To generalize Wilson's problem to D-space, combining previously posted ideas, if there are N points inside a D-dimensional ball, N large, D fixed, then the number of possible orderings of the N points along the X-coordinate (when the ball is rotated) is upper bounded by: if D=1: 2. if D=2: N*(N-1). if D>2: Any point-pair defines a "great-circle" (hyperplane intersect sphere) from bisector hyperplane. The arrangement defined on the sphere-surface by these H=N*(N-1)/2 hyperplanes has F faces, and the generic value of F upper bounds the number of orderings. And obviously F(H,D) is upper bounded by F(H,D) <= F(H-1,D-1) + F(H-1,D) which allows us to compute a table of upper bounds starting from F(H,2) = 2*H and F(1,D) = 2. (This is, of course, the Pascal triangle recurrence...) Note that F(H,D) with H large and D fixed is bounded by a degree=(D-1) polynomial in H, you can prove this inductively. Its leading (highest degree) term should have coefficient=4/D! when D>1.