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I'm a math student (last year undergraduate) and I really enjoy solving math problems but sometimes I face problems that I don't see how to solve even after trying every idea I had. When I look up at the solution I understand it perfectly but I can't figure out how the author actually found his solution. Their thinking process is never explained. Let me take an example:

Let $P \in \Bbb R[X]$. Show that if $\forall x \in [0,1], P(x) \ge 0$ then $\exists A,B,C,D \in \Bbb R[X], P=A^2+XB^2+(1-X)C^2+X(1-X)D^2$.

What I've done:

  • I tried to look at some special cases ($\deg P = 0,1,2$).
  • tried to see if there is a structure behind this (by introducing $E= \{A^2+XB^2+(1-X)C^2+X(1-X)D^2 \ | \ A,B,C,D \in \Bbb R[X] \}$ and looking if $E$ is stable by product) but this was hard.
  • tried to reinforce the hypothesis by assuming $P \ge 0$, this worked well because I remembered another problem I solved ($P\ge 0 \implies \exists A,B \in \Bbb R[X], P=A^2 + XB^2$). I tried to adapt the proof (what I already tried previously) but it didn't help.
  • tried to use coefficients (really bad idea here).
  • tried Lagrange interpolation (not helpful either).

I posted this problem here and @orangeskid advised me to show that, in fact, $P(x)= A^2(x) + x(1-x)D^2(x)$ ie. we could let $B=C=0$, it was a very good hint. Even in a parallel world I'd never have thought of doing so. But why didn't I think of doing this? I don't know, it wasn't obvious for me or maybe I don't see far enough.

Another example:

Let $A, B \in \Bbb Z[X]$ such that $(\gcd(A(n), B(n)))_n$ is periodic. What can we say about $A$ and $B$?

What I tried:

  • I supposed that $\deg A = \deg B = 1$ even with that $\gcd(A(n),B(n))$ is something that scares me because I don't have a lot of intuition on this sequence. So I applied, Bézout's theorem which says that we can find two sequences $a_n,b_n$ such that $(a_n A(n) + b_n B(n))_n$ is periodic. And I got stuck here.

I saw a solution here using the resultant of two polynomials, something I never heard of.

There are other examples here, here and here: they all need a little trick, something close to magic (is that cleverness?).

So my questions are how could I find these tricks myself and make them natural? Why the origin of these tricks isn't explained? Should I just learn them "on the job"? Should I be worried (as a student in mathematics) if I don't see them? Thanks for your time!

Alex Ravsky
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Michelle
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  • That’s not answering your question, but the resultant turns out to be pretty useful in commutative algebra for various purposes. Example: let $p,q$ be two polynomials in one variable, construct a nonzero polynomial $r(u,v)$ such that $r(p,q)=0$. But unless the corresponding problem statement is wrong, it can be solved without – basically, it forbids the polynomials being not coprime, and the converse isn’t too hard. – Aphelli Jan 23 '21 at 18:37
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    "Why the origin of these tricks isn't explained?" Good question! For some reason, many math authors make no attempt to explain how someone might have thought of the ideas they are explaining. One reason for this could be that explaining a thought process might take them much longer, and people are busy. Also, perhaps they want their solution or explanation to be as elegant as possible, like a work of art. Still, to me, explanations of a potential thought process that can lead to a discovery are super helpful. – littleO Jan 23 '21 at 18:41
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    experience, experience and experience. Unfortunately, the poisonous idea that math should be easy if only just my teachers tell us exactly what to do is very pervasive. You say you don't want to memorize tons of tricks, but a large part of being a mathematician is just that. There are other things obviously but this is a requirement. – dezdichado Jan 23 '21 at 19:04
  • @dezdichado I didn't know that! Is there a book that lists some of them or you just have to learn them "on the job"? – Michelle Jan 23 '21 at 19:13
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    @Michelle There's a website called artofproblemsolving.com devoted to building this type of problem solving skill. They have a bunch of books. I recommend The Art and Craft of Problem Solving by Zeitz. There is also a wiki called the "Tricki" that is an interesting thing to be aware of: http://www.tricki.org/ – littleO Jan 23 '21 at 19:23

2 Answers2

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I think that a typical situation of a working mathematician is described by Nicholas Bourbaki, who said that “the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquintance has made him as familiar as with the beings of the real world”.

This vision should be helpful to solve the most of usual problems.

The base of a general method of hard problem solving is sketched in a book “Mathematical discovery: on understanding, learning and teaching” by George Polya.

Should I just learn them "on the job"?

I think this is a usual way.

Why the origin of these tricks isn't explained?

I think there is no need for that. Also there is no need to memorize tons of tricks, because they come naturally, when you have knowledge. I think a trick usually emerges while thinking on the problem. I recommend Chapters 10–12 of Polya’s book for details.

how could I find these tricks myself and make them natural?

I can add to the above a few finding hints. Try to see a problem from different points of view, search familiar elements in the problem. Return to the problem time after time.

Should I be worried (as a student in mathematics) if I don't see them?

I think that although there can be specific mind problems blocking trick vision, usually a student in mathematics should not be worried about that. Leonardo da Vinci said: “There are three classes of people: those who see, those who see when they are shown, those who do not see”. A problem solver sometimes sees some tricks. More advanced see more, less advanced see less. But trick vision often requires a luck and even a very good problem solver can miss a simple trick.

Is there a book that lists some of them

I think a source for the tricks should be not a list but a general mathematical knowledge. Remark that “Mathematical quickies” by Charles Trigg is a collection of problems with tricky solutions.

Alex Ravsky
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    Thanks for last paragraph. This post of mathoverflow is also listed some tricks. – C.F.G Jan 30 '21 at 16:07
  • Why you closed your account? – nonuser Feb 20 '21 at 22:32
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    @Aqua OK, I shall write my last comments here. My mathematical investigation is a quest for interesting and beautiful things, as is quoted in my profile. Living in a mind realm, I would like to support people with similar values and a free knowledge sharing, for instance, like these. This is why I joined Stack Exchange communities. – Alex Ravsky Feb 22 '21 at 04:37
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    But while spending some years, I discovered here too many things which I consider bad, for instance, a cult of the reputation and the badges, a respect of copyright, askers’ thankless, incompetently and aggressively closed questions and deleted answers, and a speech, twisted by political correctness. But the last straw was arbitraryness of Mathematics Stack Exchange moderators, who authoritarianly suspended my account. But if this is a good thing then it can be continued, :-) so I left Stack Exchange. It is a pity that I left a huge pile of unfinished answers at my computer, but c'est la vie – Alex Ravsky Feb 22 '21 at 04:37
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    My mind vision allows me to research not being a member of a community, so I continue my hopefully endless journey. There are many other things to see. I left with my best wishes to all knowledge seekers. May the Force be with you! :-) PS. Whereas I left Stack Exchange sites, I decided not to delete my profiles in order to keep my answers and comments tied together, with a hope that it can be helpful to people looking for them. PPS. I plan not to respond to queries from Stack Exchange, but can be contacted via an e-mail address from my papers, the last one is [email protected] – Alex Ravsky Feb 22 '21 at 04:37
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    Thank you for all answers/ideas given to me (and community). Thanks for leaving open window for possible contact with you. I wish you best. – nonuser Feb 22 '21 at 07:50
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    @AlexRavsky Thanks! you were a great helper. –  Nov 22 '22 at 16:42
  • @AlexRavsky These are not your last comments as you have posted others since then. Right? – Тyma Gaidash Oct 19 '23 at 20:32
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I've thought a lot about my questions and I think I've found my answer.

The source of my problem is the existence of proofs that aren't natural (to me), and this disturbed me. By natural I mean whose main idea is obvious, or could be found with a little bit of work (on special cases, with a drawing, etc.) or by using a few elementary results of the courses for a student. For example showing that if $u_n \to l$ and $v_n \to l'$ then $u_n + v_n \to l+l'$ is natural. So a proof that is natural (to me) is one that starts from the hypothesis and arrive at the conclusion without using an idea that is far to see from the hypotheses. For example, I don't know natural proofs of the fundamental theorem of algebra or the Schröder-Bernstein theorem. They both use some kind of trick.

I was too accustomed to courses in which one proposal led to another in a quick and smooth way. And I think that I never actually realized that math isn't hard, it's very hard. Tricks are the witness of this. Most courses are designed not to scare students too much (e.g. when something is difficult to prove, we admit it, etc.).

Michelle
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