In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert have only a few tricks which they use over and over again.
Assuming Rota is correct, what are the few tricks that mathematicians use repeatedly?
$$ \sum_{i=1}^m\sum_{j=1}^n a_{i,j}=\sum_{j=1}^n\sum_{i=1}^m a_{i,j} $$
(and its variants for other measure spaces).
I still get misty-eyed whenever I read something that capitalizes on this trick in an unpredictable way.
A very useful generic trick:
If you can't prove it, make it simpler and prove that instead.
An even more useful generic trick:
If you can't prove it, make it more complicated and prove that instead!
Dennis Sullivan used to joke that Mikhail Gromov only knows one thing, the triangle inequality. I would argue that many mathematicians know the triangle inequality but not many are Gromov.
In combinatorics: shove it into OEIS, and see what's up. Also, add more parameters!
Note: the Macdonald polynomials were introduced by adding more parameters to the Jack and the Hall-Littlewood polynomials. The introduction of Macdonald polynomials unified a lot of cool stuff, and they are now essential in the field of Diagonal harmonics.
Integration by parts has allegedly earned some people big medals.
For a finite set of real numbers, the maximum is at least the average and the minimum is at most the average.
Of course this is just the real version of the Pigeonhole Principle, but Dijkstra had an eloquent argument as to why the usual version is inferior.
https://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EWD1094.html
Although Erdős was mentioned in the comments as perhaps having prompted this whole discussion, I'm surprised not to see the basic trick of "try a random object/construction" posted as an answer, which he used so often to such great success.
If an integer-valued function is continuous, it has to be constant.
This trick shows up in many places, such as the proof Rouché's theorem, and basic results about the Fredholm index.
Whenever you find yourself trying to implement inclusion–exclusion by hand ... stop immediately and start over using the Möbius $\mu$-function.
Those of us who are old enough may remember http://www.tricki.org/
Localize + complete, taking a hypersurface section, and using the socle are useful tricks in commutative algebra.
Not sure if... well, what the...
Find a duality. Play duals against each other.
If $1-x$ is invertible, then its inverse is $1 + x + x^2 + \cdots $. This is the second most useful "trick" I know, after "look for the [symmetric] group acting on you thing", but someone else already mentioned it.
I couldn't resist adding one of my own: "Apply linearity of expectation".
For example in Barbier's incredibly elegant approach (Buffon's Noodle) to Buffon's Needle Problem.
What worked very well for the French school of algebraic geometry (but it seems to predate them!) is the "French trick" of turning a theorem into a definition. See e.g. this post for some examples and background on the term.
If $r,s $ are elements of a ring, then $1-rs$ invertible implies $1-sr$ is invertible (and it is a trick: you can make an educated guess for the formula for the inverse of $1-sr$ from that for $1-rs$). This can be used to find quick proofs of: (a) in a Banach algebra, ${\rm spec\ } rs \cup \{0\} = {\rm spec}\ sr \cup \{0\}$ (which in turn yields the nonsolvability of $xy-yx = 1$---all one needs is boundedness and nonemptiness of the spectrum); (b) the Jacobson radical (defined as the intersection of all maximal right ideals) is a two-sided ideal; and probably some other things I can't think of right now ...
In the course of working with Hervé Jacquet and reading many of his papers on automorphic forms and the relative trace formula, I feel like he got an amazing amount of mileage out of clever use of change of variables.
I remember a conference where all the speakers gave extremely hard-to-follow talks using very sophisticated machinery, and then Jacquet gave a talk with a very nice result and about 45 minutes of it was going through an elementary proof (once you knew the setup) that boiled down to a clever sequence of change of variables.
Maybe more than a "trick," but if you want to investigate a sequence $a_0,a_1,\dots$, then look at a generating function such as $\sum a_nx^n$ or $\sum a_n\frac{x^n}{n!}$. If you are interested in a function $f:\mathrm{Par}\to R$, where $R$ is a commutative ring and $\mathrm{Par}$ is the set of all partitions $\lambda$ of all integers $n\geq 0$, then look at a generating function $\sum_\lambda f(\lambda) N_\lambda b_\lambda$, where $\{b_\lambda\}$ is one of the standard bases for symmetric functions and $N_\lambda$ is a normalizing factor (analogous to $1/n!$). For instance, if $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, then $\sum_\lambda f^\lambda s_\lambda = 1/(1-s_1)$, where $s_\lambda$ is a Schur function. If $f(\lambda)$ is the number of square roots of a permutation $\lambda\in\mathfrak{S}_n$ of cycle type $\lambda$, then $$ \sum_\lambda f(\lambda)z_\lambda^{-1} p_\lambda = \sum_\lambda s_\lambda = \frac{1}{\prod_i (1-x_i)\cdot \prod_{i<j} (1-x_ix_j)}, $$ where $p_\lambda$ is a power sum symmetric function and $z_\lambda^{-1}$ is a standard normalizing factor.
The Renormalization Group trick:
Suppose you have some object $v_0$ and you want to understand a feature $Z(v_0)$ of that object. First identify $v_0$ as some element of a set $E$ of similar objects. Suppose one can extend the definition of $Z$ to all objects $v\in E$. If $Z(v_0)$ is too difficult to address directly, the renormalization group approach consists in finding a transformation $RG:E\rightarrow E$ which satisfies $\forall v\in E, Z(RG(v))=Z(v)$, namely, which preserves the feature of interest. If one is lucky, after infinite iteration $RG^n(v_0)$ will converge to a fixed point $v_{\ast}$ of $RG$ where $Z(v_{\ast})$ is easy to compute.
Example 1: (due to Landen and Gauss)
Let $E=(0,\infty)\times(0,\infty)$ and for $v=(a,b)\in E$ suppose the "feature of interest" is the value of the integral $$ Z(v)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}}\ . $$ A good transformation one can use is $RG(a,b):=\left(\frac{a+b}{2},\sqrt{ab}\right)$.
Example 2: $E$ is the set of probability laws of real-valued random variables say $X$ which are centered and with variance equal to $1$. The feature of interest is the limit law of $\frac{X_1+\cdots+ X_n}{\sqrt{n}}$ when $n\rightarrow\infty$. Here the $X_i$ are independent copies of the original random variable $X$.
A good transformation here is $RG({\rm law\ of\ }X):={\rm law\ of\ }\frac{X_1+X_2}{\sqrt{2}}$.
Andre Weil's slogan that where there is a difficulty, look for the group (that unravels it).
I take this to mean something more aggressive than a truism to note and use group structure; more like "exploit the full potential of representation theory in all its manifestations after seeking out whatever obvious and hidden symmetries exist in the problem".
The chapter ‘A Different Box Of Tools’ of Surely You're Joking, Mr Feynman was named for a particular trick Richard Feymnan used:
[Calculus For The Practical Man] showed how to differentiate parameters under the integral sign — it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasise it. But I caught on how to use that method, and I used that one damn tool again and again.
(pp.86–87)
(1) Double-counting, which can also be described as counting the same thing in two ways. Very useful, and at least as powerful as interchanging summation order.
(2) Induction. When there is a natural number size parameter, one can always consider trying this.
(3) Extremal principle, which is ultimately based on induction, but looks very different. For example, the Sylvester-Gallai theorem has an extremely simple proof using this.
Existence as a property: You want to find an object that solves a given equation or a given problem. Generalize what you mean by object so that existence becomes easy or at least tractable. Being an object is now a possible property you might prove about your generalized object. Having already something you can prove properties about is often both mathematically and psychologically easier than searching in the void.
Some examples:
A common trick is compactification. First prove that a space admits a compactification, e.g.
Once one has a compact space, one can analyze the objects one is interested in by taking infinite sequences, extracting a subsequence in the limit, and analyzing this limit, sometimes obtaining a contradiction if the limit does not lie in the original space one was considering. E.g. I used this approach to analyze exceptional Dehn fillings of cusped hyperbolic 3-manifolds.
The second derivative test (i.e. "a smooth function has a local maximum at a critical point with non-positive second derivative.") is endlessly useful.
When you first see this fact in Calculus, it might not seem so powerful. However, there are countless generalizations (e.g. the maximum principle for elliptic and parabolic PDEs), which play an important role in analysis.
There's the quote in Bell's Men of Mathematics attributed to Jacobi: "You must always invert", as Jacobi said when asked the secret of his mathematical discoveries. Sounds apocryphal but it is certainly a nice suggestion.
Scott Aaronson has taken a stab at articulating his own methodology for upper-bounding the probability of something bad. He was inspired by a blog post by Scott Alexander bemoaning how rarely experts write down their expert knowledge in detail.
If, on a probability space, $\int_\Omega X\,dP = x$, then there is some $\omega$ such that $X(\omega)\ge x$.
In homotopy theory: if something is hard to compute, build an infinite tower that converges to it and induct your way up the tower. This includes spectral sequences, Postnikov towers, and Goodwillie calculus.
In category theory: apply Yoneda's Lemma.
Other common tricks in category theory:
In an old mathoverflow answer, I wrote several more common tricks in category theory, including
My favorite is perhaps the "commutator trick", i.e. "take commutators and see what happens". Some general things that may happen 1) the commutator touches less than the commutatorands 2) the commutator defies your abelian intuition.
I'm mostly familiar with 1) in the context of infinite groups, in particular finding generators for complicated groups, and 2) blew my mind to pieces as Barrington's theorem before I even knew any math.
I counted that a seventh of my papers use some type of commutator trick, but what really sold commutators to me was when I got a Rubik's cube as a christmas present.