Self defining numbers

[TallPaul]

Ever hear of the "self defining number"?

Here's one:

6 2 1 0 0 0 1 0 0 0

It mean that there are

six "0"s
two "1"s
one "2"
zero 3s, 4s, 5s, 7s, 8s and 9s
and
one "6"

0 1 2 3 4 5 6 7 8 9
6 2 1 0 0 0 1 0 0 0

Also interesting is that it can be remembered (defined) by only the first three digits. Actually, it appears it can be defined by the first two digits!

I don't think you can have anything but 0 in the 9's slot. Here is a very simple self defined number:

8 0 0 0 0 0 0 0 1 0

Can you come up with others?

 

[TallPaul]

For the record (or if anyone cares

):

Er ah., my number (8 0 0 0 0 0 0 0 1 0) is no good. Says there are zero 1s, but there is a 1. I toyed around with this last night and am thinking there is only one 10 digit self defining number (SDN). But the trick is to prove it.

Now if we shorten the number using say 0 1 2 3 4 5 6 7, we can get one like this: 4 2 1 0 1 0 0 0

Or how about 2 1 2 0 0 on a five digit SDN, of course always beginning at zero.

So I wonder if there is more than one SDN per number of digits.

Any mathematicians out there want to comment?

 

[John K]

Are you snowed in and dying of boredom?

 

[XENON12003]

How about
7 1 0 0 0 0 0 1 0 0

 

[XENON12003]

sorry no good

 

[TallPaul]

quote:

---

Originally posted by John K:
Are you snowed in and dying of boredom?

---

Actually, having fun in my own strange way.

So can anyone fill in the remaining two self-defining numbers (SDNs)?

SDN(1) through SDN(10):

0
0 1
0 1 0
2 0 2 0
2 1 2 0 0

3 2 1 1 0 0 0
4 2 1 0 1 0 0 0

6 2 1 0 0 0 1 0 0 0

But we still don't know if there is more than one solution for any given n of SDN.

 

[tom slick]

your numbers are sort of like the fibinachi (sp?) sequence.
1 1 2 3 5 8 13 21 34.......

it stangely shows up in nature, in flowers, pine cones, etc.

 

[TallPaul]

quote:

---

Originally posted by tom slick:
your numbers are sort of like the fibinachi (sp?) sequence.
1 1 2 3 5 8 13 21 34.......

it stangely shows up in nature, in flowers, pine cones, etc.

---

Fascinating sequence. Now that I have revised my table (see below) I don't think it fits the Fibonacci (I looked up the spelling) sequence, nor does the self-defined number have anywhere near the significance. As you say Fibonacci's numbers are throughout nature, the self-defined, barely more than a figment.

Anyway, I was wrong in part. Here is the revised table for the SDN, where n is the number of digits, from n=1 to n=10:

SDN(1) does not exist.
SDN(2) does not exist.
SDN(3) does not exist.
SDN(4)= 2 0 2 0
SDN(5)= 2 1 2 0 0
SDN(6) does not exist.
SDN(7)= 3 2 1 1 0 0 0
SDN(8)= 4 2 1 0 1 0 0 0
SDN(9)= 5 2 1 0 0 1 0 0 0
SDN(10)=6 2 1 0 0 0 1 0 0 0

Now we can see a pattern from SDN(7) through SDN(10) and if we look at n as the number base, we will see that this pattern continues infinitely, always with the first digit n-4, the second digit a 2, the third digit a 1, and the 4th from last digit a 1. For example, in hexadecimal (base 16) the number would be:

SDN(16)= C 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0
where C in base 16 equals 12 in base 10, i.e., the first 16 numbers are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

However, this pattern is becoming rather uninteresting to me. More interesting is SDN(4) and SDN(5) and the way SDN(6) cannot form because if you follow the pattern back from SDN(7) you see that the two "1"s clash.

Now, I think there are no other SDN's than what I have defined above, but I could be wrong, as I was before. I am pretty certain SDN(1) through SDN(3) do not exist, and mostly certain the same for SDN(6), but maybe I will be proved wrong on that one.

BTW, all this got going because of a kid's math puzzler book that had only the one ten-digit self defined number.

[ January 20, 2005, 12:52 AM: Message edited by: TallPaul ]