MTH6109 |
Combinatorics |
Course Material | Autumn 2011 |
Important information |
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Notes for lectures 1-30 (i.e. the whole course) are now available from the link below. Exercises for week 12 are available now. All uncollected coursework has been placed in the School Office, from where you may collect it. Serial non-collectors may find some of their work has not been marked: if you want this work marked, or more detailed feedback on other work, you are welcome to bring it to one of my office hours. |
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Administrative information
Your working week
Each module is designed to fill one-quarter of a full-time working week. That is, you are expected to work roughly ten hours a week on each module. Only four hours are timetabled. The rest is your responsibility. It should include, at the very least:Your career
Employers value mathematics graduates for one quality above all others: they can think. In a recession, this is more important than ever. A 2(i) Honours degree on its own is not a passport into a good job. At interview you will be expected to show you can think for yourself as well. A module like this one (Combinatorics) is designed in particular to train you to think, at a high level. Make sure you use this opportunity well.Why mathematics?
Mathematics is the most empowering of all disciplines. In many disciplines, one has to appeal to higher authority to decide what is right and what is wrong. But in mathematics, you are given the tools to decide for yourself, using logic alone. As Richard Hamming said"In science and mathematics we do not appeal to authority, but rather you are responsible for what you believe."
Lectures
What is the purpose of lectures, if the notes are available online? Experience has shown that it is actually quite difficult to learn mathematics by reading notes or books, even if you are conscientious about doing the exercises (which most people are not). If you are studying music, it is much better to hear it in real time, rather than read the score. A mathematical proof is like a piece of music: there is really no substitute for seeing it develop in real time. A good lecturer tells you not only the facts, but also why they have to be that way, and provides explanations tailored to the class that you could never find in a book.Notes
Lecture notes in some form will be put here in due course. These are however not a substitute for taking your own notes during lectures. You may wish to look at Prof. Cameron's Notes for the previous version of the course Notes on combinatorics.Notes: please let me know if you find any errors, obscurities or inconsistencies. The current version should be more or less complete up to lecture 26 (6th December).
Reading list
The recommended course text is
P. J. Cameron, Combinatorics: topics, techniques, algorithms. Cambridge University Press (1994).
Other books you may find useful:
Prerequisites
Combinatorics is often thought of as an `easy option', as it has little in the way of formal pre-requisites. However, this is a mistake. The lack of formal pre-requisites means that you have to rely instead on native intelligence, imagination, pure thought and logic. This makes it potentially a hard subject. But the problem-solving aspect of it can be very satisfying, especially if you're good at it.
Results versus methods
Learning results ("know-what", also called theorems) is inefficient compared to learning methods ("know-how", also called proofs), because a result is a fixed thing with limited applicability, whereas a method is more general and may be used to work out many different results. And if you know the method, then you can work out the result; but if you only know the result, how can you work out the method?
Exercises and tutorials
These should be regarded as compulsory. Mathematics is not about learning facts ("know-what"), it is about learning methods ("know-how"). Methods and techniques cannot be learnt without practice. As Confucius (551-479BCE) said:"I hear and I forget;
I see and I remember;
I do and I understand."
Exercises are in preparation. If you want to get ahead of the game you can look at last year's exercises in html. Also available as pdf.
I will discuss your written work with you individually in tutorials if you turn up.Joining a gym
If you paid hundreds of pounds a year for membership of a gym, would you skip the exercise classes organised for you? Would you expect to get fit if you stayed at home instead of going to the gym? Would you expect to get fit if you watched your personal trainer doing the exercises instead of doing them yourself?
No? So why do so many of you treat membership of a university in this way? There is no short cut to training your mind, just as there is no short cut to training your body. Exercise until it hurts: that is the only way. In the exercise classes organised for your benefit, your personal trainer will show you how to use the equipment, and get you started: then you need to put in the hard work yourself.
Continuous assessment and feedback
Working together
Working together can be a very useful way of learning from each other. However, it can also be an excuse for laziness, or, worse, plagiarism. Most perniciously, it can lull you into a false sense of security, thinking that you can do something when in fact somebody else is always doing it for you. Make sure that you are working together in a positive way. At the end of the day, when it comes to the exam, you have to be able to think for yourself.Solutions to selected exercises
See above for solutions of various kinds to various problems and exercises. Some will be detailed solutions which you can use as a model to emulate. Others will be sketches designed to show you the main ideas, which you then have to work up yourselves into detailed solutions. As always, if you have difficulty understand the solutions given, please ask for help in the tutorials. This is (among other things) what tutorials are there for.Solutions to certain other exercises can be found by following some of the links in this page. I won't put short-cuts here, so that by searching for yourself you gain the maximum benefit from what is on this web-page.
Model solutions
I will not provide model solutions to past examination papers, because they encourage poor quality learning. If you wish to use past exam papers as an aid to your revision, you may bring your solutions to one of my office hours, and I will give you feedback on them as far as time allows.Revision
Revision is an essential part of the learning process. Here are a few suggestions which you may find useful. First, read your notes in suitably small chunks. Make sure you understand the main points: if not, consult books or the www as necessary, and try some more exercises. Second, write new notes summarising the old ones. Third, summarise the summary. When you think you understand, try explaining it to someone else (in writing, if you can't find a willing victim to listen to you). Only when you can do this can you really say you have understood.Other course material
Web ResourcesRobert A. Wilson
Created 20 September 2011
Updated 15 December 2011