p.vii The following pages are written somewhat concisely, but as simply as possible, and are based on a
long and serious study of methods of solution. This sort of study, called heuristic by some writers, is not in fashion
nowadays but has a long past and, perhaps, some future.
p.xvi First. You have to understand the problem. Second. Find the connection between the data and
the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found.
You should obtain eventually a plan of the solution. Third. Carry out your plan. Fourth. Examine the solution obtained.
p.5 Trying to find the solution, we may repeatedly change our point of view, our way of looking
at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather
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.
p.9 Thus, it is often appropriate to start the work with the question: Do you know a related problem?
p.10 Could you restate the problem? ... Did you use all the data? Did you use the whole condition?
p.11 Do you know any problem with a similar unknown?
p.99 Examine your guess. Your guess may be right, but it is foolish to accept a vivid guess as a
proven truth... Your guess may be wrong. But it is also foolish to disregard a vivid guess altogether... Guesses
of a certain kind deserve to be examined and taken seriously: those which occur to us after we have attentively considered
and really understood a problem in which we are genuinely interested. Such guesses usually contain at least a fragment
of the truth although, of course, they very seldom show the whole truth. Yet there is a chance to extract the whole truth
if we examine such a guess appropriately.
p.101 If you cannot solve the proposed problem, try to solve first some related problem.
What is the unknown? What are the data? What is the condition? Do you know a related problem?
p.114 Do not forget that human superiority consists in going around an obstacle that cannot be overcome
directly, in devising some suitable auxiliary problem when the original one appears insoluble.
Could you imagine a more accessible related problem? You should now invent a related
problem ... Do you know a related problem?
p.121 The inventor's paradox. The more ambitious plan may have more chances of success.
p.123 Focusing our attention on our aim and concentrating our will on our purpose, we think of ways and
means to attain it. What are the means to this end? How can you attain your aim?
p.150 In order to solve a problem, we need a certain amount of previously acquired knowledge.
p.151 Our example shows that the knowledge needed and the concepts used are more complex and less sharply
defined in a practical problem than in mathematical problems... Unknowns, data, conditions, concepts, necessary preliminary
knowledge, everything is more complex and less sharp in practical problems than in purely mathematical problems. This is an
important difference, perhaps the main difference, and it certainly implies further differences; yet the fundamental motives
and procedures of the solution appear to be the same for both sorts of problems. There is a widespread opinion that practical
problems need more experience than mathematical problems. This may be so. Yet, very likely, the difference lies in
the nature of the knowledge needed and not in our attitude towards the problem. In solving a problem of one or the
other kind, we have to rely on our experience with similar problems and we often ask the questions: Have you seen the
same problem in a slightly different form? Do you know a related problem?
p.152 Did you use all the data? Did you use the whole condition? Did you use all the data which could contribute
appreciably to the solution? Could you think of other data appropriate to determine the unknown?
p.179 In a well constructed chess problem [such as: find a checkmate in 2 moves] there is no superfluous
piece. Therefore, we have to take into account all chessmen on the board; we have to use all the data.
p.182 In fact, to solve a problem is, essentially, to find the connection between the data and the unknown.
Moreover we should, at least in well stated problems, use all the data, connect each of them with the unknown.
p.185 Signs may guide our acts. Their absence may warn us of a blind alley and save us time and
useless exertion; their presence may cause us to concentrate our effort upon the right spot... signs may misguide
us in any single case, but they guide us right in the majority of them... It takes experience to interpret the signs
correctly...The expert knows by experience how the situation looks and feels when the solution is near and so he
is able to read the signs which indicate that he is approaching it. The expert knows more signs than the inexperienced,
and he knows them better; his main advantage may consist in such knowledge.
p.186 Columbus and his men conjured from the beginning that they would eventually find
land sailing westward... As they proceeded, they related every incident, major or minor, to their dominating question:
"Are we approaching land?" Their confidence rose and fell as events occurred or failed to occur..." [JLJ - this is
called by Polya a heuristic syllogism].
p.207 The intelligent problem-solver tries first of all to understand the problem as fully and as clearly
as he can. Yet understanding alone is not enough; he must concentrate upon the solution. If he cannot summon up real desire
for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole
personality into your problem.
p.209 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.
p.210 if we fail to make progress... there is danger of losing [interest in] the problem
altogether. To escape from this danger we have to set ourselves a new question about the problem... The new question reconquers our interest by varying the problem, by showing some new aspect
of it.
p.232 Going around an obstacle is what we do in solving any kind of problem...
