A certain company gives each of its clients a 10 digit number as a sort of identification code. As a precaution, any pair of used codes should differ by at least two digits so no one accidentally gives someone else's code.

How many clients can they have before adding digits? Give an example of a set of codes they might use. What if each pair of codes must differ by at least 3 digits? 4? More?

With 10 digits available and no constraints, the maximum number of employees is 10^10.

When the constraint of 2 digits in every number combination being different is added, then one digit becomes dependant on a second digit.  For every change in one place, another must change. 

Simplistically, I would arrange the numbers as follows:
x_ _ _ _ _ _ _ _x for x = 0 to 9 

This will give 10^8 possibilities for the interior numbers and 10 possibilities for the “x”

Therefore the total will be 10^9 for 2 digits differ.

Following the same logic:

3 digits differ = 10^8 maximum
4 digits differ = 10^7 maximum
. . .
10 digits differ = 10 maximum

Posted by Leming on 2005-11-29 12:00:04