Here are counters numbered 0-9 in a circle:
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?
Very interesting questions, Jer.
The total number of spaces that must be moved around the magic circle is 1+ 2 + ... + n = n*(n+1)/2.
One time around the magic circle is (n+1) spaces, so the full trip is n/2 times around.
This means that if n is even, then a magic circle is not possible, because the starting counter would need to occupy the same space as 0.
And if n is odd, then the starting point is always directly opposite the 0, halfway around the magic circle. And the starting counter cannot be (n+1)/2, because a trip which starts halfway around would end on 0 immediately.
Can any odd n be turned into a magic circle? It would not surprise me, but I have not investigated yet.