Problem solving

Gaining more experience in problem solving can be beneficial for every designer on more than one level. If design is about solving other people's problems, we need to find ways to solve our inability to solve. This will come with experience, but we can still find ways to learn more. When I heard again of the book “How to Solve It: A New Aspect of Mathematical Method” by George Polya, I thought that the time has come to see if there is something that can actually help me. Despite being somewhat repetitive, it contains some nonobvious things, who still make it worth reading. Here are my notes on things that I found thought-provoking.

“ Framework: Understand the problem deeply. Devise a plan to solve it. Carry it out. Reflect on the result. […] Don't start rashly. Look before you leap. Delay a bit for safety, but don't hesitate too long. […] There's a small discovery in the solution of every problem. A great discovery solves a great problem. Is this problem worth solving? Solve more ambitious problems. The inventor's paradox: the more ambitious plan may have more chances of success. […] Set yourself problems proportional to your knowledge. In order to solve a problem, we need a certain amount of previously acquired knowledge. […] Try not only to understand the solution, but the motives and procedures behind it. […] Many people spend a lot of time solving unpractical problems. […] Look at the unknown. Try to think of a similar problem having the same or similar unknown. […] Our conception of the problem is likely to be incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution. […] Understand the problem, but also strongly desire its solution. “Where there is a will, there is a way.” […] When stuck, draw a figure or introduce a suitable notation. […] The language you speak can both limit and empower how you feel. […] Good ideas are based on past experience and formerly acquired knowledge. Start the right train of ideas. […] Vary, transform, modify, restate the problem. […] If you can't solve the proposed problem, try to first solve a related one. Experiment with auxiliary problems, but don't go too far from the original problem or you risk losing yourself. Do we know of a related problem? Could we introduce an auxiliary element that will allow us to use an already known solution to a similar problem? […] Check each step of your solving plan carefully, especially if you're convinced that it's right. Use formal reasoning to make sure it's correct. […] Look back at a completed solution and examine it without rushing to the next problem. No problem can be completely exhausted. […] Can you check the result? Can you derive it differently? Can you check the argument? […] Imagine cases in which you can utilize a known procedure again. Can you use the result of the method for some other problem? […] A suggestion to solve a particular problem might be applicable to many others. […] What can I do with an incomplete idea? […] Analogy may reach the level of mathematical precision even if people use it for incomplete things. While trying to solve a problem, consider discovering a simpler analogous problem. […] Try to foresee the result. […] Frequent in mathematical reasoning: chains of equivalent auxiliary problems. […] Numerical results of mathematical problems can be tested by comparing them to observed numbers or to a commonsense estimate of observable numbers. […] A result is often built of intermediate results. Did we use all the available data? […] We can try solving the problem in different ways and seeing if results are the same. […] Solve problems invented by yourself. […] Try generalization, specialization, analogy, decomposition, recombination. […] Can you prove that the solution is correct? […] Sometimes the intuition is ahead, sometimes the formal reasoning. […] If a condition is expressed by more linear equations than there are unknowns, it is either redundant or contradictory; if the condition is expressed by fewer equations than there are unknowns, it is insufficient to determine the unknowns; if the condition is expressed by just as many equations as there are unknowns, it is usually just sufficient to determine the unknowns, but may be, in exceptional cases, contradictory or insufficient. […] Try to think of a familiar problem having the same or similar unknown. […] The ingenuity of a problem solver shows itself in the originality of the combination. Difficult problems demand hidden, exceptional, original combinations. […] Keep the unknown, change the data. Keep the data, change the unknown. Change both. […] Keep only part of a hypothesis, drop the other part. Is the conclusion still valid? […] In solving problems, determination and emotions play an important role (not only intellect). To solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments. Determination fluctuates with hope and hopelessness, with satisfaction and disappointment. It's easy to keep going when we think that the solution is just around the corner, but it's hard to persevere when we don't see any way out of the difficulty. […] “You can undertake without hope and persevere without success” — Rene Descartes […] Widespread deficiency in solving problems: incomplete understanding of the problem due to lack of concentration. […] In carrying out the plan, the most frequent fault is carelessness, lack of patience in checking each step. Failure to check the result (and becoming disinterested to it) is common. Another common reason for failure: we soon believe what we desire. […] Good chess problems are supposed to have but one solution and no superfluous piece on the chessboard. […] Problems are rarely absolutely new. Do you know of a related problem? […] Is it familiar to you? Have you seen it before? Could you use it/its result/its method? […] An idea by Gottfried Wilhelm Leibniz: sources of invention can be more interesting than the inventions themselves. […] Lemmas (“what is assumed”) are auxiliary theorems. […] “Respice finem” — Look at the (unknown) end. Remember your aim. Don't forget your goal. […] What causes could produce such result? […] Given … find … […] Archimedes couldn't know if a solution to “finding the area of a surface of a sphere inscribed in tetrahedron whose six edges are given” was correct, but he knew various formerly solved problems having a similar unknown. In his solution, he uses an approximation to the sphere a composite solid consisting of two cones and several frustums of cones. Collect formerly solved problems with the same or similar unknowns. […] Find common features in the way of handling all sorts of problems. […] We are elated when our progress is rapid, we are depressed when it is slow. To make progress, view the problem from different sides. What is essential to progress? How do we define it? […] A good sign should be easy to remember and easy to recognize. […] In analysis we start from what's required, we take it for granted and we draw consequences from it and consequences from the consequences, till we reach a point that we can use as a starting point in synthesis. […] You need to be able even to draw conclusions from an unproven theorem (within reason). […] What is the unknown? What is the data? What is the condition? […] As Columbus and his companions sailed westward across an unknown ocean, they were cheered whenever they saw birds. They regarded a bird as a favorable sign indicating the nearness of a land. But in this they were repeatedly disappointed. They watched for other signs too. They thought that floating seaweed or low banks of cloud might indicate land, but they were again disappointed. One day however the signs multiplied... The next day they sighted land. […] The most important signs of progress are heuristic. Follow them, but keep your eyes open. […] Treat your guiding feelings and inspirations just as you treat other visible signs. Always follow your inspiration with a grain of doubt. […] If signs multiply, you start to move with increased confidence. Concentrate effort on the right spot. […] The expert knows more signs than the inexperienced and knows them better. His main advantage is the extraordinary mental sensibility for noticing even the smallest signs for presence or absence of progress. He has, perhaps no more ideas than the inexperienced, but appreciates more what he has and uses it better. […] Specialization is often useful in the solution of problems. […] A problem, after prolonged absence, may return into consciousness essentially clarified, much nearer to its solution than it was when it dropped out of consciousness. We work at problems subconsciously. […] We can start with the unknown and work backwards to our hypothesis. “A wise man begins in the end, a fool ends in the beginning.” […] Throw your whole personality into your problem. […] Success in solving the problem depends on choosing the right aspect, on attacking the fortress from its accessible side. In order to find out which aspect is the right one, which side is accessible, we try various sides and aspects, we vary the problem. Variation of the problem is essential. Varying the problem, we bring in new points, and we create new contacts, new possibilities of contacting elements relevant to our problem. […] We cannot hope to solve any worthwhile problem with intense concentration. But we are easily tired by intense concentration of our attention upon the same point. In order to keep our attention alive, the object on which it is directed must unceasingly change. […] If we fail to make progress, our attention falters, our interest fades and we get tired of the problem. To escape such danger, we need to set ourselves a new direction about the problem and vary it by showing a new aspect of it. We need to vary the data on the problem too as this helps to make it more interesting. […] Questions are of the greatest importance for the problem solver (no matter what type of problem we have). What/Why/Where/When/How? […] Understanding proofs can reveal insights. […] Connected facts are more interesting and are better retained in memory than isolated ones. Isolated facts are hardly collected, but easily forgotten, so we need a system to connect them and organize our information. […] We must do as we may if we can't do as we would. […] A wise man changes its mind, a fool never does. He will make more opportunities than he finds. […] The end of fishing is not angling, but catching. ”

George Polya in "How to Solve It: A New Aspect of Mathematical Method"