- While studying mathematics in any form, always
do some problems at frequent intervals.
- When attempting a difficult problem, try the
methods listed on this page ((i)-(ix)).
Since one cannot be a mathematician without solving
problems, you should do some at every occasion, between
lectures, and at frequent intervals while reading. They
may be exercises from a book, or problems from a sheet
which you have been given. Where you get the problems
from is not so important, provided that they span a
range of difficulty, so that there are some which you
expect to solve, and some which will really stretch
In solving a mathematical problem, there are usually
- understanding the problem;
- experimentation (to find a method which looks
likely to work);
- verifying and writing out the solution.
These stages will be examined more closely in the
section on Applied
Part 2: Proofs
, but for the moment we will merely
make outline some general principles.
Suppose then that you are faced with a taxing
problem. Here are some guidelines as to how you might
proceed if you find the problem difficult. These
suggestions are listed in roughly the order in which
they should be tried, but not all of them will be
suitable for any given problem.
- While searching for a method, do not worry about
details; fill them in later, if you can.
- Try working backwards from the answer (if this is
given); this may show you the connection between the
assumptions and the conclusion; eventually, you will
have to turn the argument around so that it runs in
the right direction.
- Draw a diagram, if appropriate (many students are
very reluctant to do this). Although diagrams rarely
constitute a solution in themselves, they often show
you something which you would not otherwise have
- If the problem is a general one, try special
cases, or simpler versions of the problem.
- Make sure that you have read as much as possible
of the relevant theory. Re-read your lecture notes
and books; write down on a single sheet of paper all
the information which appears to be relevant.
- Look in your notes and books for similar worked
- If you are still stuck, put the problem aside and
come back to it later (e.g. the next day).
Surprisingly often, the solution will then stare you
in the face.
- If you are still stuck, ask your fellow-students
- If the worst comes to the worst, ask your
Let's be cheerful, and suppose that somehow or other
you have found a method which you think may work.
Inevitably, there will be some details which you have
skipped over, or which you are not completely confident
about, so there remains the task of writing out a
complete and accurate solution. For example, you may
have to reverse the direction of your argument, or you
may have to set out a formal proof by induction. This
stage also serves to verify your solution, in the sense
that it shows you whether the method actually works. It
may be that when you try to write the solution out, you
find there is some unexpected difficulty which you had
overlooked, in which case you will have to return to
experimentation. We shall describe some of the general
principles of writing mathematics in Writing mathematics
, and some
particular aspects will be discussed in detail in Part 2
Even if you think that you have solved a problem
correctly and arrived at the correct answer, pay
attention to any solution given in the back of the book
or in tutorials. If the other solution looks different
from yours, think about whether yours is equally valid;
if your solution now looks dubious, make sure that you
understand what is wrong.