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?

Normal:

first numbers:
1 een
2 twee
3 drie
4 vier
5 vijf (either four or five letters - there is unclarity about the ij being one letter or not - most dictionaries agree that it is two letters)
6 zes
...
this sequence will always converge to 4. This will be the case for most natural languages, where the names of the numbers are used to not count (as in the "aa aaa aaaa" invented language previously mentioned) but to shorten the names for convenience.

For Dutch ordinals:

eerste, tweede, derde, vierde, vijfde, zesde, zevende,achtste, negende, tiende, (from here on all ordinals use less letters than the value of the number). It is clear that the sequence will converge either to 7 (zevende) or to 5/6 (if 'ij' is treated as two letters) or 5 in the other case.

Posted by Robby Goetschalckx on 2006-09-10 17:59:57