11 questions linked to/from "Simple" beautiful math proof
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Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
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### Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction

I recently proved that $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$ using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation ...
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### Are the “proofs by contradiction” weaker than other proofs?

I remember hearing several times the advice that, we should avoid using a proof by contradiction, if it is simple to convert to a direct proof or a proof by contrapositive. Could you explain the ...
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### Could I be using proof by contradiction too much?

Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it ...
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### Beautiful, simple proofs worthy of writing on this beautiful glass door [closed]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
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### Why some people don't like proofs by contradiction [duplicate]

Possible Duplicate: Are the “proofs by contradiction” weaker than other proofs? I have been active on this site for two months and on a few occasions I noticed that some people judge ...
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### Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
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### $1^3 + \dotsb + n^3 = (1 + \dotsb + n)^2$: reason? [duplicate]

We have $$1^3 + \dotsb + n^3 = (1 + \dotsb + n)^2$$ as we can establish by induction. But why does this hold? Can we connect it to something else?
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### When asked to show that $X=Y$, is it reasonable to manipulate $X$ without thinking about $Y$?

When I'm asked something like "show X is equal to Y", I first try to manipulate what I know (X) into the result (Y). A lot of the time, I do not investigate the result I'm trying to conclude with. I ...
The relation $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$$ baffled me when I first found out (i.e. yesterday on a train trip). Writing an inductive proof is easy and I know that there is a ...