```On 9/6/06, Daniel Martin <martin / snowplow.org> wrote:
> "Rick DeNatale" <rick.denatale / gmail.com> writes:
>
> > Here are some results from my code for various bases.  Do these look
> > like what others are seeing?  Has anyone uncovered a base 10 number
> > which is happier than 8 steps to 1?  Unless my code has a bug in it'
> > maybe I should state DeNatale's conjecture which is "There is a
> > maximum happiness for numbers expressed in a base > 2"
>
> That conjecture is false.
>
> > Of course it might just be a lack of patience on my part.
>
> Probably.
>
> > one of the happiest 5 digit base 10 numbers is 78999, with 8 steps
> > after 1287 probes
>
> I propose that the number (10**78999 - 1)/9, that is, the number made
> by:
>
>  ("1" * 78999).to_i
>
> takes 9 steps.

Right you are.  I never really beleived my conjecture, just wanted to
stir up thoughts.

> I strongly doubt that this method is the most
> efficient way to get to a 9-step number; as a trivial adjustment, the
> number
>
>   ("1" * 24 + "9" * 975).to_i
>
> is also a 9-step number.

I think you lost me on that step.

>However, if 78999 is the smallest 8-step
> happy number, the smallest 9-step happy number must have at least
> (78999/81.0).ceil digits.  Small wonder that you didn't find one...

Efficient or not, it would seem that generating large happy numbers by
construction beats the hell out of trying to find them.
--
Rick DeNatale

My blog on Ruby