Problems to find, problems to prove We draw a parallel
between these two kinds of problems.
1.  The aim of a "problem to find" is to find a certain
object, the unknown of the problem.
    The unknown is also called "quaesitum", or the thing
sought, or the thing required.  "Problems to find" may be 
theoretical or practical, abstract or concrete, serious problems
or mere puzzles. We may seek all sorts of unknowns;
we may try to find, to obtain, to acquire, to produce, or
to construct all imaginable kinds of objects. In the problem
of the mystery story the unknown is a murderer. In
a chess problem the unknown is a move of the chess men.
In certain riddles the unknown is a word. In certain elementary
problems of algebra the unknown is a number. 
In a problem of geometric construction the unknown is
a figure. 
     2.  The aim of a "problem to prove" is to show
conclusively that a certain clearly stated assertion is true, or
else to show that it is false. We have to answer the question:
Is this assertion true or false? And we have to
answer conclusively, either by proving the assertion true,
or by proving it false.
     A witness that the defendant stayed at home a 
certain night. The judge has to find out whether this
assertion is true or not and, moreover, he has to give as
good grounds pas possible  for his finding. Thus, the judge
has a "problem to prove." Another"problem to prove"
is to "prove the theorem of Pythagoras." We do not say:
"Prove or disprove the theorem of Pythagoras". It would
be better in some respects to include in the statement of
the problem the possibility of disproving but we may 
neglect it because we know that the chances for disprove
disproving the theorem of Pythagoras are rather slight.
     3.  The principal parts of a problem to find" are the 
unknown the data and the condition
     If we have to construct a triangle with sides a,b,c, the 
unknown is a triangle, the data are the three lengths a
b,c, and the triangle is required to satisfy the condition
that its sides have the given lengths a,b,c. If we have to
construct a triangle whose altitudes are a,b,c, the unknown-
known is an object of the same category as before, the
data are the same, but the condition linking the unknown 
to the data is different.
4.    If a "problem to prove" is a mathematical problem
of the usual kind, its principal parts are the hypothesis
and the conclusion of the theorem which has to be proved
or disproved.
"If the four sides of a quadrilateral are equal, then the
two diagonals are perpendicular to each other." The
second part starting with "then" is the conclusion, the 
first part starting with "if" is the hypothesis.
  [Not all mathematical theorems can be split naturally
into hypothesis and conclusion. Thus, it is scarcely possible
to split so the theorem:"There are an infinity of prime numbers."
5.  If you wish to solve a "problem to find" you must
know, and know very exactly, its principal parts, the 
unknown, the data, and the condition. Our list contains
many questions and suggestions concerned with these
parts.
 What is the unknown? What are the data? what is the condition?
     Separate the various parts of the condition.
Find the connection between the data and the unknown
Look at the unknown! And try to think of a familiar
problem having the same or a similar unknown
    keep only a part of the condition, drop the other part;
how far is the unknown then determined, how can it
vary? Could you derive something useful from the data?
Could you think of other data appropriate to determine
the unknown? Could you change the unknown, or the
data, or both if necessary, so that the new unknown and the
new data are nearer to each other?
     Did you use all the data? Did you use the whole
condition?
6.    If you wish to solve a problem to prove you must
know, and know very exactly its principal parts, the
hypothesis, and the conclusion. There are useful questions
and suggestions concerning these parts which correspond
respond to those questions suggestions of our list
which are specially adapted to problems to find.
     What is the hypothesis? what is the conclusion?
     Separate the various parts of the hypothesis
     Find the connection between the hypothesis and the
conclusion
     Look at the conclusion! And try to think of a familiar
theorem having the same or a similar conclusion.
     Keep only a part of the hypothesis, drop the other 
part; is the conclusion still valid?  Could you derive some-
thing useful for the hypothesis? Gould you think of 
another hypothesis from which you could easily derive
the conclusion? Gould you change the hypothesis, or the
conclusion, or both if necessary, so that the new hypothesis
 and the new conclusion are nearer to each other?
    Did you use the whole hypothesis?
7.  "Problems to find" are more important in elementary
mathematics, "problems to prove" more important
in advanced mathematics.  In the present book, "problems
to find" are more emphasised then the other kind.
but the author hopes to reestablish the balance in a fuller
treatment of the subject.