Speed

[Gareth McCaughan]

Janos Blazi wrote:

[I said:]
>> (Have you encountered Conway's work on the problem this
>> little program is investigating?)
> 
> Thank you for your suggestion.
> I have not known this has anything to do with Conway (whom I sort of
> worship), but I would be VERY, VERY MUCH interested in learning about this
> aspect.

He wrote an article in a journal called "Eureka" (it's
the annual journal of the Archimedeans, the mostly-undergraduate
mathematics society at Cambridge University; you're unlikely
to find it in your university library...), in which he
revealed all sorts of interesting things.

Whatever sequence you start with, after a few iterations
the sequence you have is a concatenation of "atoms"; there
are only a small number[1] of different atoms. In the long
term, the proportions of the different atoms don't depend
on what sequence you started with. The length of the
sequence after n iterations is asymptotically constant*alpha^n,
where alpha is the largest root of some polynomial of
moderate degree (in the tens rather than the hundreds);
the polynomial is given explicitly in the article, as is
the approximate value of alpha.

That's about all I remember. Oh, he called the sequence
the "audioactive sequence", I think.

It's the kind of stuff Conway does better than anyone else,
of course.

If you desperately want to know more, the best suggestion
I can make is to find a copy of the relevant issue of
"Eureka". I don't remember which number it is (might
be 32, but I would strongly suggest not trusting me).
In the very likely event that you can't find one anywhere
nearby, you could get in touch with the Archimedeans;
you could try e-mailing
  archim at cam.ac.uk
but I don't know whether it will work. Failing that,
snail-mail to

    Eureka
    The Arts School
    Bene't Street
    Cambridge CB2 3PY
    England

will probably get there.

>         (I teach and one of my pupils showed me the problem stating that
> probably it was not possible to write a computer program for that [ :) ]

Such odd things people think are impossible. :-)


[1] 92.

-- 
Gareth McCaughan  Gareth.McCaughan at pobox.com
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