Numbers which describe themselves

[Emmanuel Marin]

Describing here means :
134567 means '1 time 3, 4 times 5, and 6 times 7'

22 describes himself
So does 14233221.

I 'd like to know if there are already funny things on this
'property' I have re-invented from a puzzle
(The puzzle was: 1,11,21,1211,?,?...)

Because if you do a program Un+1 = describe(Un)
there are strange things :
Sometimesit ends on 14233221, or 14331231,
or .. sometimes two numbers which describe at one another.

I don t know if there are triplets describing and so.

Any comment ?

[Glenn Chappell]

In article <91132.1444...@FRECP12.BITNET> MARI...@FRECP12.BITNET (Emmanuel Marin) writes:
>Describing here means :
>134567 means '1 time 3, 4 times 5, and 6 times 7'
>
>22 describes himself
>So does 14233221.

And so does 1514332231
161524232241 and 161524134231 descibe each other

But things are perhaps a little ambiguous here.

131221 describes 1231
Does 211213 describe 1231?
Or 121321?
What about 11121311?

Does 1321 describe 311, or must we use 130221?
(There's also 13022100....)

>I 'd like to know if there are already funny things on this
>'property' I have re-invented from a puzzle
>(The puzzle was: 1,11,21,1211,?,?...)

The puzzle is ambiguous, too.
The next entry could be 1231, if we put the digits in descending order,
3112, if we put them in order of first occurrence,
or 111221, if we want to be able to reconstruct the previous entry.
All of these are in some sense consistent with the previous entries.

We run into real trouble if we try to describe 1111111111.
That gives us 101...or does it?
And if we describe that, we get 2110, which suggests that "0" should
have been included in the first place.

An alternative is to assume that we are working in "base infinity",
and we have as many digits as we need. Then we could describe 1111111111
with something like A1.

Another way to think about it is not as numbers
but as finite sequences of numbers. Thus, we can describe
1,1,1,1,1,1,1,1,1,1 with 10,1...then we describe that with
1,10,1,1 then 1,10,3,1...1,10,1,3,2,1...1,10,1,3,1,2,3,1...etc....
I think I like this last way best: numbers listed in descending order,
no zeroes allowed.

So, 1,100,2,3,3,2,2,1 describes itself
and so does 1,200,1,100,3,3,3,1
and 1,300,1,200,1,100,2,4,2,3,3,2,4,1

This gives an easy proof that there are an infinite number of
self-describing sequences:
for x > 3, 1,x,2,3,3,2,2,1 describes itself.

There are also an infinite number of pairs that describ each other:
for x > 5,
1,x,1,5,2,4,1,3,4,2,3,1 and 1,x,1,5,2,4,2,3,2,2,4,1
describe each other.

But then maybe that wasn't what you were looking for....

>I don t know if there are triplets describing and so.

I didn't find any either.

GGC <><

[amine...@gmail.com]

17161534132251
17162514232251
17162514134241