Did you use all the data?   Owing to the progressive mobilisation
of our knowledge, there will be much more in
our conception of the problem at the end than was in it 
at the outset(PROGRESS AND ACHIEVEMENT ,1).
But how is it now? Have we got what we need?
Is our conception adequate?  Did you use 
all the data? Did you use the whole condition? The
corresponding question concerning "problems to prove" 
is :  Did you use the whole hypothesis? 
1.     For an illustration, let us go back to the 
"parallelepiped problem" stated in section 8 ( and followed up in 
sections 10,12,14, 15). It may happen that a student
runs into the idea of calculating the diagonal of a face, s
sqrt(a2+b2), but then he gets stuck.  The teacher may help
him by asking:  Did you use all the data? The student
can scarcely fail to observe that the expression sqrt(a2+b2)
does not contain the third datum c.  Therefor, he should
try to bring c into play. Thus, he has a good chance to 
observe the decisive right triangle whose legs are 
sqrt(a2+b2) and c, and who's hypotenuse is the desired
diagonal of the parallelepiped. (For another illustration
see AUXILIARY ELEMENTS,3.)
    The questions we discuss here are very important.
Their use in constructing the solution is clearly shown
by the foregoing example. They may help us to find the 
weak spot in our conception of the problem. They may
point out a missing element.  When we know that a certain
element is still missing, we naturally try to bring it
into play.  Thus we have a clue, we have a definite line
of inquiry to follow, and have a good chance to meet
with the decisive idea.
2.    The questions we discussed are helpful not only in 
constructing an argument but also in checking it.  In
order to be more concrete, let us assume that we have to 
check the proof of a theorem whose hypothesis consists
of three parts, all three essential to the truth of the
theorem. That is, if we discard any part of the 
hypothesis, the theorem ceases to be true. Therefor, if the proof 
neglects to use any part of they hypothesis, the proof must
be wrong. Does the poof  use the whole hypothesis? Does
it use the first part of the hypothesis? Where does it use
the first part of the hypothesis? Where does it use the 
second part? Where the third? Answering to all these 
questions we check the proof.
    This sort of checking is effective, instructive and almost
most necessary though understanding if the argument 
is long and heavy as  THE INTELLIGENT READER 
should know.
3.  The questions we discussed aim at examining the 
completeness of our conception of the problem. Our
conception is certainly incomplete if we fail to take into
account any essential datum or condition or hypothesis
But it is also incomplete if we fail to realize the meaning
of some essential term. Therefor, in order to examine
our conception, we should also ask :  Have you taken into 
account all essential notions involved in the problem>? 
See DEFINITION,7.
4. The foregoing remarks, however, are subject to caution
and certain limitations. In fact, their straight for
ward application is restricted to problems which are
"perfectly stated?" and "reasonable."
    A perfectly stated and reasonable "problem to find"
must have all necessary data and not a single superfluous 
datum; also its condition must be just sufficient, neither
contradictory nor redundant. In solving such a problem, 
we have to use, of course, all the data and the whole
condition. 
     The object of a "problem to prove" is a mathematical
theorem. If the problem is perfectly stated and the reason
able, each clause in the hypothesis of the theorem must
be essential to the conclusion. In proving such a theorem
we have to use, of course, each clause of the hypothesis.
    Mathematical problems proposed in traditional text
books are supposed to be perfectly stated and reasonable. 
We should however not rely  too much on this; when
there is the slightest doubt, we should ask 
IS IT POSSIBLE TO SATISFY THE CONDITION? Trying to answer 
this question-or a similar one, we may convince 
ourselves, at least to a certain extent, that our
problem is as good as it is supposed to be.
     The question stated in the title of the present article
and allied questions may and should be asked without 
modification only when we know that the problem before
us is reasonable and perfectly stated or when, at least, 
we have no reason to suspect the contrary.
     5. There are some non-mathematical problems which 
may be, in a certain sense, "perfectly stated." For in
stance, good chess problems are supposed to have
but one solution and no superfluous piece on the chess board,
etc. PRACTICAL PROBLEMS 
however are usually far from being perfectly stated and 
require a thorough reconsideration  of the questions 
discussed in the he present article.