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From: Phil Carmody <pc+usenet@asdf.org>
Newsgroups: rec.puzzles
Subject: Re: Repeated digits in Pi -- the Feynman point
Date: Thu, 03 Jul 2025 15:30:02 +0300
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richard@cogsci.ed.ac.uk (Richard Tobin) writes:
> In article <103k5nq$3ju79$1@dont-email.me>,
> David Entwistle  <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:
>
>>> 3.14159 26535 89793 23846 26433 83279       41971 69399 3751...
>>  3.14159 26535 89793 23846 26433 83279 50288 41971 6939...
>
> (reformatted)
>
> Presumably the first was copied from a listing in groups of 5 digits,
> and one group was missed out.
>
>>Oh, who decides?
>
> A substantial proportion of the population are capable of learning the
> necessary maths and writing a program to determine which is correct.
> A much smaller proportion are sufficiently motivated to do so.
>
> If you don't trust the computer, it would be possible to use the
> formula
>
>   pi = 16 atan(1/5) = 4 atan(1/239)
>
> to calculate it by hand to that precision.  William Shanks used it and
> obtained 527 decimal places correctly in 1853.  This was not surpassed
> (and an error found in his later digits) until 1946 using a mechanical
> calculator.

By hand, I think I'd favour the spigot method. I know Matt Parker
(/Stand-Up Maths/ on youtube) likes to organise the manual calculation
of pi on "pi day", and I'm a little disappointed he's never tried
spigot. I think its Big-Oh is superior to any of the algorithms that
effectively do arbitrary precision arithmetic, as you never need to deal
with numbers much bigger than a hundred.

Phil
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