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How close can you get to $\pi$ (as close as possible) using these rules?

1) You can only use at most four integers ranging from $1$ to $20$, each only once.

2) You can only use plus, minus, multiply, divide, exponential, and log as binary operators.

3) You can only use integer factorial as a unitary operator.

4) You should work in the real number region.

The best answer that I know currently is $$3 + \frac{16}{5!-7} = \frac{355}{113} = 3.141592920\ldots$$

Jiahao Fan
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  • An answer given 6 years ago in a similar Q... https://math.stackexchange.com/questions/445277/pi-estimation-using-integers – Anton Vrdoljak Mar 14 '20 at 09:16
  • That is good but the crucial point here is that you could only use four numbers and only once. – Jiahao Fan Mar 14 '20 at 09:22
  • I agree about that, and will ask: where you found (source or reference) the detail regarding the current record? – Anton Vrdoljak Mar 14 '20 at 09:39
  • Well the word “record” may not be appropriate, the above formula was given by one of my friends which relate to the famous “fractional approximation” of $\pi$. – Jiahao Fan Mar 14 '20 at 09:43
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    $355/113$ is a known convergent of $\pi$ (https://mathworld.wolfram.com/PiContinuedFraction.html) and as well the funny article (https://blogs.scientificamerican.com/roots-of-unity/don-8217-t-recite-digits-to-celebrate-pi-recite-its-continued-fraction-instead/). See also (https://en.wikipedia.org/wiki/Mathematical_coincidence). Besides, IMHO, one can loose a lot of time on those things without real mathematical interest. – Jean Marie Mar 14 '20 at 09:55
  • Yes I agree with you, I will not try to find the answer myself. Just curious about whether there is some well-known result or trick on this kind of composition. – Jiahao Fan Mar 14 '20 at 12:40
  • I am not experienced to program the tree of the possible expressions but someone should be able to determine the best approximation by brute force. – Peter Mar 15 '20 at 15:00

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