Re: series puzzle

From	richard@cogsci.ed.ac.uk (Richard Tobin)
Newsgroups	rec.puzzles
Subject	Re: series puzzle
Date	2026-08-28 19:39 +0000
Organization	HCRC, University of Edinburgh
Message-ID	<116so5c$4t1g$1@artemis.inf.ed.ac.uk> (permalink)
References	<8733vz45sz.fsf@asdf.ee>

Show all headers | View raw

In article <8733vz45sz.fsf@asdf.ee>, Phil Carmody  <pc+usenet@asdf.org> wrote:

>Bonus points for the mathmos: give an expression for how quickly this
>series grows: how many digits would you expect the 1000th, 10000th,
>100000th, and 1000000th terms to have?

    1000 167268718788585637957754176512
   10000 115561287648644129422797664757359949255136
  100000 1136141149823685669283941799894965343754256
 1000000 1114828839832196352142477728158415189411298271623636456323512734248

Spoiler space
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

[Waves hands]

Call the Nth term T(N).  It has log10(T(N)) digits.

Let's pretend that multiplying T(N) by N+1 produces a random sequence
of digits log10(N) longer than T(N).  On average a tenth of these will
be removed.  So T(N+1) has on average

  9/10 (log10(T(N)) + log10(N))

digits.  So if log10(N) is less than log10(T(N))/10 the number of digits
will tend to reduce, otherwise it will tend to increase.  We will get
equilibrium when log10(T(N)) = 10 log10(N), or T(N) = N^10.

-- Richard

Thread

series puzzle Phil Carmody <pc+usenet@asdf.org> - 2026-08-28 00:22 +0300
  Re: series puzzle David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-28 08:56 +0000
  Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-28 17:54 +0000
  Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-28 19:39 +0000
    Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-28 19:49 +0000
      Re: series puzzle David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-30 08:49 +0000
        Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-30 11:51 +0000
    Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-30 12:06 +0000
      Re: series puzzle Phil Carmody <pc+usenet@asdf.org> - 2026-09-01 12:14 +0300
        Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-09-01 10:03 +0000
          Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-09-05 10:11 +0000
            Re: series puzzle richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-09-05 10:51 +0000
              Re: series puzzle David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-09-06 07:52 +0000
    Re: series puzzle Phil Carmody <pc+usenet@asdf.org> - 2026-08-31 22:22 +0300

csiph-web