Search for permutations of objects in a circle for which any pair of objects is at a different minimal distance along the circle than originally; see http://math.stackexchange.com/questions/281094.

Question281094.java

public class Question281094 {
    static int m;
    static int [] p;
    static int count;

    public static void main (String [] args) {
        for (m = 2;;m++) {
            p = new int [m];
            p [0] = 1;
            count = 0;
            recurse (2);
            System.out.println (m + " : " + count);
        }
    }

    static void recurse (int n) {
        if (n > m) {
            count++;
            return;
        }

        outer:
        for (int i = 1;i < m;i++)
            if (p [i] == 0) {
                for (int j = 0;j < m;j++)
                    if (p [j] != 0) {
                        int d1 = Math.abs (j - i);
                        int d2 = Math.abs (p [j] - n);
                        if (d1 > m / 2)
                            d1 = m - d1;
                        if (d2 > m / 2)
                            d2 = m - d2;
                        if (d1 == d2)
                            continue outer;
                    }
                p [i] = n;
                recurse (n + 1);
                p [i] = 0;
            }
    }
}