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Let $P$ be a mathematical statement or a mathematical problem. I am looking for a couple of nice examples for $P$ that satisfy the following criteria:

  1. Given two (or more) mathematical points of view on $P$, we find that one of them views makes $P$ easy to solve/prove and the other one makes $P$ hard to solve/prove.

  2. $P$ should (at least from the easier point of view) be understandable by someone who studied maths obtained the basics and has a quiet good understanding of mathematical problems.

  3. It shouldn't take to much text to formulate, since I don't have that much time and space to present it.

It would be nice if the problem is prominent and it is ok if the problem is not pure math but must have a clear link to maths.

Any ideas? A short explanation of the problem from the different angels is welcome and appreciated.

Edit: I forgot to mention that by different point of view I meant somethink like looking at $P$ from an algebraic point of view and from an analytical point of view and maybe from a topological point of view. I want to point out the awesome properties of maths to transform a hard problem to another theory where the problem is easily solvable.

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    This is a *really* vague and broad question. "points of view" as in different proof strategies, or as in from different fields/domains of math, or as in...different theoretical stances, e.g. assuming the axiom of choice or without assuming it...This question, as it stands, could open pandora's box!2012-11-11
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    @amWhy As in different fields/domain. Thanks for asking. I edited my question.2012-11-11
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    Thanks, that helps narrow things a little bit, at least!2012-11-11
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    @Aufwind, then I'll delete my answer.2012-11-11
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    @DimitriSurinx Sorry for my unspecified original answer. Thanks for your effort!2012-11-11
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    @Aufwind, No problem!2012-11-11
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    Check Wiener's theorem on the convergence of the Fourier series of $1/f$. He did it with brilliant hard analysis techniques but Gelfand did a three line proof using Banach algebra technique. You can look at Folland's *A Course in Abstract Harmonic Analysis* Chapter 1 to see the details.2012-11-11
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    @Aufwind: It has been suggested that this question be made [Community Wiki](http://math.stackexchange.com/privileges/community-wiki). Please consider doing this.2012-11-12
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    Dear @robjohn, I consulted the link you provided to see how to make this question Community Wiki. It states: *You can choose to make any answer you own a community wiki by ticking the checkbox under the edit area.* Unfortunately I own the question and not the answer. How do I turn the question into Community Wiki? :-)2012-11-13
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    @Aufwind: if you start to edit your question, you should see the "Community Wiki" [check box](http://i.stack.imgur.com/LsVt7.png) below and to the left of the edit box.2012-11-13
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    @robjohn: Unfortunately this doesn't work. I started to edit the question, hoping for the Checkbox to appear. I added several sentences, since I thought maybe there is a minimal amount of characters, that should be changed to make an edit valid. But nothing happened. You have my blessing to use your moderator power to change the question into community wiki, if you want. :-)2012-11-14
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    @Aufwind: so done. [Your reputation](http://math.stackexchange.com/privileges/user/9850) should allow the creation of wiki posts, so I don't know why you did not get the checkbox. I apologize for all the back and forth.2012-11-14
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    @robjohn no harm done. :-)2012-11-14
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    A nice elegant example is the existence of transcendental numbers.2014-03-11

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