Preparing for the Putnam Exam

Date: 11/04/2004 at 21:24:30
From: John
Subject: Hardest Math Test in the World

I was wondering if you could give me some specific advice on how to 
study for the Putnam Exam because I will be taking it for my first 
time.  I know the test is very hard, and I would like some helpful
hints from people who are more experienced than me at mathematics.

I am reading a book called Techniques of Problem Solving and I will 
probably finish it before the Putnam Exam is given.

Date: 11/05/2004 at 04:33:55
From: Doctor Pete
Subject: Re: Hardest Math Test in the World

Hi John,

The William Lowell Putnam Mathematics Competition may not be the most
difficult math exam, but it is certainly challenging.  There are
twelve questions, divided into two sections of six each.  Each section
has a time limit of three hours.  The questions are graded on a 
scale of 0 to 10 inclusive, where full credit is given for a complete,
correct, and well-written solution.  The level of mathematics covered
typically includes that which is studied by undergraduate mathematics
majors, and draws from algebra, geometry, number theory,
combinatorics, and calculus/real analysis.

The test is typically taken by about 2000 students each year.  Many
students score zero, and a score of 25-35 usually puts you in the
80-90th percentile, so the test is not very good at discriminating
between students of lesser mathematical experience, but extremely 
good at the other end--very, very rarely has a perfect score been
achieved in its 60+ year history.  Many past winners have gone on to
distinguished careers in mathematics.

Doing well on the Putnam, in my opinion, has less to do with your
mathematics background than it does with constant practice and
exposure to similar problems.  In other words, you need to get access
to as many prior problems and other similar competition problems as
possible, as well as their solutions.  Analyze the solution as well 
as the problem--this is how you learn what strategies work and what is 
likely to lead you in the right direction.  No doubt you will 
encounter areas of math that you are unfamiliar with; research them
and learn the basic concepts.  Most often this happens with algebra 
and combinatorics, and less so with calculus or geometry.

The single best resource available online is at

  http://www.kalva.demon.co.uk/index.html 

There are more problems on that site than you will ever have time to
solve as an undergraduate.  Good luck!

- Doctor Pete, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/05/2004 at 17:20:01
From: Doctor Vogler
Subject: Re: Hardest Math Test in the World

Hi John,

Thanks for writing to Dr. Math.  I competed in the Putnam exam three
times, and I thoroughly enjoy math puzzles, so I still like to work on
the problems even though I am no longer an undergraduate and therefore
no longer eligible to compete.

If you look at some old exams, I think you'll notice that in each of
the two sections (A and B), the six questions are approximately in
order of difficulty.  So the first question is generally within reach
of anyone who's taken calculus and maybe linear algebra, although most
such students will still find the question very challenging. 
(Remember that the median score on any question is zero.)  The second
question is harder, but still feasible.  I found question three
usually doable but very challenging.  Then question four was only
sometimes doable, and five and six rarely.  The first four or five
problems usually only require exposure to normal undergraduate
mathematics (like Doctor Pete listed off for you) and a lot of
creativity or cleverness.  The sixth (and sometimes the fifth)
question will usually require some advanced math that most
undergraduates haven't learned yet.  That might seem unfair, but it's
supposed to be the most challenging problem on (arguably) the most
challenging math test in the world.  And, even if not all students
learned that subject, they all had the chance.  (You have a university
library with a math section that you can use, right?)

So it is probably better to spend more time on earlier problems until
you get those than wasting time trying to work the sixth problem,
which you're not likely to get.  (Unless you're really *really*
smart.)  In other words, a half hour per problem is not very
realistic.  If you want to finish all six problems, then you had
better get the first one or two done very quickly so that you have
more time on the sixth.  A more realistic approach might be:  a half
hour for the first problem, an hour for the second, an hour and a half
for the third, and never mind the other three.  Or perhaps even: 
three hours for the first problem, and never mind the rest.

I would agree with Doctor Pete that you should practice by working
problems on old exams, but I would not say that the ability to do well
on this exam is not related to the ability to do math in general.  In
trying to work old problems, and in reading solutions to the ones you
can't get, you will learn mathematical techniques that are very useful
in many fields of mathematics, and which will come up in your courses
later.  You can learn some really neat things.  Better yet, you learn
these techniques with a use in mind, so that makes them seem very
worthwhile.  If you learned the same thing in a class, you might
think, "Why do we have to learn this?  When will I ever use this?"

By the way, you can find the questions for previous Putnam exams,
along with complete solutions for each question, in a fall issue of
the American Mathematical Monthly, around the October or November
issue of the next year.  (Does it take professional mathematicians
that long to get good solutions to all of the problems?)  You can
probably find that math journal in the periodicals section of your
university library.  Ask the librarian if you need assistance.

Be familiar with writing clear proofs, because this is the style of
writing that they will be looking for, even when it isn't a proof they
ask for.  Be familiar with mathematical rigor, such as using
definitions as they are given to you, and not making incorrect
assumptions.  You can use other theorems that you've learned in your
courses, but you usually won't need obscure theorems, just the ones
normally taught in undergraduate courses.

And one other thing:  They will often ask a question with a fairly
large number, like "What is the 30th term in this sequence?"  Almost
always it will require *way* too much time to generate 30 terms in the
sequence, and that's not what they want you to do.  You should
generate a few terms, find a pattern, write out a formula for the n'th
term, and then prove it by induction.  They only say "30'th" to
disguise the fact that it's easier to get the general n'th term than
it is to get 30 terms.  Then you just substitute n=30 into the 
formula.  (Notice that this type of question is an example of a
question that doesn't ask for a proof, but you should give one anyway:
a proof by induction of your formula.)

So that's the advice that comes to mind.  It looks like you don't have
too much time left to prepare.  But, unless you're about to graduate,
don't worry; you'll find out what it's all about this December, and
then you'll get another chance to try again next year.  And I hope you
have as much fun working on these problems as I did.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/