Can you use the result? To find the solution of a problem
by our own means is a discovery. If the problem is
not difficult, the discovery is not so momentous, but it is
a discovery nevertheless. Having made some discovery
however modest; we should not fail to inquire whether
there is something more behind it , we should not miss the
possibility opened up by the new result, we should try
to use it against the procedure used. Exploit your success!
Can you used the result, or the method, for some other
problem?
1. We can easily imagine new problems if we are some
what familiar with the principal means of varying a
problem as GENERALISATION, SPECIALISATION ANALOGY,
DECOMPOSING AND RECOMBINING. We start from a proposed
problem, we derived from it others by the means we just
mentioned, from the problems we obtained we derive
still others, and so on, The process is unlimited in theory
but in practice, we seldom carry it very far, because the
problems that we obtain so are apt to be inaccessible.
On the other hand we can construct new problems
which we can easily solve using the solution of a problem
previously solved; but these easy new problems are apt
to be uninteresting.
To find a new problem which is both interesting and
accessible, is not so easy; we need experience, taste, and
good luck, Yet we should not fail to look around for
more good problems when we have succeeded in solving
one. Good problems and mushrooms of certain kinds
have something in common; they grow in clusters. having
found one, you should look around; there is a good
chance that there are some more quite near.
2. We are going to illustrate some of the foregoing
points by the same example that we discussed in section
8, 10, 12, 14, 15. Thus we start from the following
problem:
given the three dimensions (length, breadth, and
height) of a rectangular parallelepiped, find the diagonal.
If we know the solution of this problem, we can easily
solve any of the following problems (of which the first two
were almost stated in section 14).
Given the three dimensions of a rectangular parallelepiped,
find the radius of the circumscribed sphere
The base of a pyramid is a rectangle of which the centre
is the foot of the altitude of the pyramid. Given the
altitude of the pyramid and the sides of its base, find lateral
edges.
Given the rectangular coordinates (x1,y1,z1), (x2,y2,z2)
of two points in space, find the distance of these
points.
We solve these problems easily because they are scarcely
different form the original problem whose solution we
know. In each case, we add some new motion to our original
problem, as circumscribed sphere, pyramid, rectangle
coordinates. These notions are easily added and
easily eliminated, and, having got rid of them, we fall
back upon our original problem.
The foregoing problems have a certain interest be
cause the notions that we introduced intro the original
problem are interesting. The last problem, that about the
distance of two points given by their coordinates, is even
a important problem because rectangular coordinates
are important.
3. Here is another problem which we can easily solve
if we know the solution of our original problem: Given
the length, the breadth, and the diagonal of a rectangular
parallelepiped, find the height.
In fact, the solution of our original problem consists
essentially in establishing a relation among four quantities,
the three dimensions of the parallelepiped and its
diagonal. If any three of these four quantities are given,
we can calculate the fourth from the relation.
Thus we can solve the new problem.
We have here a pattern to derive easily solvable new
problems from a problem we have solved; we regard the
original unknown and given and one of the original data
as unknown. The relation connecting the unknown and
the data is the same in both problems, the old and the
new. Having found this relation in one, we can use it
also in the other. this pattern of deriving new problems by
interchanging the roles is very different form the pattern
followed under 2.
4. Let us now derive some new problems by others means.
A natural generalization of our original problem is the
following: Find the diagonal of a parallelepiped being
given the three edges issued from an end-point of the
diagonal, and the three angles between these three edges.
By Specialization we obtain the following problem:
Find the diagonal of a cube with given edge.
We ay be led to an inexhaustible variety of problems
by analogy Here are a few derived from those considered
under 2. Find the diagonal of a regular octahedron with
given edge. Find the radius of the circumscribed sphere
of a regular tetrahedron with given edge.
Given the geographical coordinates, latitude and longitude, of two
points on the earth's surfaces, which we regard as a
sphere find their spherical distance.
These problems are interesting but only the one
obtained specialisation can be solved immediately on
the basis of the solution of the original problem.
5. We may derive new problems from a proposed one
by considering certain of its elements as variable.
A special case of a problem mentioned under 2 is
to find the radius of a sphere circumscribed about a cube
whose edge is given. Let us regard the cube, and he common-
centre of the cube and sphere as fixed, but let us vary
the radius of the sphere. If this radius is small, the sphere
is contained in the cube. As the radius increases, the
sphere expands(as a rubber balloon in the process of
being inflated). At a certain moment, the sphere touches
the faces of the cube, a little later, its edges; still later the
sphere pusses though the vertices. Which values does the
radius assume at these three critical moments.?
6. The mathematical experience of the student is incomplete
if he never had an opportunity to solve a problem invented
by himself. The teacher may show the derivation
of new problems from one just solved and, doing
so , provoke the curiosity of the students. The teacher
may also leave some part of the invention to the students;
For instance, he may tell about the expanding sphere we
just discussed(under 5) and ask: "What would you try
to calculate? Which value of the radius is particularly
interesting?"