In Self-Descriptor, we found a number ABCDEFGHIJ such that A is the count of how many 0's are in the number, B is the number of 1's, and so on.

I wonder... what if the number didn't have to be 10 digits long?

Find the smallest whole number such that the left-most digit describes the number of 0s in the number, the next digit describes the 1s, etc. Prove that it's the smallest.

I've been trying all night to upload this from a lousy browser on a crappy computer. I'm finally home on my own.

First some definitions:

A describable number is one that can be transformed into a descriptor.

A descriptor is a number associated with a descibable number such that the first digit describes how many 0s are in the descibable number, the second digit, how many 1s, etc.

A self-descriptor is a describable number which is its own descriptor.

Any number N of d digits =
  d
 ∑[n(d - a)10ª], where 0 ≤ n ≤ 9     [n(a) = the value of the ath n; not n times a]
a=0

---------

Now some rules:

1) For describable numbers: For all a, 0 ≤ n(a) ≤ (d - 1)

2) For descriptors:
  d
  ∑[n(a)] = d
a=0

Self-descriptors must follow both rules, plus:
3) n(0) ≠ 0

Posted by TomM on 2002-10-12 06:26:19