If you take any number you can think of, any integer whatsoever, count the number of letters it takes to write it out fully (with or without any 'and's you may need), then take THAT number and repeat the procedure infinitely, what number (or numbers) does this strange series converge to? Is there a unique solution?

Let's face it, this won't be a challenge. But here's an extra thing or two for you. Do you know if this series converges in every language? If the series converges, what number does it converge to? Do they have a unique solution? Can you tell of any language(s) in which does this series not converge?

Note: What if you converted this series into cardinal numbers instead (34 = thirty fourth [12 letters] then 12 = etc.)? How many possible convergence values are there in English or any other language you know?

Now I am not certain if there exist a language where this series does not converge but I can certainly invent one.

now take any form of "representation" i.e a way of naming the integers and associate with it a function L(n) which gives the number of "letters" in n.  Now what we are looking for is a speration of this representations into 2 categories.

(1)  those that contain a cyclic convergence

(2) those that do not contain a cyclic convergence

now a representation can eaisly be made that falls into category 2 namely one where you reperesent the number N with N+1 a's for example

1->   aa
2->   aaa
3->   aaaa
...

thus in this example it is easy to see that on convergence can exist.

Posted by Daniel on 2006-09-10 11:04:27