A shrinkable number is one that can be reduced to a single digit by the following process:

Create a new number where each digit is the absolute difference between consecutive digits.
Repeat until you have a single digit.
No digit is allowed to be a zero at any point in the process.

Examples:
7624 is shrinkable. It reduces to 142 which reduces to 32 which reduces to 1.
4131 is not shrinkable because it reduces to 322 which reduces to 10 which contains a zero.

Your goal is to create the smallest possible n digit shrinkable number for n = 2, 3, 4, ...
Is there a shrinkable number for any value of n?

n : smallest shrinkable number [shrinks to ->]
2 : 12 -> 1
3 : 124 -> 1
4 : 1248 -> 1
5 : 12492 -> 1
6 : 124921 -> 2
7 : 1249216 -> 1
8 : 12712912 -> 1
9 : 139219216 -> 1


The largest single digit number that a shrinkable number may be reduced to is 8 [19 & 91]. A number [in base-10] can never shrink to 9.
A number with the digit 9 may be shrunk, but the reverse process may not be performed.
A number with the digit 8 may be shrunk, yet the reverse process for base-10 requires consecutive numbers of (1 and 9) in the next larger shrinkable number.  With digit 7, consecutive numbers of the next larger shrinkable number must have consecutive digits (1 and 8) or (2 and 9). And so forth.
Due to the constriction created with each additional digit with what digits are possible, the largest value of n is 9.
[number for n=9 supplied by Charlie]. 

Edited on April 6, 2011, 3:35 am

Posted by Dej Mar on 2011-04-05 15:19:39