weird stuff

[gabriel_ro]

Hello there math people!

A friend of mine has recently introduced me to a little math puzzle.

You have the following series of numbers:

1
11
21
1211
111221
312211
13112221
1113213211

The question is, what is the algorithm used for generating these series.

-SPOILER FOLLOWS-

Take the sixth series, for example (312211).
You get the seventh series based only on the previous series, in the following manner:
You have one "3" digit, so the first two digits of the seventh series are 13.
Then you have one "1" digit, so the next two digits of the seventh series are 11.
Then you have two "2" digits, so the next two digits of the seventh series are 22.
And then you have two "1" digits, so the last two digits of the seventh series are 21.

OK. That was fun enough, so I thought I could write a little PHP script to see what interesting properties I can find. For example I wondered if there was ever foing to be a 4 anywhere, and stuff like that. I was also curious whether there was a pattern for the number of digits in each string, or for their sum.

Here is a link to it:
http://www.cozmoslabs.com/projects/tbd/ ... 00&rows=15

Results?
Well, I found out that 4 never comes up, and I could find no pattern for the number of digits or for their sum, BUT something really unexpected came up: I calculated the average digit, I mean the sum of the digits divided by the number of digits in every string. AND apparently, as you iterate further and further, this average converges to a certain value of about 1.689

Here is another link, with more iterations:
http://www.cozmoslabs.com/projects/tbd/ ... 40&rows=40

SO, at last, my question: How can I get that value? I suspect it might be something like sqrt(a)*b/c, where a, b and c are integers, or whatever.

Regards, Gabriel

[CRGreathouse]

This is Conway's Look-and-Say sequence.

Can you explain the calculation of the 'average digit' a little more?

[gabriel_ro]

WOW! The "Look and Say Sequence". That's so cool. Thanks.

There sure is a wealth of information at the page you indicated.

But what surprised me is the lack of information regarding the sum of all digits for each term of the sequence.

Perhaps 'average digit' was not exactly the smartest choice of words, since obviuously digits must be integers, not real numbers.

Anyhow, you can add all the digits of a term and get an "X".
And you can count all the digits of a term and get a "Y".

And then you can divide X by Y and get a Z.

Z=X/Y

Z is what I called the 'average digit'.

And strangely enough, the higher the term of the sequence, the more Z converges to a value of about 1.689

see http://www.cozmoslabs.com/projects/tbd/alg/index.php?start=1&displimit=40&rows=40

I was just wondering if there is a way to determine it, like maybe it's the root of a polynomial or something like that.

[CRGreathouse]

I was just wondering if there is a way to determine it, like maybe it's the root of a polynomial or something like that.

You can probably find it as a consequence of the so-called "Cosmological Theorem".

[gabriel_ro]

I'd love it if you could elaborate a bit.

I mean, I can't find any reference to the sum of digits for each term in the Cosmological Theorem...

[CRGreathouse]

I mean, I can't find any reference to the sum of digits for each term in the Cosmological Theorem...

That's because it's not about the sums of digits. But it gives you a fair amount of information on the structure of the numbers, and that could probably be used to prove the exact 'average' of the digits.

[gabriel_ro]

ok. thanks

[CRGreathouse]

Good luck with your problem. If you solve it (or make large strides) post here so we can celebrate with you!