On Problem Solving 31: On Challenges

"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, "How to solve it", 1944)

"Throughout the evolution of mathematics, problems have acted as catalysts in the discovery and development of mathematical ideas. In fact, the history of mathematics can probably be traced by studying the problems that mathematicians have tried to solve over the centuries. It is almost disheartening when an old problem is finally solved, for it will no longer be around to challenge and stimulate mathematical thought." (Theoni Pappas, "More Joy of Mathematics: Exploring mathematical insights & concepts", 1991)

"[...] all problems can be reperceived as challenges, or 'opportunities' to change, grow or learn." (Robert B Dilts, "Sleight of Mouth: The Magic of Conversational Belief Change", 1999

"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) 

"Many of the great mathematical problems stem from deep and difficult questions in well-established areas of the subject. They are the big challenges that emerge when a major area has been thoroughly explored. They tend to be quite technical, and everyone in the area knows they’re hard to answer, because many experts have tried and failed. The area concerned will already possess many powerful techniques, massive mathematical machines whose handles can be cranked if you’ve done your homework - but if the problem is still open, then all of the plausible ways to use those techniques have already been tried, and they didn’t work. So either there is a less plausible way to use the tried-and-tested techniques of the area, or you need new techniques." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)

"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)

"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)