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There are a number of theorems or lemmas or mathematical ideas that come to be known as eponymous tricks, a term which in this context is in no sense derogatory. Here is a list of 10 such tricks (the last of which I learned at MO):

Edit: List augmented from the comments and answers:

Further Edit. And although my original interest was in eponymous (=named-after-someone) tricks, several non-eponymous tricks have been mentioned, so I'll gather those here as well:

Some of those listed above do not yet have Wikipedia pages (hint, hint—Thierry).

I (JOR) am not seeking to extend this list (although I would be incidentally interested to learn of prominent omissions), but rather I am wondering:

Is there some aspect or trait shared by the mathematical ideas or techniques that, over time, come to be named "tricks"?

I am aware this is a borderline question; feel free to close if it unduly distracts.

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    $\begingroup$ "An idea which can be used only once is a trick. If one can use it more than once it becomes a method." Quoted from books.google.co.uk/… $\endgroup$ – Andrey Rekalo Dec 4 '10 at 4:45
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    $\begingroup$ That construction of Whitney ought to have a more dignified name. $\endgroup$ – Tom Goodwillie Dec 4 '10 at 4:58
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    $\begingroup$ You could add the Eilenberg Swindle to your list. (A Swindle sounds even more disreputable than a Trick.) $\endgroup$ – Charles Rezk Dec 4 '10 at 5:36
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    $\begingroup$ Why is the page linking to the Whitney trick linking to a "Global Oneness" site? <br> <br> Why do they even have a page on Whitney embedding on a site about spirituality? I'm very confused. $\endgroup$ – Simon Rose Dec 4 '10 at 5:42
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    $\begingroup$ Are you expecting a definitive answer to this? If not, then CW would surely be in order? $\endgroup$ – Alex B. Dec 4 '10 at 6:22

13 Answers 13

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How about the following (which I think applies to some of these tricks but not others): a trick is something whose usefulness is not fully captured by any particular set of hypotheses, so it would limit its usefulness to write it down as a lemma.

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    $\begingroup$ This is my favorite answer. It captures the notion that a trick is one of the essentially human contributions to math, a creative touch that isn't part of some axiomatic or algorithmic approach that something more mechanical might invent. It's generalizable but not categorizable. $\endgroup$ – Ryan Reich Dec 4 '10 at 22:04
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    $\begingroup$ I like this answer, but I'm not sure it really distinguishes between tricks and methods. For instance, I don't know of a lemma one could write down that would do justice to the probabilistic method in combinatorics. But actually I'm tempted to say that "Take a random X" feels like a trick that can be applied over and over again, perhaps for this very reason. $\endgroup$ – gowers Dec 5 '10 at 22:24
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    $\begingroup$ @Ryan: I must also leap to the defence of the poor old computer. I don't see any reason in principle that a clever program could not invent mathematical tricks. (I think a proof that a program couldn't invent tricks would prove that humans couldn't either.) $\endgroup$ – gowers Dec 5 '10 at 22:26
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I'll take a stab at this.

I think that the term "trick" is used to connote a technique that achieves something as if by magic. If I make a cake by combining flour, sugar, and eggs and baking, that is simply a standard technique, but if I make the cake by putting the ingredients into a top hat and waving a wand over it, that is a magic trick. The way that the Weyl unitary trick makes complex groups behave like compact ones seems like a magic trick. (For those of you trying to follow this half baked analogy, the cake is complete reducibility of representations, the oven is integration, and the hat is ... uhhh.... )

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    $\begingroup$ Ha ha, half baked :) $\endgroup$ – Zev Chonoles Dec 4 '10 at 7:56
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    $\begingroup$ I think this is the answer. $\endgroup$ – timur Dec 4 '10 at 15:29
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    $\begingroup$ The hat is orthogonality? $\endgroup$ – Qiaochu Yuan Dec 4 '10 at 16:12
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    $\begingroup$ The hat is Fourier transform, of course! $\endgroup$ – Paul Siegel Dec 6 '10 at 12:20
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One well-known trick is a way to evaluate the Gaussian integral $G = \int_\mathbb{R} e^{-x^2}dx = \sqrt{\pi}$ by writing $$G^2 = \left(\int_\mathbb{R} e^{-x^2}dx\right)\left(\int_\mathbb{R} e^{-y^2}dy\right) = \int_{\mathbb{R}^2} e^{-(x^2+y^2)}dxdy$$ which when transformed to polar coordinates becomes $$G^2 = 2\pi \int_0^\infty e^{-r^2} r dr = \pi \int_0^\infty e^{-u} du = \pi$$ via the substitution $u=r^2$. It appears this idea is due to Poisson.

In a 2005 note in the American Mathematical MONTHLY, R. Dawson has observed that this is a trick that only works once; there are no other integrals that can be evaluated by this method. Specifically:

Theorem. Any Riemann-integrable function $f$ on $\mathbb{R}$, such that $f(x)f(y) = g(\sqrt{x^2+y^2})$ for some $g$, is of the form $f(x)=ke^{ax^2}$.

See: Dawson, Robert J. MacG. On a "singular" integration technique of Poisson. American Mathematical Monthly 112 (2005), 270-272.

So if a technique is a trick that works twice, this one is definitely still a trick.

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    $\begingroup$ Note that this trick has something in common with the Rabinowitsch, Cayley and Eilenberg tricks and probably some others on the list: in order to solve a $k$-dimensional problem, you go into more than $k$ dimensions. This also seems to be a distinguishing feature of things called tricks. $\endgroup$ – darij grinberg Dec 5 '10 at 22:04
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    $\begingroup$ I think Dawson was anticipated by James Clerk Maxwell. A result called Maxwell's theorem says that if $X_1, \dots, X_n$ are independent real-valued random variables and their joint density is spherically symmetric, then all of them are normally distributed, i.e. the probability density of each of them is a Gaussian function. $\endgroup$ – Michael Hardy Dec 5 '10 at 22:17
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    $\begingroup$ Constantine Georgakis, "A Note on the Gaussian Integral", Mathematics Magazine, February, 1994, page 47 This paper gives what its author considers "a better alternative to the usual method of reduction to polar coordinates" for evaluating this integral. See en.wikipedia.org/wiki/Gaussian_integral . $\endgroup$ – Michael Hardy Dec 5 '10 at 22:19
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    $\begingroup$ While the specific form $f(x) f(y) = g(\sqrt{x^2+y^2})$ applies only to Gaussians, there are further uses of this kind of transformation: in one direction, to the volumes of Euclidean spheres in higher dimension (imagine you already know $\Gamma(1/2)$ but not the area of a circle); in another direction, to the usual proof of $B(x,y) = \Gamma(x) \Gamma(y) / \Gamma(x+y)$; combining these, to Dirichlet integrals; and by analogue, to the relation between Gauss sums and Jacobi sums — and probably others that don't come to mind right now. $\endgroup$ – Noam D. Elkies Jul 9 '11 at 6:11
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    $\begingroup$ @Michael Hardy: that "better alternative" in the paper by Georgakis in fact is due to Laplace. See york.ac.uk/depts/maths/histstat/normal_history.pdf. $\endgroup$ – KConrad Mar 26 '12 at 3:15
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To my way of thinking, the other answers are missing an important element, a necessary feature for a mathematical tool or method to be called "trick."

Namely, in order to be called a "trick," a method or technique must involve artifice or misdirection of some kind. When we treat a mathematical object as something that it isn't really or when we pretend that something is other than it is in order to advance an argument (which is not to suggest that the mathematics is not correct), then we are using trickery. When we solve a problem by placing our focus on something else, in which we aren't actually interested as such and which may even be silly in some way—a kind of misdirection—but by doing so we become successful in the original problem, then we are using trickery. When we replace a robust concept, in which we are really interested, with a modified version of it, perhaps even an absurd version of it, but which makes the argument work, then we are using trickery.

For example, with Craig's trick, we replace a formula $\varphi$ with the conjunction with itself $\varphi\wedge\varphi\wedge\dots\wedge\varphi$ repeated many times over. The new assertion is just silly and we don't actually care about it as such, although of course it is logically equivalent to $\varphi$. How could it possibly help? The point is that we can use the new assertion to code some extra information into an axiomatization or presentation: the number of times it was repeated. By this artifice, we can deduce that every computably enumerable theory has a computable set of axioms. The same idea works in many other contexts. For example, every c.e. presentable group has a computable presentation, by sufficiently repeating relations suitably in the presentation.

With Scott's trick, the issue to be solved is that the equivalence class of an object forms a proper class, which can cause certain problems, and so we replace that equivalence class with the set of rank-minimal members of the class. If we think of this fake equivalence class as the real thing, then everything works great! This trick is surprisingly robust, and can be used to find small canonical sets of representing structures in almost any situation. For example, in ZFC there is a definable manner of choosing a set of groups from each group isomorphism class: the rank-minimal groups from that class. This is a trick, because we don't really care much about that particular collection as such.

With Rosser's trick, we replace the concept of a theory $T$ proving a sentence $\sigma$, with: $T$ proves $\sigma$ by a proof for which there is no shorter proof of $\neg\sigma$. When you think of "proof" using this concept, then Gödel's incompleteness theorem is improved to the Gödel-Rosser theorem, where one can drop Gödel's extra hypotheses about $\omega$-consistency. This is a trick, because we don't actually want to think about "proof" using Rosser's concept, except that it makes the argument work.

In many of the other tricks, we do something that seems a little absurd at first, misdirecting our attention from the original problem to this other thing, which may seem irrelevant at first, but when we follow it more fully it provides the answer we seek.

In each case, we replace the concepts or objects in which we are truly interested by concepts or objects that we don't actually care about as such and which in several cases are comical versions of the original, except that they make the argument work.

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    $\begingroup$ In particular, I don't agree with the "used only once" joke idea for distinguishing tricks and methods. We have already many counterexamples to that view mentioned on this thread of posts. Some of these tricks are robust mathematical methods; but they are still tricks, because they involve artifice. $\endgroup$ – Joel David Hamkins Feb 11 '16 at 14:39
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http://en.wikipedia.org/wiki/Rosser%27s_trick

"A technique is a trick that works twice"

Note that Grothendieck never published his proof of the Grothendieck-Riemann-Roch theorem because he felt that the proof depended on an "astuce" (trick) rather than flowing naturally.

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    $\begingroup$ It is interesting that the accepted English translation of "astuce" is "trick", whereas that of the adjectival form "astucieux/euse" is "clever" or "astute". This seems to give the concept of a trick a better connotation in French than in English. (I am tempted to load up Lord of the Rings with the French subtitles on to see whether Gollum accuses Frodo and Sam of being "astucieux".) $\endgroup$ – Pete L. Clark Dec 4 '10 at 16:32
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    $\begingroup$ @Pete: you're absolutely right. For instance, to translate the expression "dirty trick" into French, you would not use the word "astuce" because "astuce" has a uniformly positive connotation of praise (yes, even in that Gollum context). $\endgroup$ – Thierry Zell Dec 4 '10 at 16:53
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    $\begingroup$ In French, one says "un artifice de calcul", "l'artifice de Legendre". By the way, there is also the van der Waerden trick (associativity of the composition of reduced words). $\endgroup$ – Panurge Feb 11 '16 at 14:26
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Scott's Trick is called a "trick" because it is not actually necessary for the completion of the proof in which it is involved; however, without the trick the proof is massively more tedious. Although the other tricks may not have a widely-agreed-upon-reason for being a trick, I suspect that they may be called such for similar reasons.

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    $\begingroup$ Same for the Rabinowitsch trick. Also as pointed in another answer, the Rabinowitsch trick works only once (although similar ideas must be used, albeit in less famous circumstances). $\endgroup$ – Thierry Zell Dec 4 '10 at 16:10
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    $\begingroup$ Well, it’s only unnecessary if you’re assuming AC (which was not, if I understand right, quite as entrenchedly automatic among set theorists back in the ’60s as it is now). It’s necessary if you want to talk about cardinalities — and more generally, construct quotients of classes by equivalence relations — in ZF and many other set theories. $\endgroup$ – Peter LeFanu Lumsdaine Dec 6 '10 at 5:31
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I've long known the adage that a "trick" works only once whereas a "method" works in multiple instances, or maybe is expected to work in yet unanticipated future instances.

But there's another POV: a trick is something whose efficacy cannot be anticipated, but only by hindsight is seen to work. All methods I've seen of finding $\int \sec x \ dx$ are "tricks". I've always leaned toward viewing unanticipatability as the essence of trickhood.

But I also like Qiaochu Yuan's answer.

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    $\begingroup$ There is an algorithm to find the integral of any rational function of trigonometric functions: use a rational parameterization (e.g. using tan x/2), then use partial fractions (or a residue computation, etc.). I don't see how this is a "trick" in any sense. It is a direct corollary of the fact that the circle is birational to the projective line. $\endgroup$ – Qiaochu Yuan Dec 4 '10 at 17:37
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    $\begingroup$ Yes, I agree that that argument is trick-like. The rational parameterization of the circle is not. It is a natural and beautiful geometric argument (just taking the slope of a line between two points) and I think it could easily be presented to a first-year calculus class. $\endgroup$ – Qiaochu Yuan Dec 4 '10 at 18:05
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    $\begingroup$ I had a student come up with a different way to do this, on the fly, on the final exam. Write $$ \sec(x) = {1\over \sqrt{1 - \sin^2(x)}}, $$ then substitute $u = \sin(x)$ and use partial fractions! $\endgroup$ – Jeff Strom Dec 5 '10 at 2:31
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    $\begingroup$ I usually integrate this function differently (using something trick-like, but natural one): $$\frac{dx}{\cos(x)} = \frac{\cos(x)dx}{{\cos^2(x)}} = \frac{d(\sin(x)}{1 - {\sin^2(x)}}$$ $\endgroup$ – Ostap Chervak Jun 14 '11 at 15:00
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    $\begingroup$ A fairly new Wikipedia article: en.wikipedia.org/wiki/Integral_of_the_secant_function Raise your hand if you knew that this was a famous conjecture in the 17th century. Or that it was originally done for the purposes of cartography (I think maybe lots of people know that). Or that Isaac Newton was aware of the conjecture and wrote about it, and Isaac Barrow was the first to prove it. $\endgroup$ – Michael Hardy Jun 14 '11 at 18:43
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While tricks have names because they wind up being associated with with some particular mathematician, tricks are tricks because something important goes on "behind the curtain."

For instance, to prove $$ (a_1 b_1 + \cdots + a_n b_n)^2 \leq ({a_1}^2 + \cdots + {a_n}^2) ({b_1}^2 + \cdots + {b_n}^2), $$ write \begin{align*} A &= ({a_1}^2 + \cdots + {a_n}^2)\\\ B &= (a_1 b_1 + \cdots + a_n b_n)\\\ C &= ({b_1}^2 + \cdots + {b_n}^2), \end{align*} then we must show $$ B^2 \leq AC. $$ Equality clearly holds when $A = 0$. Otherwise, since $\mathbb{R}$ has no negative squares, for all $x \in \mathbb{R}$, $$ 0 \leq (a_1 x - b_1)^2 + \cdots + (a_n x - b_n)^2. $$ Expanding the squares, $$ 0 \leq Ax^2 - 2Bx + C. $$

The quadratic expression vanishes whenever $$ x = \frac{B}{A} \pm > \sqrt{\left(\frac{B}{A}\right)^2 - > \frac{C}{A}}. $$

If $x = \dfrac{B}{A}$, then $$ 0 \leq A\left(\frac{B}{A}\right)^2 - 2 B\left(\frac{B}{A}\right) + C = \frac{B^2}{A} - 2 \frac{B^2}{A} + C = - \frac{B^2}{A} + C, $$ thus $$ B^2 \leq AC. $$

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Would regarding a scalar as the trace of a $1\times1$ matrix be considered a "trick"?

Here's an occasion where that's useful:

http://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Maximum-likelihood_estimation_for_the_multivariate_normal_distribution

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There's always Feynman's trick. It's more common amongst physicists; Feynman, afterall was a physicist. Although quite standard now, back in Feynman's hayday he was able to solve many complicated integrals by adding another variable, using an auxiliary function, solving a basic integral involving the auxiliary function, and then differentiating said variable to get the correct integral. This is common now, but when Feynman would use it, people wouldn't understand how he'd solved such complicated integrals so fast. It's very diverse too, in that it applies in many scenarios. One example I always found interesting, off the top of my head, was using it to prove that the Gamma function interpolates the factorial in one line:

$$\int_0^\infty e^{-x}x^{n}\,dx = (-1)^{n}\frac{d^{n}}{dt^{n}}\Big{|}_{t=1}\int_0^\infty e^{-tx}\,dx = \frac{d^{n}}{dt^{n}}\Big{|}_{t=1} \frac{(-1)^{n}}{t} = n!$$

though is of no way limited to this single, obvious, instance. One can solve the $\text{sinc}$ integral using this. One can simplify the amount of work one needs by using this little trick.

It is a trick in the sense that it's very simple. It's a little magic, where persons usually go "where'd you think of using that extra variable and that second function and taking the derivative?" It's basic and easy to understand, isn't a theorem or a lemma really (putting down hypotheses or conditions blurs its simplicity). It's just a technique one tries when solving integrals; like partial fractions or trigonometric substitution; except it takes a while to get a hang of using, and at first seems extraneous. I think that's an important fact to something being a trick: it isn't obvious, but it's easy, and it takes a while to get in the hang of using it. As a kicker, Feynman's trick can go very far with little work; Feynman made a living off using this trick.

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    $\begingroup$ I wouldn't attribute this "trick" (method?) to Feynman, although he helped to raise awareness of it. en.wikipedia.org/wiki/Leibniz_integral_rule#Examples $\endgroup$ – Todd Trimble Mar 2 '17 at 3:49
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    $\begingroup$ @ToddTrimble I know it isn't technically Feynman's (maybe I should've made that clearer), but people have gotten into the swing of calling it Feynman's trick (or Feynman's method; it goes by both names), especially physicists. That's its nomer, and Feynman used it notoriously. I like to call it a trick, because it's really simple and reminds me of a card trick, there's some sleight of hand involved; and that's how it was first called when I heard of it. I also think it's a bit more involved than just Leibniz's integral rule. $\endgroup$ – user78249 Mar 2 '17 at 3:53
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    $\begingroup$ This is essentially the "differentiation under the integral sign" trick. I like its application to $\int_0^\infty \sin xdx/x$ best. $\endgroup$ – Fan Zheng Mar 2 '17 at 5:05
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    $\begingroup$ @FanZheng It is essentially this. But by adding the "Feynman" it makes a closer semantic connection to: how effective it is, how it can be used in complicated ways, and that it's even applicable in very advanced scenarios. Especially when you see how Feynman uses it. $\endgroup$ – user78249 Mar 2 '17 at 6:31
  • $\begingroup$ Another example on introducing an extra variable-the integral of the bell function over $\ \mathbb R\ $ was (is?) popular on American tests and exams. One computes its square as a product of two such integrals but each w.r. to a different variable. Then a student treats it as one integral over $\ \mathbb R^2 $ Then switch to the polar coordinates. $\endgroup$ – Włodzimierz Holsztyński Mar 2 '17 at 17:36
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Looooong before I was old (then I was ancient, and now I am archeological), this is what I said:

        The difference between a method and a trick is understanding.

That's all.

Trick => method (evolution):   Observe, that once a "trick" is understood it becomes a method. Often, it is still called a trick for various reasons like tradition, recreational value, inertia, etc.

And just a distant association (poetry): once a kenning becomes common, it gets simplified to a metaphor--the metaphor is then like a hint (of that kenning). Perhaps this is one of the reasons why literary critics often are confused, they say that kenning is a kind of a metaphor (hmmm....). Thus we get another evolution: 1.trick => method, 2.kenning => metaphor. (Didn't I say distant?)

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    $\begingroup$ @NoahSchweber, there are vague intentions of what people expect from a word. Those expectations are not consistent. That's how natural languages are. And then people ask about the so-called "real meaning" of a word. But there is no such thing. Thus, the best, if we care, we should hopefully agree on the definition of the term which is the most useful. This in particular means simplicity(!). Other considerations are avoiding a new synonym, etc. Of course, one needs to sacrifice some examples. *** There is need of definitions of terms also outside mathematics, e.g. in poetry. $\endgroup$ – Włodzimierz Holsztyński Mar 2 '17 at 3:36
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    $\begingroup$ @NoahSchweber, even in pure mathematics, we do have definitions but we hardly ever have anything like "real meaning". Even great mathematicians were fooling themselves when they expected, say, "true real numbers"--again, there was never anything like "true real numbers". Thus, the definitions should be profound and useful but shouldn't attempt "real meaning". Ironically, the same definition (but for trivial linguistic variations) may be meaningful to different chapters of mathematics, e.g. probability and mechanics. Then how can it have one true meaning (in the usual sense)? Funny. $\endgroup$ – Włodzimierz Holsztyński Mar 2 '17 at 3:45
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    $\begingroup$ @WłodzimierzHolsztyński That's a very functional linguistics thing to say on a math forum. It's nice to see mathematicians engage in a healthy dialogue of the ideal semantics of a word; the blurred line between signifiers and the signified. After reading your comment; it framed your answer much more clearly. +1 $\endgroup$ – user78249 Mar 2 '17 at 4:36
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    $\begingroup$ And your edit made it even more poignant! $\endgroup$ – user78249 Mar 2 '17 at 6:43
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    $\begingroup$ As I mentioned under James Nixon's answer, I think that in promoting the trick of differentiating under the integral sign to a method, Gert Almkvist and Doron Zeilberger understand something like what is said in this answer. $\endgroup$ – Todd Trimble Mar 5 '17 at 22:10
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I would like to mention an important trick in Vanishing theorem in algebraic geometry-The "cyclic cover trick", which is in the MO question: what is the cyclic cover trick?

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I am very fond of the device by Landau by means of which one can establish Bertrand's Postulate for every natural number $n$ less than some given constant $C>0$.

Let us suppose that we need to show the validity of the "postulate" for every $n<4000$ (as in Erdös's famous 1932 paper). According to Erdös, in the case under consideration, Landau's teachings imply that one does not have to look for a prime number in all of the intervals

$$(1,2], (2,4], (3,6], (4,8], \ldots, (3999, 7998],$$

and that it suffices to consider the following list of fourteen primes in which each of them is smaller than twice the other:

$$2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 4001.$$

Indeed, if $N \in [2,4000) \cap \mathbb{N}$, let us denote by $p_{N}$ the greatest prime in the list that is smaller than or equal to $N$; then, if $p_{N+1}$ is the prime in the list that comes right after $p_{N}$, it holds that $p_{N+1} \in (N,2N]$ and we are done.

If a trick is an idea which can be used only once, then the previous "Bemerkung" of Landau (as Erdös refer to it in the aforementioned paper) is definitely deserving of being declared as one, right?

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