"I do believe that problems are the heart of mathematics, and I hope that as teachers, in the classroom, in seminars, and in the books and articles we write, we will emphasize them more and more, and that we will train our students to be better problem-posers and problem-solvers than we are." Paul Halmos in his Problems for Mathematicians, Young and Old, Mathematical Association of America, 1993. Here are two Paul Halmos related photographs: his central place on our open departmental notice board, and birthday greetings to him in 2001 from my (then mathematics) colleague Olivia Bree (Olivia became the College's Assistant Registrar in 2003) and myself.
The Moscow University Mathematics Undergraduate Problem Seminar
- conducted annually by
the great Russian mathematician V.I.Arnold
- presents unsolved
problems to students (and staff) to think about.
Possible solutions are presented, discussed, accepted or rejected.
Solvers usually develop into outstanding leading mathematicians.
I have a much more modest aim in my course. I wish to draw the attention of my students to the rich resource of some well-known collections of solved problems (see recommended reading list below), and to study (by discussion, hints, etc) a selection of problems.
In considering the types of problems that I present to my students it is uppermost in my mind that a good number of those problems to be ones they can tell to their non-mathematical friends (as challenges or fun); problems that do not require knowing any mathematical theorems or theories. Some of the topics are unashamedly 'recreational mathematics' (and why not!)...
Learning Outcomes. At the end of this course students should have added a number of new problem-solving strategies to their repertoire, and have developed (I hope!) a life-long passion for problem solving.
teach my course. I sit with my students in my office; we chat, ask questions,
give hints, struggle towards solutions. I tell them stories about famous
mathematicians (for a break from time to time), I pull my hair out; we play
Nim (great fun!); we walk into Dublin one afternoon and have coffee and
cakes in Panem - or
somewhere nearby - and
talk mathematics en route (I want them to experience that doing mathematics is
not confined to sitting at a desk, and yes, I mention the classic
in-joke that a mathematician is a machine for turning coffee into
selection of problems that I have in mind, some of which are classics I
will add more details in time, but for the moment just a flavour. If I don't
attribute a problem it's simply that it's 'well-known' (it's been around for a
long time...), but otherwise I will attribute when I know an actual source
(which may not itself be 'original').
I will add more details in time, but for the moment just a flavour. If I don't attribute a problem it's simply that it's 'well-known' (it's been around for a long time...), but otherwise I will attribute when I know an actual source (which may not itself be 'original').
1. An 8 by 8 chess board may, of course, be covered with 32 non-overlapping dominoes; argue that if two opposite corners are removed then the resulting board cannot be covered with 31 dominoes. (That simple sounding problem leads to others...)
2. Play Nim (and subsequently understand the mathematical theory behind
it), a game for two people: Place any number of matches in any number of
piles; a player's 'move' entails removing whatever number of matches from
whichever pile he/she wishes (an entire pile may be removed). It is then the
other player's move, and the game continues until there are no matches left.
The 'winner' is the one who removes the last match (which may be
several in one move). So, get out your matches, and play...
the extraordinary card trick of William Fitch Cheney (this email
that I sent to All Users in my college will tell you what it is if you don't
already know) and study its wonderful ()
generalization... . In connection with the
trick's generalization I will need to introduce my students to the 'base
of whole numbers, and more besides... . Michael
(Brandeis university) Mathematical Intelligencer article - The Best Card
Trick - is available here,
and a Math Forum article about it here.
classic 6-people-at-a-party problem: in a group of 6 people, argue that there must be
three (at least) of them, every two of whom know each other, or there must be
three of them (at least) no two of whom know each other.
(n+1) different numbers from the 2n numbers (1, 2, 3, ...
, 2n); argue that that at least one of those must divide ('evenly')
into another (different one, of course) of those. (One should keep in mind
that this result is 'best possible' in the sense that the same conclusion does
not necessarily hold if 'n+1' is replaced with 'n'.)
I have prepared a quite detailed Maple worksheet on this topic: Fermat-Thue (mws format (62KB), or in zip format (15KB) - in those I have removed the output which you may recreate by executing Maple commands - and for readers without Maple I have made a html format version (76KB, plus an 'images' folder)
6. Let there be any number of boys and girls at a party, and suppose no boy dances with every girl, and that every girl dances with some boy; argue that there must be a boy-girl pair (b, g) who dance, and another such pair (B, G), but that 'b' does not dance with 'G', nor does 'g' dance with 'B'. (This was problem A-4 in the 1965 William Lowell Putnam Mathematical Competition). Here is a solution that I typed up, in two formats: Word zip (18KB), rtf zip (16KB).
7. Some (positive) whole numbers can be expressed as a sum of consecutive (positive) whole numbers; e.g. 7 = 3+4, 14=2+3+4+5, ... whereas 8 cannot be so expressed (check to see). Find (with proof, of course!) those which can, and those which can't. This is an old problem, and was the subject of a paper of mine: A Halmos Problem and a Related Problem, American Mathematical Monthly, 993-996, Vol. 101, No 10, December 1994. It may be obtained by anyone with access to JSTOR, or whose university subscribes to it. If you can't access it then send me an email and I'll send you a hard copy.
8. A well known problem (dear to Paul Erdös): for any n non-collinear points in the plane, prove that some two of them lie on a line that passes through no other one of them (answering a question first posed by Sylvester). Here is a solution that I typed up, in two formats: Word zip (19KB), rtf zip (16KB).
9. A problem (with a related proof-idea): Let A be a set of 2n points in the plane, no three of which are collinear. Suppose n of them are coloured red and the remaining n blue. Prove or disprove: there are n closed line segments, no two with a point in common, such that the endpoints of each segment are points of A having different colours (This was problem A-4 in the 1979 William Lowell Putnam Mathematical Competition). Here is a solution that I typed up, in two formats: Word zip (19KB), rtf zip (16KB).
10. Prove that if n is any whole number greater than 1, then n does not divide (). This was problem A-5 in the 1972 William Lowell Putnam Mathematical Competition, and clearly it provides lots of fodder as far as understanding is concerned: why does this work with '2' (so what happens if '2' is replaced with 3, 4, 5, ... ?), and what happens if (e.g.) one replaces () with ()?
fractions (The outstanding US mathematician Ron Graham did his PhD - with D.H.
Lehmer (no less!) - on egyptian fractions; so this is a non-trivial
topic, with many unsolved problems, e.g.: the Erdös-Strauss
conjecture, or the odd greedy algorithm, or ...).
+ ... +
for some distinct whole numbers , ... , . Thus, for example:
I never realised until I began this topic with my students just how fascinating it would turn out to be... Do a Google web search on egyptian fractions, and delight in the exceptional egyptian fractions site (which has many fine related links) of David Eppstein. I have prepared a substantial (67KB, output removed) Maple worksheet on Egyptian fractions , and also in html format.
12. A problem of Ducci (I first heard of this from Richard Dubsky when I visited the wonderful Hockaday School of Dallas in March 2002)...
problems based on a 1989 paper of mine, A Remark on Euclid's Proof of the
Infinitude of Primes, American Mathematical Monthly, 339-341, Vol. 96, No.
4, April 1989. (It may be obtained by anyone with access to
JSTOR, or whose university subscribes to
it. If you can't access it then send me an email and I'll send you a hard
copy.) A typical problem: choose any number of primes whatever, from the first
one (2), up to and including any other one p (say) (it must be chosen
to be at least the fourth prime). Finally choose any one of
those primes - q (say) - and form the product of all the original
primes (with q excluded) plus 1. Now prove that number cannot
be a power of the excluded prime.
that the number can
be expressed as the product of no less than n prime factors (not
I will add more when time allows...
Hungarian Problem Books 1 and 2 (Eötvös Competitions), 1984-1905 and
1906-1928, Mathematical Association of
(Compiler and Solver), International Mathematical Olympiads, 1959-77,
Mathematical Association of America, 1978.
(Compiler and Solver),
International Mathematical Olympiads (and 40 supplementary Problems),
1978-85, Mathematical Association of
Yaglom A.M. and Yaglom I.M., Challenging Mathematical Problems, Holden-Day. 1967.
Problem Solving Through Problems, Springer-Verlag,
Mathematical Discovery, Vols. I and II, John Wiley, 1962.
Mathematics and Plausible Reasoning, Vols. I and II, Princeton University Press,
Ingenuity in Mathematics, Mathematical Association of America, 1970.
Honsberger R., Mathematical Gems, Vols. I, II and III, Mathematical Association of America, 1973, 1976 and 1985.
Honsberger R., Mathematical Morsels, Mathematical Association of America, 1978.
Honsberger R., Mathematical Plums, Mathematical Association of America, 1979.
(You can tell that I admire Honsberger!!)
G.L., Klosinski L. and Larson L. (Compilers),
The William Lowell Putnam Mathematical Competition,
Problems and Solutions for 1965-84, Mathematical Association of America,
(Compiler), U.S.A. Mathematical Olympiads, 1972-86, Mathematical Association of America,
Klee V. and Wagon S., Old and New Unsolved Problems, Mathematical Association of America, 1991.
Conway J. and
Guy R., The Book of Numbers, Copernicus, 1996.
Halmos P., Problems for Mathematicians, Young and Old, Mathematical Association of America, 1993.
Aigner M. and Ziegler G., Proofs from the Book, Springer-Verlag (1998, 2000)
The American Mathematical Association of Two-Year Colleges maintains a Problem Solving site.
There is a lovely Pigeonhole Principle site (with some nice problems) maintained here (it's a subset of the AB site above).
Another wonderful problems site: Purdue Problem of the Week
After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com
This page was last updated 08 March 2007 10:43:42 -0000