This is the home page for the Mathematical problem-solving module, to be given in the second semester of 2011–12 at Queen Mary, University of London.
Overview
This module is about solving tricky mathematical problems, rather than learning new material. The way the module works is that you'll be given a problem to work on. While there will be classes where you can work on the problems, the idea is that you do them in your own time. When you solve a problem, you submit a written solution to the module organiser, who will then give you another problem. The aim is to solve roughly one problem a week, so that at the end you have twelve problems.
After the end of term, you won't receive any more problems, and you won't be able to get any help, but you can keep working on the problems and submit solutions until the start of the exam term. After that, your solutions are marked, and there's also an oral exam where we can ask you about your solutions. (This is nothing to be afraid of – it's just to help to explain anything we don't understand in your solutions, and also to check that the solutions are really your own work.) After that, the module is over, and you can feel smug about having fewer exams than everyone else.
The problems
The twelve problems you get will be on a variety of topics and will have a range of difficulties. (They'll start easy, and gradually get harder.) Each problem has a 'difficulty rating', which is a number between 4 and 15. At the end of the module, each of your solutions is marked, and the mark is multiplied by the difficulty rating for that problem. The best ten solutions then give you your final mark.
The subject areas that the problems come from include algebra, geometry, combinatorics and number theory. However, none of the problems requires specialised knowledge, so there are no prerequisite modules. You shouldn't need to know anything beyond what you learnt in the first year. If there is anything you don't understand in a problem you've been given, then you can ask for clarification during the class. However, you won't generally get help with solving the problem – you're really on your own. However, you're free to use resources such as the library and the internet. If you want a hint from a member of staff, then you can ask (though any member of staff might say they're too busy). If a member of staff gives you a hint, they will tell me, and this will be taken into account when marking your solution.
Note that these problems are generally longer than coursework problems for other modules. In particular, almost all the problems require you to prove things, so you need to be skilled at writing proofs and explaining your reasoning. The solution for a problem will usually be at least a page long, and may be several pages.