writers almost forgot, but some older writers, such as Euler and Laplace, clearly perceived) that the role of inductive evidence in mathematical investigation is similar to its role in physical research.
—  George Polya
If you can't solve a problem, then there is an easier problem you can solve: find it.
—  George Polya
A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
—  George Polya
He was the only student that ever scared me (in reference to John von Neumann).
—  George Polya
The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.
—  George Polya
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself.
—  George Polya
The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them.
—  George Polya
Beauty in mathematics is seeing the truth without effort.
—  George Polya
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
—  George Polya
There are many questions which fools can ask that wise men cannot answer.
—  George Polya
When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.
—  George Polya
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
—  George Polya
To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important.
—  George Polya
If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
—  George Polya
[Hilbert] once had a student in mathematics who stopped coming to his lectures, and [he] was finally told the young man had gone off to become a poet. Hilbert is reported to have remarked: 'I never thought he had enough imagination to be a mathematician.'
—  George Polya
To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.
—  George Polya

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