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.
Beauty in mathematics is seeing the truth without effort.
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.
He was the only student that ever scared me (in reference to John von Neumann).
If the learning of mathematics reflects to any degree the invention of mathematics, it must have a place for guessing, for plausible inference.
If you can't solve a problem, then there is an easier problem you can solve: find it.
Mathematics consists of proving the most obvious thing in the least obvious way.
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.
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.
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.