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. I have a Paul Halmos section in another corner of my web site.
Aims/Objectives. The Moscow University Mathematics Undergraduate Problem Seminar - conducted annually by the great Russian mathematician V. I. Arnold (I met him once, here in Dublin) - 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.
How I 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 (I have a special note about Panem in this corner of my web site) - or else somewhere near the college - 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 theorems...).
I lend my students personal copies of problem books from the reading list (below); I expect each one to read through the book, identify a problem that appeals to them, try to solve it, and if stuck consult the sketch solution. If still stuck then talk about it with the others, and finally with me. The student will eventually make a presentation of that problem to the others and myself.
Sometime we will (later: we did) watch the Paul Erdos (I don't know how to get the double acute accent over the 'o') N is a number video ('The story of a wandering mathematician obsessed with unsolved problems'), and the Andrew Wiles BBC Horizon video...
A 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').
Note concerning occurrences of Maple worksheets below. Almost everything at this page may be read as just text, but there are instances where links are to 'Maple worksheets'. Those sheets - in their original .mws formats may only be read by someone who is expert in such matters (those which I have converted to html format may be read by anyone), but even those should read the note at the top of my Maple page.
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...
Do a Google web search on 'Nim'. Play Nim online (with 3 piles and a maximum of 21 'matches').
Some 40 years ago when I first encountered 'Nim' in Hardy and Wright's great classic I rather looked down my nose at it, but now my attitude is that if it was good enough for Hardy and Wright to include in their Theory of Numbers then I shouldn't feel too badly about introducing it to my students.
3. Encounter 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 factorial' representation of whole numbers, and more besides... .
If you want to know a lot more about the mathematics of card tricks, you need look no further than the irish mathematician Colm Mulcahy.
I've prepared a related Maple worksheet: (n!+n-1).mws, and for readers without Maple I have made a html version.
Here was a great reaction I had in September (2002): I mentioned this trick to a second year BA student (Aoibheann Reilly), and I told her that she would study it in the third year... A few days later she said she had told it to her father who was out of his mind wanting to know how to solve it; I was horrible and said she'd have to wait until next year... Then, at the start of the second term I displayed Ian Stewart's New Scientist article about the trick, Aoibheann read and understood it, and finally took her father out of his agony. This is precisely the sort of thing I hoped would happen...; (Later Aoibheann went on to my beloved London college - Royal Holloway College - to study for the Masters in Information Security.)
4. The classic 6-people-at-a-party problem: in a group of 6 people, argue that there must be at least three of them, every two of whom know each other, or there must be at least three of them no two of whom know each other.
This elementary, but non-trivial problem allows one a first step into the wonderful world of Ramsey numbers. How quickly one then gets into really deep waters and unsolved problems... Look up Frank Ramsey at the monumental St Andrews University History of Mathematics site.
Frank Ramsey featured in the wonderful BBC Great Lives series in December 2020.
Here are some Ramsey numbers notes that I typed up in pdf format.
Stanislaw Radziszowski maintains an invaluable Small Ramsey Numbers site.
5. Choose any (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'.)
People 'in the know' will know that a very beautiful solution to this problem exploits the Dirichlet box principle (aka the pigeonhole principle: if m 'objects' are placed in n 'boxes', and m is greater than n, then some 'box' must contain at least two 'objects'); in fact I wish to use this problem to introduce my students to this incredibly powerful principle, and to demonstrate its power I will use it to prove (using an idea of Axel Thue's) Fermat's great classic that every prime number that leaves remainder 1 on division by 4 is expressible as a sum of two squares of integers:
I have prepared a quite detailed Maple worksheet on this topic:
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.
+ ... +
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)...
After August 31st 2007 please use the following Gmail address: jbcosgrave at gmail.com