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Use 1, 2, 2, 2 (Each number used exactly once) to form π.
I want an exact value of π, not any approximation.

Available operators:

+ - * / ( ) ^ !
You can use any one of them for any times.

Hint:

Think about a special feature of the "!".

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  • $\begingroup$ MathJax is very slow here. $\endgroup$ – Scratch---Cat Oct 27 '19 at 13:03
  • $\begingroup$ Yes, I wouldn't recommend using MathJax unless it's necessary for things like fractions - for some people, it takes a while to load. (Here, you could easily just use the π character rather than switching to MathJax.) $\endgroup$ – Deusovi♦ Oct 27 '19 at 15:06
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    $\begingroup$ I want an exact value of π, not any approximation - I wholeheartedly endorse this message. $\endgroup$ – HTM Oct 27 '19 at 20:35
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    $\begingroup$ Instead of the factorial do you mean the gamma function? It seems that the only answer provided erroneously uses the factorial rather than the gamma function (the factorial is only defined for positive integers). $\endgroup$ – N. Bar Oct 27 '19 at 22:43
  • $\begingroup$ Well, not exact, but I remember it to 19 digits: 3.141592653589793238 $\endgroup$ – S.S. Anne Oct 27 '19 at 23:52
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It is

$(2 \times (\frac{1}{2})!)^2$

This video tells why.

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    $\begingroup$ By my calculations, this adds up to 4 times domain error squared. $\endgroup$ – Bass Oct 27 '19 at 14:10
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    $\begingroup$ Factorial is only defined for positive integers. This answer clearly means to use the gamma function (one of - but not the only - way to extend factorials to the complex numbers), but that was not one of the allowed functions in the question... $\endgroup$ – BlueRaja - Danny Pflughoeft Oct 27 '19 at 21:30
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    $\begingroup$ @BlueRaja-DannyPflughoeft But we all know it's only possible if factorial is interpreted as gamma function - which is the only point of this question... $\endgroup$ – WhatsUp Oct 28 '19 at 4:02
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    $\begingroup$ @BlueRaja-DannyPflughoeft The gamma function is the standard way to extend factorials to the complex numbers. If you write $\alpha!$ for $\alpha\not\in\mathbb{N}$ in mathematics, this is assumed to mean $\Gamma(\alpha+1)$. See for example Wolfram Alpha and my experience as a mathematician in a field where gamma functions are used all over the place. $\endgroup$ – Rand al'Thor Oct 28 '19 at 6:15
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    $\begingroup$ @Bass I disagree. Saying "the gamma function" explicitly would make the puzzle trivial and pointless. The exclamation point is not strictly integer-only in mathematics, and using it in this way makes for a very clever puzzle - kudos to the OP. If you don't believe me, all I can do is point to the evidence in my previous comment. I work with factorials and gamma functions every day, and use $\alpha!$ completely interchangeably with $\Gamma(\alpha+1)$. $\endgroup$ – Rand al'Thor Oct 28 '19 at 21:30

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