"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)
"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, 1945)
"In picking that problem be sure to analyze it carefully to
see that it is worth the effort. It takes just as much effort to solve a
useless problem as a useful one." (Charles F. Kettering, 1955)
"Solving a problem simply means representing it so as to make the solution transparent." (Herbert A Simon, "The Sciences of the Artificial", 1968)
"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)
"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)
"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)
"Problem solving starts off from an initial given situation or statement of a problem (known as the initial state of the problem). Based on the problem situation and your prior knowledge you have to work towards a solution. When you reach it you are in the goal state of the problem. On the way from the initial state to the goal state you pass through a number of intermediate problem states." (S Ian Robertson, "Problem Solving", 2001)
"This does not mean that the representation has to be 'complete' before any problem solving can take place. If you had a 'complete' representation of a problem then you wouldn't have a problem, as you would know exactly how to get from where you are now to where you want to be. A problem only exists when it is not immediately obvious how to get from where you are now to your goal. An adequate representation should at least allow you to see what moves you can possibly make and allow you to start heading towards your goal." (S Ian Robertson, "Problem Solving", 2001)
"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)
"[...] a general-purpose universal optimization strategy is theoretically impossible, and the only way one strategy can outperform another is if it is specialized to the specific problem under consideration." Yu-Chi Ho & David L Pepyne, "Simple explanation of the no-free-lunch theorem and its implications", Journal of Optimization Theory and Applications 115, 2002)
"Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley." (Yōko Ogawa, "The Housekeeper and the Professor", 2003)
"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)"Every problem has a solution; it may sometimes just need another perspective.” (Rebecca Mallery et al, “NLP for Rookies”, 2009)
"Framing the right problem is equally or even more important than solving it." (Pearl Zhu, "Change, Creativity and Problem-Solving", 2017)
"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)
"A problem thoroughly understood is always fairly simple." (Charles Kettering)
"A problem well-defined is a problem half solved." (John Dewey)
"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves)
"Finding the right answer is important, of course. But more important is developing the ability to see that problems have multiple solutions, that getting from X to Y demands basic skills and mental agility, imagination, persistence, patience." (Mary H Futrell)
"I have not seen any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." (Paul Anderson)
"I knew nothing, except how to think, how to grapple with a problem and then go on grappling with it until you had solved it." (Sir Barnes Wallis)
"Man is not born to solve the problems of the universe, but to find out where the problems begin, and then to take his stand within the limits of the intelligible." (Johann Wolfgang von Goethe)
"One is always a long way from solving a problem until one actually has the answer." (Stephen Hawking)
"One measure of our understanding is the number of independent ways we are able to get to the same result." (Richard P Feynman)
“Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice.” (George Polya)
"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner)
"The best way to escape from a problem is to solve it." (Brendan Francis)
"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)
"The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems." (Charles W Churchman)
"The mere formulation of a problem is often far more essential than its solution. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances in science." (Albert Einstein)
"The worst thing you can do to a problem is solve it completely." (Daniel Kleitman)
"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)
"To ask the right question is harder than to answer it." (Georg Cantor)
"When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems." (David Hilbert)
"You are never sure whether or not a problem is good unless you actually solve it." (Mikhail Gromov)
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