A nine digit number has the property where the first digit equals the number of zeros and ones used in the number, the second digit equals the number of ones and twos used in the number, the third digit equals the number of twos and threes used in the number, etc. through the ninth digit equals the number of eights and nines used in the number. What could the number be?

A ten digit number has a similar property to the nine digit number. The first digit equals the number of zeros and ones used in the number, the second digit equals the number of ones and twos used in the number, etc. through the ninth digit. And also, the tenth digit equals the number of zeros and nines used in the number. What could this number be?

I was curious about Avin's very interesting iterative method.  Is it guaranteed to find a solution, if one exists? 

It turns out that it does not!

For example, is there a self-referential 2 digit number? i.e., a number where the first digit equals the number of zeros and ones used in the number and the second digit equals the number of ones and twos used in the number.

There is one:  12

However, if we try the iterative method of finding it:
00 --> 20 -->  11 --> 22 -->  02 --> 11 --> 22 -->  02, etc.

The only starting points which avoid this loop are 21 and 12.

Edited on September 24, 2005, 12:13 am

Edited on September 24, 2005, 2:49 pm

Posted by Steve Herman on 2005-09-24 00:12:03