[math-fun] Sum of last ten digits

[Michele Dondi]

On Fri, 23 Dec 2005, Alexandre Wajnberg wrote:

>> b=10, and examine all 10^7 starting states.  Amazingly, ALL states fall
>> into the same loop, of length 461.  (That is, all 9,999,999 states.  The
>> all-zeros state is the simple zero-loop of length 1.)

While I'm generally averse to base-dependent sequences, if(f) 461 really 
has some sort of precisely definable "universal" property in connection 
with the algorithm described here for base 10, then it may be interesting 
to investigate whether there are equivalent ones for other bases and then 
a mathematically interesting sequence may be found.

> I wrote:
> The  same loop of lenght 461, is it
> ‹ the very "same" loop, fully identical (all same terms) and beginning by
> the same "first term" of the loop?

Since you write "first term" in quotes yourself, you must be aware that 
almost by definition, in a loop there's not a "first term". Personally I 
think that the entry point into the loop may not be that relevant after 
all.

Have you considered trying the opposite path? That is, you _do_ what the 
loop is like, don't you? Well, then for all of its entries try to devise 
the inverse image, i.e. where they do come from: the previous entry of the 
loop must be in it, of course; is it its only member? At least for one 
element of the loop this must not be the case, but it all boils down to: 
for how many of them this is not the case?


Michele
-- 
I would never die for my beliefs, because I might be wrong.
- Bertrand Russell