Tricki
a repository of mathematical know-how

What is the Tricki?

The primary content of the Tricki

The main body of the Tricki will be a (large, if all goes according to plan) collection of articles about methods for solving mathematical problems. These will be everything from very general problem-solving tips such as, “If you can’t solve the problem, then try to invent an easier problem that sheds light on it,” to much more specific advice such as, “If you want to solve a linear differential equation, you can convert it into a polynomial equation by taking the Fourier transform.” Some articles will be written at a very elementary level and some will be quite advanced, though, obviously enough, the usefulness of an article will increase dramatically if it can be made widely accessible. Some will concern particular areas of mathematics, such as algebraic geometry or probability, whereas others will concern techniques that are relevant to many different areas.

Because clarity and accessibility are so important to the success of the Tricki, we have decided to have a recommended format for the articles. However, we also want to be as unprescriptive as possible, so this format is very loose. Originally we had thought of having articles divided into three parts: a brief summary of the technique in question, ideally in a highly memorable sentence or two; a general discussion of what the technique is and the kinds of situations in which it applies; and a collection of examples of the technique in action. However, once we started to write sample articles we came to realize that examples and general discussion were often best mixed. So instead we merely ask that the beginnings and ends of examples should be clearly labelled, and we recommend that there should not be too much general discussion without illustrative examples. In many cases, it may be best to start with examples and then move on to general discussion.

Here are a few examples of articles that are written in the recommended format.

Just-do-it proofs

Dimension arguments in combinatorics

Once written, an article will be open to editing by other people, just as it would be on Wikipedia. This has two important consequences. First, if you have an idea for a good article but not the time to write the perfect article that expresses your idea, then you should still write it: if it is unclear in places, or a bit short of good examples, or imperfect in some other way, then others will be able to improve it, especially if you explicitly say in what way you think it falls short. The second consequence is that if you are reading an article and see an obvious way of improving it, then you should: we would like the Tricki to converge article-wise to a collection of crystal-clear expositions.

For more detailed advice on writing and editing Tricki articles, go to one of the following pages

How do I create a Tricki article?

What is a Tricki article allowed to be about?

How do I edit or comment on an existing article?

The secondary content of the Tricki

An important issue for the Tricki is how to organize the articles in such a way that they can be easily found. Our aim is an ambitious one: to produce a website that people can go to with their mathematical problems and be able to find good explanations of all known techniques that are likely to be relevant. For some techniques this will be easy to achieve. For instance, if one wanted to solve a quadratic equation and didn't know how, one could do a search for "quadratic equation" and one would find techniques such as completing the square, applying the formula, searching for two numbers with given sum and product, and so on. But things get more difficult when the technique is more widely applicable. For instance, a very useful technique for solving inequalities is this: often, a good way of showing that A\leq B is to show that B-A can be written as a sum of squares. The most obvious way of discovering this one would be to search for "inequalities" and find a page with quite a long list of techniques, many of which would be irrelevant, but this would already be much less direct than searching for techniques that are designed for a specific problem and known to work. And for other techniques, even this method does not obviously work. Suppose for instance that somebody found a problem difficult to solve because they were unaware of Zorn's lemma. How would they find this article on how to use it? What is needed is a route to the article via a description of the mathematical difficulty dealt with by the technique, rather than via a phrase such as "Zorn's lemma".

For these more challenging navigational problems, considerable ingenuity may be needed on the part of Tricki contributors. If you are writing an article, you are strongly encouraged to think about how those who you hope will benefit from it could be expected to find it. Of course, they might just stumble on it by means of a lucky key-word search, or because the range of applicability of the technique is narrow enough that one can find it by looking through a fairly short list of techniques relevant to the area in question. But we also envisage a large collection of "secondary" articles that will be intended to help people find the primary ones. For instance, as suggested in the previous paragraph, there will probably be a main page devoted to techniques for solving inequalities. The idea of such a page would be to help the reader decide which of many existing techniques is likely to be the one that will help them prove the inequality they want to prove. Thus, it would mainly be devoted to a classification of inequalities, and would contain many links to techniques that work for inequalities of different kinds. As for Zorn's lemma, it would have to be referred to from a more general page entitled something like "Techniques for building up mathematical structures bit by bit".

We call these super-pages navigation pages. For a (short) list of existing navigation pages, see this page.

We hope that eventually the distinction between primary and secondary material on the Tricki will become somewhat blurred. After all, a page that tells you how to classify a certain kind of problem in order to find appropriate technique for solving the particular example you have in front of you is itself putting forward a kind of higher-level technique.

As well as navigation pages of this kind, we shall have different varieties of tags that will lead to a number of methods for searching for articles. For instance, articles will have subject-matter tags and key-word tags. We also hope to institute a system of tags that describe general problem types. These would constitute vague or partial descriptions of problems. An example might be "Prove–there exists–group–such that", which would take one to a page such as this with annotated links to pages describing techniques for constructing groups with prescribed properties.

For more details about tags, see this page.

It is highly unlikely that we have thought in advance of the best solutions to the navigational problems that we will face. There will be a forum for discussing both the issue in general and specific challenges that arise when articles are written that are problematic in this respect.

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