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Do you have proofs without word?

Your proofs are not necessary has zero word, you may add a bit explanations.

As an example, I has a "Speechless proof" for $$\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+...=\frac{1}{3}$$ enter image description here

I welcome all aspects of mathematical proofs. Thank you.

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    $\begingroup$ These are usually called proofs without words. MathOverflow has a nice list of them. $\endgroup$ – Rahul Nov 5 '12 at 9:05
  • $\begingroup$ Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange. $\endgroup$ – A. Chu Nov 5 '12 at 9:58
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    $\begingroup$ For this question, what's really important are imaginative answers. $\endgroup$ – A. Chu Nov 5 '12 at 10:27
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    $\begingroup$ In fact, this particular example is not obvious without some words (at least to me)! $\endgroup$ – NoChance Nov 5 '12 at 10:40
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    $\begingroup$ @NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul. $\endgroup$ – Cyriac Antony Jan 17 at 5:14
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Part of the proof in my blog post is done with my own ad-hoc diagrams..

grid with tour

The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)

If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)

sketch

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The best one I have ever seen is to prove $$1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}2$$

enter image description here

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    $\begingroup$ Should we just copy the MO answers here - or link to them from here? $\endgroup$ – Old John Nov 5 '12 at 9:50
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    $\begingroup$ Could somebody give me the "with words" version of this proof? $\endgroup$ – littleO Nov 5 '12 at 10:05
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    $\begingroup$ @littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $\frac{n(n+1)}{2}$ as $\binom{n+1}{2}$ $\endgroup$ – EuYu Nov 5 '12 at 10:09
  • $\begingroup$ Ohh, I see, crazy. Thanks for the hint. $\endgroup$ – littleO Nov 5 '12 at 12:00
  • $\begingroup$ @OldJohn I don't know. I anyway made my post CW when I posted it. $\endgroup$ – user17762 Nov 5 '12 at 21:48
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Found this great one surfing the web recently. $$ \displaystyle 1/2+1/4+1/8+1/16+\ldots =1 $$ Classic geometric sum

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Convergence of reciprocal of squares

Reciprocals of squares converge.

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Here's one showing the area of a disk is $\pi R^2$.

enter image description here

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  • $\begingroup$ This happens to end with the correct answer, but the method is invalid, because it assumes that $\lim_{n\to\infty} a(n)=a(\infty)$ where $a(n)$ is the total area of the $n$ strips after division into $n$ parts, and $a(\infty)$ is the circle's area. $\endgroup$ – Rosie F May 15 at 19:05

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