EXAMINE YOUR GUESS Your guess may be right, but it is 
foolish to accept a vivid guess as a proven truth as
primitive people often do. Your guess may be wrong. But 
it is also foolish to disregard a vivid guess altogether
as pedantic people sometimes do. 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 which we are genuinely 
interested. Such guesses usually contain at least a
fragment of 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 
many a guess has turned out to be wrong but never-the-less 
useful in leading to a better one.
     No idea is really bad, unless we are uncritical. What is 
really bad is to have no idea at all.
1.   Don't. Here is a typical story about Mr. John Jones.
Mr. Jones works in an office. He had hoped for a little
raise but his hope, as hopes often are, was disappointed.
The salaries of some of his colleagues were raised but not
his. Mr. Jones could not take it calmly. He worried and
worried and finally suspected that Director Brown was
responsible for his failure in getting a raise.
      We cannot blame Mr. Jones for having conceived such
a suspicion. There were indeed some signs pointing to 
Director Brown. The real mistake was that, after having
conceived that suspicion. Mr Jones became blind to all
signs pointing in the opposite direction. He worried himself
into firmly believing that director brown was his
personal enemy and behaved so stupidly that he almost
succeeding in making a real enemy of the director.
     The trouble with Mr. John Jones is that he behaves 
like most of us.  He never changes his major opinions. He 
changes his minor opinions not infrequently and quite 
suddenly but he never doubts any of his opinions, major 
or minor, as long as he has them.  He never doubts them,
or questions them or examines them critically. He would 
especially hate critical examination, if he understood 
what that meant.
   Let us concede that Mr. John Jones is right to a certain 
extent.  He is a busy man; he has his duties at the office
and at home.  He has little time for doubt or examination
At best, he could examine only a few of his
convictions and why should he doubt one if he has no time
to examen that doubt?
     Still, don't do as Mr. John  Jones does. Don't let your 
suspicion, or guess, or conjecture, grow without examination
till it becomes ineradicable.  At any rate, in theoretical
matters, the best of ideas is hut by uncritical 
acceptance and thrives on critical examination. 
2.  A mathematical example Of all quadrilaterals with
given perimeter, find the one that has the greatest area.
 What is the unknown A quadrilateral.
 What are the data? The perimeter of the quadrilateral is given.
What is the condition? The required quadrilateral 
should have a greater area than any other quadrilateral
with the same perimeter.
     This problem is very different from the usual problems
in elementary geometry therefor , it is quite natural to start
guessing.
     Which quadrilateral is likely to be the one with the
greatest area? What would be the simplest guess? We may
have heard that of all figures with the same perimeter the 
circle has the greatest area; we may even suspect 
some reason for the plausibility of this statement. 
Now which quadrilateral comes nearest to the circle?
which one comes nearest to it in symmetry?
     The square is a pretty obvious guess. If we take
this guess seriously, we should realize what it means.
We should have the courage to state it: ?"Of all
quadrilaterals with the given perimeter the square
has the greatest area." If we decide ourselves to
examine this statement, the situation changes. 
Originally, we had a "problem to find" after having
formulated our guess,s we have a "Problem to prove";
we have to prove or disprove the theorem formulated.
     If we do not know any problem similar to ours that has 
been solved before, we may find our task pretty tough. If
you cannot solve the proposed problem, try to solve first
some related problem. Could you solve a part of the 
problem? It may occur too us that if the square is privileged
among quadrilaterals it must, by that very fact
also be privileged i among rectangles. A part of our problem 
would be solved if we could succeed in proving the 
following statement": of all rectangles with the given perimeter
the square has the greatest area"
     This theorem appears more accessible than the former;
it is, of course, weaker. At any rate, we should realize
what it means; we should have the courage to restate it 
in more detail. We an restate it advantageously in the 
language of algebra.
     The arc of a rectangle with adjacent sices a and b is ab. its perimeter is 2a+2b.
    One side of the square that has the same perimeter as
the rectangle just mentioned is (a+b)/2.  Thus the area of 
this square is ((a+b)/2)^2.  It should be greater than the
area of the rectangle, and so we should have

              ((a+b)/2)^2   >   ab

Is this true?  The same assertion can be written in the
equivalent form
               a^2+2ab+b^2 > 4ab
This, however is true, for it is equivalent to

               a^2-2ab+b^2 > 0
or to
               (a-b)^2 > 0

and this inequality certainly holds, unless a=b, that is
the rectangle examined is a square.
     We have not solved our problem yet, but we have
made some progress just by facing squarely our rather
obvious guesses.
3.     A non mathematical example.  In a certain crossword
puzzle we have to find a word with seven letters, and the 
clue is: "Do the walls again, back and forth"4(footnote:the nation, june 9, 1945, Crossword puzzle, no 119)
    What is the unknown? A word
    What are the data? The length of the word is given; it has seven letters.
 what is the condition?  It is stated in the clue. It has
something to do with walls, yet it is still very hazy.
     Thus we have to reexamine the clue.  As we do so, the 
last part may catch our attention: "... again, back and
forth."  Could you solve a part of the problem? Here is a 
chance to guess the beginning of the word. Since the 
repetition is so strongly emphasised, the word, quite 
possibly, might start with "re." This is a pretty obvious
guess. If we are tempted to believe it., we should realize
what it means. The word required would look thus: 

                    R E - - - - - 

Can you check the result? If another word of the puzzle
 crosses the one just considered in the first letter, we 
have an R to start that other word. It may be a good idea
to switch to that other word and check the R. If we 
succeed in verifying that R or if, at least we do not find
any reason against it, we come back to our original word.
We ask again; What is the condition?  As we reexamine
the clue, the very last part may catch our attention
"...back and forth."  Could this imply that the word
we seek can be read not only forward but backward?
This is a less obvious guess (yet there are such cases, see
DECOMPOSING AND RECOMBINING,8).
     At any rate, let us face this guess; let us realize what 
it means. The word would look as follows:
     
                     RE --- ER

    Moreover, the third letter should be the same as the f fifth;
it is very likely a consonant and the forth or middle letter a vowel
     The reader can now easily guess the word by himself.
If nothing else helps , he can try all the vowels, one after 
the other, for the letter in the middle.