[G.Polya]
Decomposing and recombining are important operations
of the mind.
You examine an object that touches your interest or
challenges your curiosity: a house you intend to rent, an
important but cryptic telegram, any object who's purpose
pose and origin puzzle you, or any problem you intend
to solve. You have an impression of the object as a hole
but this impression, possibly, is not definite enough. A
detail strikes you, and you focus your attention upon it
Then, you concentrate upon another detail; then again,
upon another. Various combinations of details may present
themselves and after a while you again consider the
object as a whole but you see it now differently. You decompose
the whole into its parts, and you recombine the
parts into a more or less different whole.
1. If you go into detail you may lose yourself in details.
Too many or too minute particulars are a burden
on the mind. They may prevent you from giving sufficient
attention to the main point, or even from seeing the main
point at all. Think of the man who cannot see the forest
for the trees.
Of course we do not wish to waste our time with the
unnecessary detail and we should reserve our effort for
the essential. The difficulty is that we cannot say beforehand
which details will turn out ultimately as necessary and
which will not.
Therefor, let us, first of all, understand the problem
as a whole. Having understood the problem, we shall be
in a better position to judge which particular points may
be the most essential,. Having examined one or two
essential points we shall be in a better position to judge
which further details might deserve closer examination
Let us go into detail and decompose the problem gradually,
but not further than we need to.
Of course, the teacher cannot expect hat all students
should act wisely in this respect. On the contrary, it is a
very foolish and bad habit with some students to start
working at details before having understood the problem
as a whole.
2. We are going to consider mathematical problems,
"problems to find."
Having understood the problem as a whole, its aim, its
main point, we wish to go into details. Where should we
start? In almost all cases, it is reasonable to begin with
the consideration of the principal parts of the problem
which are the unknown, the data, and the condition. In
almost all cases it is advisable to start the detailed
examination of the problem with the questions: What is
the unknown? What are the data? What is the condition?
If we wish to examine further details, what should we
do? Fairly often, it is advisable to examine each datum
by itself, to separate the various parts of the condition,
and to examine in each party itself.
We may find it necessary, especially if our problem is
more difficult, to decompose the problem still further,
and to examine still more remote details. Thus it may
be necessary to go back to the definition of a certain term
to introduce new elements involved by the definition
and to examine the elements so introduced.
3. After having decomposed the problem, we try to
recombine its elements in some new manner. Especially
we may try to recombine the elements of the problem
into some new, more accessible problem which we could
possibly use as an auxiliary problem.
There are of course, unlimited possibilities of recombination.
Difficult problems demand hidden, exceptional, original
combinations, and the ingenuity of the problem-solver
shows itself in the originality of the combination.
There are, however, certain usual and relatively simple
sorts of combinations, sufficient for simpler problems,
which we should know thoroughly first, even if
we may be obliged eventually to resort to less obvious
means.
There is a formal classification in which the most usual
and useful combinations are neatly placed. In constructing
a new problem from the proposed problem, we may
1) Keep the unknown and change the rest (the datas and the condition) ; or
2) Keep the data and change the rest (the unknown and the condition); or
3) change both the unknown and the data.
We are going to examine these cases.
The cases (1) and (2) overlap. In fact, it is possible
to keep both the unknown and the data, and transform
the problem by changing the form of the condition alone.
For instance, the following problems, although, visibly
equivalent, are not exactly the same:
Construct an equilateral triangle, being given a side.
Construct an equiangular triangle, being given a side.
The difference of the two statements which is slight in
the present example may be momentous in other cases.
such cases are even important in certain respects but it
would take up too much space to discuss them here.
Compare AUXILIARY PROBLEMS, 7, last remark.)
4. Keeping the unknown and changing the data and
the condition in order to transform the proposed problem
is often useful. The suggestion LOOK AT THE UNKNOWN
aims at problems with the same unknown. We
may try to recollect a formerly solved problem of this
kind: And try to think of a familiar problem having the
same or a similar unknown. Failing to remember such a
problem we may try to invent one: Could you think of
other data appropriate to determine the unknown?
A new problem which is more closely related to the
proposed problem has a better chance of being useful.
Therefor, keeping the unknown, we try to keep also
some data and some part of the condition, and to change,
as little as feasible, only one or two data and a small part
of the condition. A good method is one which we omit
something without adding anything; we keep the unknown,
keep only a part of the condition, drop the other part,
but do not introduce a new clause or datum.
Examples and comments on this case follow under 7, 8.
5. Keeping the data. We may try to introduce some useful
and more accessible new unknown. Such an unknown
must be obtained from the original data and we have
an unknown in mind when we ask: COULD YOU DERIVE SOMETHING
USEFUL FROM THE DATA?.
Let us observe that two things here are desirable. First, the
new unknown should be more accessible, that is
more easily obtainable from the data than the original
unknown. Second, the new unknown should be useful,
that is, it should be, when found, capable of rendering
some definite service in the search of the original unknown.
In short, the new unknown should be a sort of
stepping stone. stone in the middle of the creek is
nearer to me than the other bank which I wish to arrive at
and, when the stone is reached, it helps me on toward
the other bank.
The new unknown should be both accessible and useful
but, in practice, we must often content ourselves with
less. If nothing better presents itself, it is not unreasonable
to derive something from the data that has some
chance of being useful; and it is also reasonable to try a
new unknown which is closely connected with the original one,
even if it does not seem particularly accessible
from the outset.
For instance, if our problem is to find the diagonal of
a parallelepiped (as in section 8) we may introduce the
diagonal of a face as the new unknown. We may do so either
because we know that if we have the diagonal of the face
we can also obtain the diagonal of the solid. (as in section 10);
or we may do so because we see that the
diagonal of the face is easy to obtain and we suspect that
it might be useful in finding the diagonal of the solid
(Compare DID YOU USE ALL THE DATA?)
If our problem is to construct a circle, we have to find
two things, its centre and its radius; our problem has
two parts, we may say. In certain cases, one part is more
accessible than the other and therefor, in any case, we
may reasonably give a moments consideration to this
possibility; Could you solve a part of the problem?
Asking this, we weigh the chances; would it pay to concentrate
just upon the centre, or just upon the radius, and
to chose one or the other as our new unknown? Questions
of this sort are very often useful. In more complex
or in more advanced problems. The decisive idea often
consists in carving out some more accessible but essential
part of the problem.
6. Changing both the unknown and the data we deviate more
from our original course than in the foregoing
cases. This, naturally, we do not like; we sense the danger
of losing the original problem altogether. Yet we may
be compelled to such an extensive change if less radius
changes have failed to produce something accessible and
useful, and we may be tempted to proceed so far from our
original problem if the new problem has a good chance
of success. 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?
An interesting way of changing both he unknown and
the data is interchanging the unknown with one of the
data. (See CAN YOU USE THE RESULT
7. Example Construct a triangle, being given a side a,
the altitude h perpendicular to a, the angle a opposite
to a.
What is the unknown? A triangle.
What are the data? Two lines a and h, and an angle a
Now if we are somewhat familiar with problems of
geometric construction, we try too reduce such a problem
to the construction of a point. We draw the line BC equal to
the given side a ; then all that we have to find is the
vertex of triangle A, opposite to a, See Fig 16. We
have in fact, a new problem
A
.
. | .
. *-|-* .
. | .
. h | .
. | .
._________________L___________.
B a C
What is the unknown? The point A.
What are the data? A line h, an angle a, and two points
B and C given in position.
What is the condition? The perpendicular distance of
the point A from the line BC should be h and /_BAC is a.
In fact, we have transformed our problem, changing
both the unknown and the data. The new unknown is a
point, the old unknown was a triangle. Some of the data
are the same in both problems, the line h, and the angle
a, but in the old problem we were given a length a and
now we are given two points, B and C, instead
The new problem is not difficult. The following suggestion
brings us a quite near to the solution.
Separate the various parts of the condition. The
condition has two parts, one concerned with the datum h,
the other with the datum a. The unknown point is
required to be.
(I) at distance h from the line BC; and
(II) the vertex of an angle of given magnitude a
whose sides pass through the given points B and C.
If we keep only one part of the condition and drop the
other part. The unknown point is not completely determined.
There are many points satisfying part (I) of the
condition, namely all points of a parallel to the line BC
at the distance h from BC2(footnote: The plane is bisected
by the line through B and C. We chose
one of the half-planes to construct A in it, and so we may consider
just one parallel to BC; otherwise we should consider two
such parallels.). This parallel is the locus of
the points satisfying part (II) is a certain circular are
whose end-points are B and C. We can describe both loci;
their intersection is the point that we desired to
construct.
The procedure that we have just applied has a certain
interest; solving problems of geometric construction, we
can often follow successfully its parter; Reduce the
problem to the construction of a point, and construct the
point as an intersection of two loci.
But a certain step of this procedure has a still more
general interest; solving "problems to find" of any kind
we can now follow its pattern; Keep only a part of the condition,
droop the other part. Doing so, we weaken the
condition of the proposed problem, we restrict less the
unknown. How far is the unknown then determined,
how can it vary? By asking this we set, in fact, a new
problem. If the unknown is a point in the plane (as it
was in our example) the solution of this new problem
consists in determining a locus described by the point.
If the unknown is a mathematical object of some other
kind (it was a square in section 18) we have to describe
properly and to characterise precisely a certain set of
objects. Even if the unknown is not a mathematical object
(as in the next example under 8) it may be useful
to consider, to characterise, to describe, or to list those
objects which satisfy a certain part of the condition imposed
upon the unknown by the proposed problem.
8. Example In a crossword puzzle that allows puns
and anagrams we find the following clue;
"Forward and backward part of a machine(5 letters)."
What is the unknown? A word.
What is the condition? The word has 5 letters. It has
something to do with some part of some machine. It
should be, of course, an English word, and not a too
unusual one. Let us hope.
Is the condition sufficient to determine the unknown?
No. Or, rather, the condition may be sufficient but that
part of the condition which is clear by now is certainly
insufficient. There are too many words satisfying it, as
"lever," or "screw," or what not.
The condition is ambiguously expressed-on purpose
of course. If nothing can be found that could be plausibly
described as a forward part" of a machine and would be
a "backward part" too, we may suspect that forward
and backward" reading might be meant. It may be a good idea
to examine this interpretation of the clue.
Separate the various parts of the condition?. The condition
has tow parts, one concerned with the meaning of
the word, the other with its spelling. The unknown word
is required to be.
(I) a short word meaning some part of some machine
(II) a word with 5 letters which spelled backward
give again a word meaning some part of some machine.
If we keep only one part of the condition and drop the
other part the unknown is not completely determined.
There are many words satisfying part (i) of the condition
we have a sort of locus. We may "describe this" locus
(I), follow it to its "intersection with locus"
(II). The natural procedure is to concentrate upon part
(I) of the condition, to recollect words having the prescribed
meaning and, when we have succeeded in recollecting
some such word, to examine whether it has or has
not the prescribed length and can or cannot be read
backward. We may have to recollect several words before
we run into the right one. Lever,screw,wheel,shaft,hinge motor.
Of course, "rotor"!
9. Under 3, we classified the possibilities of obtaining
a new "problem to find" by recombining certain elements
of a proposed "problem to find" If we do not
introduce just one new problem, but two or more new
problems, there are more possibilities which we have to
mention but do don't attempt to classify.
Still other possibilities may arise. Especially, the solution
of a "problem to find" may depend on the solution
of a "problem to prove." We just mention this important
possibility; considerations of space prevent us from
discussing it.
10. Only a few and short remarks can be added concerning
"problems to prove"; they are analogous to the foregoing
more extensive comments on "problems to find" (2 to 9).
Having understood such a problem as a whole, we should,
in general, examine its principal parts. The
principal parts are the hypothesis and the conclusion
of the theorem that we are required to prove or to
disprove. We should understand these parts thoroughly;
what is the hypothesis? what is the conclusion? If there
is need to get down to more particular points
we may separate the various parts of the hypothesis
and consider each part by itself. Then we may proceed
to other details, decomposing the problem further and further.
After having decomposed the problem, we may try to
recombine its elements in some new manner. Especially,
we may try to recombine the elements into another
theorem. In this respect, there are three possibilities.
(1) we keep the conclusion and change the hypothesis.
We first try to recollect such a theorem: Look at the
conclusion! And try to think of a familiar thereom having
the same or a similar conclusion. If we do not
succeed in recollecting such a theorem we try to invent
one: Could you think of another hypothesis from which
you could easily derive the conclusion? we may change
the hypothesis by omitting something without adding
anything: Keep only a part of the hypothesis, drop the
other part; is the conclusion still valid?
(2) We keep the hypothesis and change the
conclusion: Could you derive something useful
from the hypothesis?
(3) We change both the hypothesis and the conclusion.
we may be more inclined to change both if we
have had no success in changing just one. Could 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?
We do not attempt to classify here the various
possibilities which arise when, in order to solve the
proposed "problem to prove" we introduce two or more new
"problems to prove" or when we link it up with appropriate
"problems to find."