Here are counters numbered 0-9 in a circle:

                  0

        1                   2




  8                               7





  4                               5




        6                   3

                  9

The property that makes it a magic circle is: there is a specific starting counter (in this case the 9) so that if you proceed clockwise by that number of steps and repeat the process with each new number you eventually visit every number and end at zero.

Given numbers 0-n:
For what values of n does a magic circle exist?
For a given value of n, how many magic circles exist?
Are there infinite families of magic circles?