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."
Here is Polya's problem solving process, in simplified form:
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."
In order to keep the problem interesting and to avoid getting stuck,
Polya suggests that we keep trying to look at the problem in different ways:
p.6"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."
Sometimes an educated guess can lead to progress towards a solution
- one should not fear a guess being wrong if solving a problem is important to you:
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."
Solving a related problem can lead to progress towards a solution
of the desired problem:
p.101"If you cannot solve the proposed problem, try to solve first
some related problem."
Polya constantly asks himself these questions about the problems
he faces:
"What is the unknown? What are the data? What is the condition?
Do you know a related problem?"
Real world problems are not quite the same as math problems, but
the process towards a solution is the same:
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?"
Did you use all the data? What an interesting question. Progress
towards a solution may not be possible until all pieces of the puzzle are located and brought into play:
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?"
Polya notes that certain mate-in-two chess problems require the
solver to consider all the pieces on the board, as each one plays a part in the solution:
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."
Find the connection between the data and the unknown, and you will
find the solution to your problem.
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."
