"Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution." (David Hilbert [Paris International Congress], 1900)
"To find a new problem which is both interesting and accessible, is not so easy; we need experience, taste, and good luck. Yet we should not fail to look around for more good problems when we have succeeded in solving one. Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near." (George Pólya, "How to Solve It", 1945)
"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"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)
"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future." (Andrew Wiles)
"No one should pick a problem, or make a resolution, unless he realizes that the ultimate value of it will offset the inevitable discomfort and trouble that always goes along with the accomplishment of anything worthwhile. So, let us not waste our time and effort on some trivial thing." (Charles F Kettering)
"Problems worthy of attack prove their worth by fighting back." (Piet Hein)
"The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver." (Israel N Herstein)
"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)
"You are never sure whether or not a problem is good unless you actually solve it." (Mikhail Gromov)
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