```Jacob Fugal wrote:
> On 9/1/06, Hans Fugal <fugalh / xmission.com> wrote:
>> And what stops it from being a 1x10^50 length period, or any arbitrary
>> length?
>
> The period cannot be any larger than 1000. The largest period is
> probably quite a bit smaller than this, but this is a upper bound.
> Why?

<mathematical proof proof />

> So, for those of you glossing over the math, the conclusion is that
> applying the "happy function" to any number greater than or equal to
> 1000 will necessarily result in a smaller number.
>
> Since the sequence can't loop if we're constantly decreasing (until we
> get down to 999 or less), none of the numbers in a period can be
> greater than 999 -- there's no way to get back *up* to any of those
> numbers.
>
> So, for any loop resulting from a number, the space of numbers liable
> to be in that loop is no larger than 1000 number (0 through 999). So
> no loop can have a period greater than 1000.

And that, my friends, is what stops it from being a 1x10^50 length
period, or any arbitrary length.

If the intent of the quiz was to be mathemeticians and figure all this
mathy stuff out, I apologize for dragging it into the open.

```