11 questions linked to/from "Simple" beautiful math proof
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Can every proof by contradiction also be shown without contradiction?

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 ...