This is an attempt to give a quick guide to the top few levels of the Tricki. It may cease to be feasible when the Tricki gets bigger, but we might perhaps be able to automate additions to it. Clicking on arrows just to the right of the name of an article reveals its subarticles. If you want to hide the subarticles again, then you should click to the right of them rather than clicking on the name of one of the subarticles themselves, since otherwise you will follow a link to that subarticle.

What kind of problem am I trying to solve?

- Techniques for finding algorithms and algorithmic proofs
- How to use Euclid's algorithm
- Approximation algorithms
- Greedy algorithms
- Algorithms associated with polynomials
- Randomized algorithms front page
- Elementary randomized algorithms Brief summary ( This article is about algorithms that exploit the fact that repeated trials of the same experiment almost always give rise to the same approximate behaviour in the long term. A famous example of such an algorithm is the randomized algorithm of Miller and Rabin for testing whether a positive integer is prime. )
- Random sampling using Markov chains Brief summary ( Suppose that you want to generate, uniformly at random, some combinatorial structure or substructure. If the structure is simple enough, then there may be an easy way of converting random bits into the structure you want, with the correct distribution. For example, to choose a random graph one can just pick each edge independently at random with probability 1/2. However, a trivial direct approach like this is often not possible: how, for example, would you choose a (labelled) tree uniformly at random? A commonly used technique in more difficult situations is to take a random walk through the "space" of all objects of interest and to prove that the walk is
*rapidly mixing*, which means that after not too long the distribution of where the walk has reached is approximately uniform. ) - Property testing

- Derandomization

- Techniques for comparing sets and mathematical structures
- Techniques for classifying mathematical structures
- Techniques for counting
- Techniques for solving equations
- How to solve linear equations in one variable Brief summary ( These are equations like . This equation is called linear because the graph of is a straight line. )
- How to solve linear equations in two variables Brief summary ( Things get slightly more complicated if you have
*two*unknowns, and two equations to help you to determine them. For example, the equations could be and . However, there are various systematic ways of solving such pairs of equations and determining and . ) - How to solve linear equations in many variables Brief summary ( The methods used to solve linear equations in two variables can be generalized to any number of variables. At this point it becomes fruitful to think of the unknown as a vector rather than as a sequence of many individual variables. )
- How to solve quadratic equations Brief summary ( A quadratic equation is something like , which involves as well as . There are several techniques for solving them: which is best varies from example to example. )
- How to solve cubic and quartic equations Brief summary ( These are like quadratic equations, but now they also involve cubes and fourth powers, respectively. For instance, the equation is a quartic equation. Cubic and quartic equations can be solved systematically, but they are much harder than quadratic equations. )
- What makes some equations so much easier to solve than others? Brief summary ( There are several answers one might give to this question but here is one: if you can work something out on your calculator without using any memory, then the resulting equation will be easy to solve. A few examples will make clear what this means. )
- How to solve polynomial equations in several variables
- How to solve linear Diophantine equations
- How to solve quadratic Diophantine equations
- Diophantine equations front page
- Differential equations front page
- How to solve functional equations

- Techniques for obtaining estimates
- Techniques for proving existence
*Organized by methods of proof:*- Technique 1: off-the-shelf examples
- Examples of functions defined on the complex numbers
- Examples of functions defined on the real numbers
- Some useful examples of graphs
- Basic examples of groups
- Some examples of manifolds
- Examples and counterexamples in metric spaces
- Some important classes of polynomials
- Some interesting sets of integers
- Some interesting sets of real numbers
- Examples of rings
- Some important solutions of differential equations
- Examples and counterexamples in topological spaces

- Technique 2: building a bespoke example
- How to use Zorn's lemma Brief summary (If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn's lemma may well be able to help you.)
- Just-do-it proofs Brief summary (If you are asked to prove that a sequence or a set exists with certain properties, then the best way of doing so may well be just to go ahead and do it: that is, you build the set/sequence up one element at a time, and however you do it you find that it is never difficult to continue building.)

- Technique 3: building complicated examples out of simple ones
- Technique 4: making one mathematical structure out of another
- Turning groups into algebras
- Turning topological spaces into algebras
- Using ordinals to build Banach spaces
- Algebraic constructions of graphs
- Geometrical constructions of graphs
- Building manifolds using polynomials
- Algebraic constructions of sets of integers with given properties
- Building topological spaces out of groups

- Technique 5: universal examples with given properties
- Technique 6: indirect proofs of existence
- Technique 7: the probabilistic method
- Technique 8: cardinality, measure and category
*Organized by subject matter:*- Proving the existence of groups with certain properties
- Proving the existence of subsets with certain properties
- Just-do-it proofs
- How to use Zorn's lemma Brief summary (If you are building a mathematical object in stages and find that (i) you have not finished even after infinitely many stages, and (ii) there seems to be nothing to stop you continuing to build, then Zorn's lemma may well be able to help you.)
- Transfinite induction
- How to use the continuum hypothesis
- Finding small nets
- Probabilistic combinatorics front page

- Proving the existence of sets of integers with certain properties
- Proving the existence of finite subsets of groups with certain properties
- Proving the existence of graphs with certain properties
- Proving the existence of functions with certain properties

- Techniques for producing explicit examples
- Techniques for proving identities
- Technique 1: proving that two polynomials are the same by looking at the roots of their difference
- Technique 2: double counting
- Technique 3: using the law of trichotomy
- Technique 4: Axiomatics
- Technique 5: To prove that two objects are equal, show that in enough circumstances they behave in the same way

- Techniques for proving impossibility and nonexistence
- Techniques for proving inequalities
- Techniques for maximizing and minimizing
- Techniques for proving "for all" statements
*Methods:*- Method 1: Convert "every " into a single arbitrary
- Method 2: Induction
- Method 3: Classification
- Method 4: Prove the result for some cases and deduce it for the rest
*Problem type:*- Deducing one property from another

- Don't start from scratch
- Hunt for analogies
- Mathematicians need to be metamathematicians
- What can a lower bound say about potential proofs of the upper bound?
- Which techniques lead to which kinds of bounds?
- When does reformulating a problem count as progress?
- If you are getting stuck, then try to prove rigorously that your approach cannot work
- Sharp results need sharp lemmas
- Two proofs of the same theorem usually have deep similarities

- Think about the converse
- Try to prove the opposite
- Look for related problems
- Work on clusters of problems
- Look at small cases
- Try to prove a stronger result
- Think axiomatically even about concrete objects

Front pages for different areas of mathematics

- Algebra
- Group theory
- Basic examples of groups Quick description (This page contains descriptions of a number of groups that can be used as tests for the truth or otherwise of general group-theoretic statements.)
- How to build groups
- Presentations of groups
- How to use group actions
- Proving results by letting a group act on a finite set Quick description (There are a number of group theoretic statements that do not mention group actions, but which can be solved by defining a set on which a given group acts and studying that action. Typically, the set is defined in terms of the group itself. This article gives several examples, and tips for choosing an appropriate action.)
- Representation theory
- Actions on topological spaces Quick description (If you want to study a group , try to realize as the fundamental group of a topological space. This works best when is infinite and discrete, especially if is finitely presented and torsion-free.)

- Group theory
- Algebraic geometry
- Counting constants
- Dimension counting via incidence varieties Quick description (There are many situations in algebraic geometry in which one wishes to compute dimensions of various spaces, such as the space of all lines contained in a given surface in projective space. One approach to doing this is via so-called incidence varieties. An incidence variety is essentially the graph of a relation (for example, the collection of all ordered pairs of lines and surfaces of a given degree , where the line lies on the surface), but thought of itself as a variety. Since the incidence variety is a set of ordered pairs, it admits two natural projections, onto the first or second member of each ordered pair. Playing off the information that one obtains from considering these two projections, it is often possible to make non-trivial deductions.)
- How to compute the dimensions of the fibres of a map of varieties
- Use finite fieldsQuick description (If you have an algebraic problem about the complex numbers, it might be possible to solve it by solving a similar, but easier, problem about finite fields.)

- Analysis
*Areas of analysis*- Real analysis
- How to use the Bolzano-Weierstrass theorem Quick description ( The Bolzano-Weierstrass theorem asserts that every bounded sequence of real numbers has a convergent subsequence. More generally, it states that if is a closed bounded subset of then every sequence in has a subsequence that converges to a point in . This article is not so much about the statement, or its proof, but about how to use it in applications. If you come across a statement of a certain form (explained in the article), then the Bolzano-Weierstrass theorem may well be helpful. )
- Constructing exotic sets and functions using limiting arguments Quick description ( If you want to construct a set or function with a strange property (for instance, you might want a function that was continuous everywhere and differentiable nowhere) then a good way of doing so is often to define your object as a limit of a sequence of objects that exhibit the behaviour you want on smaller and smaller distance scales.)
- I have a problem to solve in real analysis Quick description ( This is a page that tries to understand what kind of problem you are trying to solve, so that it can take you to appropriate advice on the Tricki.)
- I have a problem to solve in real analysis and I do not believe that a fundamental idea is needed
- I need to find a real number with a certain property
- I have a problem about the convergence of a sequence
- I have a problem about a supremum or an infimum
- I have a problem about an infinite sum
- I have a problem about a continuous function
- I have a problem about open or closed sets
- I have a problem about differentiation
- I have a problem about integration
- I have a problem about uniform convergence
- I have a problem about differentiation in higher dimensions

- Metric spaces
- Topological spaces
- Complex analysis
- Harmonic analysis
- Partial differential equations
- Functional analysis
- Geometric measure theory
- Dynamics
*Analytic techniques*- Discretization front page
- Techniques for proving inequalities

- Combinatorics
- Extremal combinatorics
- Probabilistic combinatorics
- What is the probabilistic method and when can it be applied?
- Useful heuristic principles for guessing probabilistic estimates
- Unusual choices of probability distribution
- Averaging arguments
- The second-moment method
- How to use martingales
- How to use the Lovász local lemma
- How to use Talagrand's inequality
- How to use Janson's inequality
- Stein's method
- How to use correlation inequalities
- How to use the inclusion-exclusion principle

- Graph minors etc.
- Additive combinatorics

- Geometry
- Logic and set theory
- Mathematical physics
- Number theory
- Probability
- Elementary probability
- Use linearity of expectation
- Use the fact that the variances of independent random variables add together
- Bounding probabilities by expectations
- One way of working out the probability of a conjunction of dependent events
- How to use the inclusion-exclusion formula
- Condition on the first thing that happens
- Don't forget the law of total probability
- How to use Bayes's theorem
- How to reason with conditional expectations

- Stochastic processes

- Elementary probability
- Statistics
- Theoretical computer science
- Topology

How to use mathematical concepts and statements

- How to use the Baire category theorem
- How to use Bayes's theorem
- How to use the Bolzano-Weierstrass theorem
- How to use the Cauchy-Schwarz inequality
- How to use the central limit theorem
- How to use the classification of finite simple groups
- How to use cohomology
- How to use compactness
- Convergent subsequences and diagonalization Quick description ( A topological space is
*sequentially compact*if every sequence has a convergent subsequence. One form of the Bolzano-Weierstrass theorem states that a closed bounded subset of is sequentially compact. More generally, compact metric spaces are sequentially compact. These facts have many applications. Also, some useful diagonalization techniques can be interpreted as saying that certain topological spaces are sequentially compact – as a result, one often hears the phrase "by compactness" when no topology has been specifically mentioned. ) - Discretization followed by compactness arguments Quick description ( Often one proves a theorem about a continuous structure by finding a fairly dense finite subset of it and proving a finite statement about that instead. For this one needs the continuous structure to have a suitable compactness property. )
- Using the fact that a continuous function on a compact set attains its bounds Quick description ( A continuous real-valued function defined on a compact topological space is bounded and attains its bounds. This fact has many applications, as this article demonstrates with some varied examples. )

- Convergent subsequences and diagonalization Quick description ( A topological space is
- How to use the continuum hypothesis
- How to use correlation inequalities
- How to use duality
- How to use entropy
- How to use fixed point theorems
- How to use the Fourier transform
- How to use generating functions
- How to use group actions
- How to use the Hahn-Banach theorem
- How to use the inclusion-exclusion principle
- How to use Janson's inequality
- How to use the Lovász local lemma
- How to use martingales
- How to use the max-flow-min-cut theorem
- How to use the mean value theorem
- How to use ordinals
- How to use the pigeonhole principle
- How to use the Riemann-Roch theorem
- How to use spectral gaps
- How to use Szemerédi's regularity lemma
- How to use Talagrand's inequality
- How to use ultrafilters
- How to use Zorn's lemma.